geometry and combinatorics

Geometry and Combinatorics

Circles Γ1 and Γ2 have centers X and Y respectively. They intersect at points Aand B, such that

angle XAY is obtuse. The line AX intersects Γ2 again at P, and the line AY intersects Γ1 again at Q.

Lines PQ and XY intersect at G, such that Q lies on line segment GP.

If GQ=255, GP=266 and GX=190, what is the length of XY?No answer given See worked solution


ABCD is a trapezoid with AD∥BC and AB=AD. 0∘<θ<180∘ is an angle such

that ∠BAC=θ, ∠CAD=3θ and ∠ACD=5θ. What is the measure (in degrees) of θ?No answer given See worked solution


Bernoulli flips a coin 15 times and writes down the sequence of heads (H) and tails (T) that he gets.

He notices that he has 5 instances of consecutive TH, 4 instances of consecutive HT, 3 instances of

consecutive HH and 2 instances of consecutive TT. How many different sequences could he have

written down?No answer given See worked solution


Bernoulli places a mouse token at the point 0 on the real line and two cheese tokens at the

points 3 and −3. Bernoulli then begins flipping a coin repeatedly. Each time the coin is heads, he

moves the mouse 1 to the right, and each time the coin is tails, he moves the mouse 1 to the left.

Let p be the probability that the mouse will eventually get a piece of cheese. Determine the value

of ⌊100p⌋.No answer given See worked solution


P is a point in triangle ABC such that ∠PAB=∠PAC=22∘, ∠PBA=∠PBC=33∘. What is the measure

(in degrees) of ∠APC ?Your answer 123 See worked solution


A triangle is subdivided into 25 smaller triangles as shown in the figure below. Two of these 25

triangles are selected at random. The probability that the two selected triangles have exactly one

vertex in common can be expressed as abwhere a and b are coprime positive integers. What is the

value of a+b?

No answer given See worked solution


Dhelia wants to grow rose, orchid, and jasmine trees in her garden. Planting nrose trees

requires n2 square meters, planting n orchid trees requires 5n square meters, and

planting n jasmine trees requires n3 square meters. If she plants 10trees, what is the minimum

amount of space that her garden will occupy (in square meters)?

This problem is posed by Dhelia D .



ABCD is a convex quadrilateral with ∠ABD=26∘,∠ACD=42∘. E is a point such that EA is the angle

bisector of ∠BAC and ED is the angle bisector of ∠BDC. What is the measure (in degrees)

of ∠AED?No answer given See worked solution

Number Theory and Algebra

Find the largest positive integer n<100, such that there exists an arithmetic progression of

positive integers a1,a2,...,an with the following properties.

1) All numbers a2,a3,...,an−1 are powers of positive integers, that is numbers of the

form jk, where j≥1 and k≥2 are integers.

2) The numbers a1 and an are not powers of positive integers.No answer given See worked solution


A and B are positive 4-digit palindrome numbers. C is a positive 3-digit palindrome number. Given

that A−B=C, what is the value of C?No answer given See worked solution


Find the number of triples of integers a<b<c with a≥−10, such

that a(b−c)3+b(c−a)3+c(a−b)3=0.No answer given See worked solution


Let S(n) denote the sum of digits of the integer n. Over all positive integers, the minimum and

maximum values of S(n)S(5n) are X and Y, respectively. The value of X+Y can be written as ab,

where a and b are coprime positive integers. What is the value of a+b?

This problem is proposed by Zi Song .



For a set of red, blue and yellow cards, we receive the points according to the following system:

1. For each red card, we receive 1 point. 2. For each blue card, we receive points equal to twice the number of red cards. 3. For each yellow card, we receive points equal to thrice the number of blue cards.

What is the maximum number of points we can receive for a set of 15 cards?

This problem is posed by Minimario M.



The real number x satisfying the equation

x+2x−1−−−−−√−−−−−−−−−−√+x−2x−1−−−−−√−−−−−−−−−−√=10

can be written as ab where a and b are positive coprime integers. Find a+b.Your answer 53 See worked solution


N is a 3-digit number that is a perfect square. When the first digit is increased by 1, the second

digit is increased by 2, the third digit is increased by 3, the result is still a perfect square.

Determine N.Your answer 361 See worked solution


Find the smallest positive integer e such that the polynomial ax4+bx3+cx2+dx+e has integer

coefficients and has roots at −2, 3, 5/3, and −1/2.Your answer 30 See worked solution

Two weeks ago July 29 to August 4


Let ABCD be a square with side length 2 and let X be a point outside the square such

that XA=XB=2√. The length of the longest diagonal in pentagon AXBCD can be expressed as x√.

What is the value of x?




Determine how many 1000 digit numbers A have the following property:

When any digit of A, aside from the first, is deleted to form a 999 digit number B, then B divides A.No answer given See worked solution


Let a, b, and c be positive real numbers such thata4+b4+c4+16=8abc.

What is the value of abc?

This problem is posed by Ed M.

Your answer 8 See worked solution


How many ordered sets of integers (x,y) are there such that there exists a (positive) prime

number p satisfying x(x+y)=2(y2+15p)?

This problem is posed by Ahmad.



It can be shown that for any positive integer n, the infinitely nested radical expressionn+n+n+⋯−−−−−−√−−−−−−−−−−√−−−−−−−−−−−−−−−√

equals a finite number. What is the largest positive integer n≤999 such that this expression is equal to a positive integer?Your answer 992 See worked solution


Consider all polynomials f(x) with integer coefficients, such that f(n) is a multiple of n, for all

integers 1≤n≤6. What is the smallest possible positivevalue of f(0)?Your answer 60 See worked solution


N is an even 3-digit number such that the last 3 digits of N2 are N itself. What is the value of N?Your answer 376 See worked solution


You are taking a Math class, where your grade is determined by your performance in 5 tests. The

maximum score on each test is 100. Your first 4 test scores have been 63,80,90 and 85. To receive

a B in the class, the average of the five test scores must be greater than or equal to 80 and strictly

smaller than 90. If only integer test scores are allowed, how many different test scores could you

receive for the last test and get a grade of B?Your answer 18 See worked solution


Alice can complete a project in 20 days and Bob can complete the same project in 30 days. Alice

and Bob start working on the project together, with Alice quitting after some time, and it takes Bob

another 10 days to complete the project. What is the total number of days it took to complete the

project?

This problem is posed by Nidhi T.



A construction worker is building a straight walking path that is 5m long from a client's front door

to the sidewalk. The path is to be built out of planks that are 60 cm, 70cm, and 100 cm in length.

How many different paths could be constructed using these plank lengths?



ABC is a right triangle with ∠ABC=90∘. Point D is the midpoint of BC, and point E is the foot of the

perpendicular from B to AC. AD and BE intersect at F. If AB=36 and ∠BAD=∠BCA, what is the

value of FE2?



Three 10-sided dice, each numbered 1 through 10, are rolled. One die is green, one die is blue, and

one die is red. Kareem rolls the three dice and writes down the numbers showing on the green die,

the blue die, and the red die, in that order. There are 1000 different ordered triples that Kareem

could have written down. For how many of those triples is the number on the blue die strictly

between the numbers on the red and green dice?



Right triangle △ABC has side lengths AB=51,AC=24, and CB=45. If the area of the inscribed

circle of △ABC can be written as aπ, what the value of a?

This problem is posed by Alex R.



A triangle has side lengths that are all integers. The length of one side is four times as long as the

second side, and the third side has length 20. What is the largest possible perimeter of the

triangle?



A squirrel is collecting food for the winter. Every pine cone he collects will last him 3 days and

every acorn he collects will last him 2 days. If the squirrel collects enough food for exactly 150

days, how many different combinations of acorns and pine cones could he have?



How many positive integers n less than 1000, have the property that the number of positive

integers less than n which are coprime to n, is exactly n3?No answer given See worked solution


Suppose f(x) is a quadratic polynomial satisfyingf(0)f(4)−f(2)f(6)−f(4)=1=2=3

What is the value of f(8)?Your answer 11 See worked solution


Let x,y,z be the real roots of the cubic equation2u3−799u2−400u−1=0

and let ω=tan−1x+tan−1y+tan−1z. If tanω=ab, where a and b are positive coprime integers, what

is the value of a+b?

This problem is posed by Russelle G .



For what positive integer value N<1000 is N2−755N+56 also an integer?No answer given See worked solution


a,b and c are integers such that(x−a)(x−20)+1=(x+b)(x+c).

What is the sum of all possible values of a?No answer given See worked solution


Let a be the minimum of x(x−4)−2y(x−y) as x and y range over all real numbers. What is the

value of |a|?

This problem is posed by Zi Song Y.



The function cos(5θ) can be written as acos5(θ)−bcos3(θ)+ccos(θ) where a, b, and c are positive

integers. Find a+b+c.Your answer 41 See worked solution


In a sequence of four positive numbers, the first three are in geometric progression and the last

three are in arithmetic progression. The first number is12 and the last number is 452. The sum of

the two middle numbers can be written as ab where a and b are coprime positive integers.

Find a+b.Your answer 139 See worked solution

Four weeks ago July 15 to 21


Let ABCDEF be an equiangular hexagon with perimeter 98 such that EF=2CD=4AB and BC=2FA.

What is the length of EF?

This problem is posed by Shyan A.



An ant starts a random walk on the real number line at 0. At each step, the ant moves

by +1 or −1 with equal probability. After 6 moves, the probability that the ant is on a positive

number can be expressed as ab, where a and b are positive coprime integers. What is the value

of a+b?

This problem is posed by Michael T.



What is the remainder, when (2013101) is divided by 101?No answer given See worked solution


Let x,y be complex numbers satisfyingx+yxy=a,=b,

where a and b are positive integers from 1 to 100 inclusive. What is the sum of all possible distinct

values of a such that x3+y3 is a positive prime number?

This problem is posed by Joe T.



Let x and y be real numbers satisfying 4x2+5y2=1. Over all such pairs, let the maximum and

minimum values of 2x2+3xy+2y2 be M and Nrespectively. If M+N+MN=ab, where a and b are

coprime positive integers, what is the value of a+b?

This problem is posed by Christian L.



Winston must choose 4 classes for his final semester of school. He must take at least 1 science

class and at least 1 arts class. If his school offers 4 (distinct) science classes, 3 (distinct) arts

classes and 3 other (distinct) classes, how many different choices for classes does he have?



How many ordered pairs (x,y) are there with x∈{0,1,2,…,25}, y∈{0,2,4,…,50} and x<y?Your answer 494 See worked solution


A square is cut by two parallel lines into three pieces of equal area as shown below.

The perpendicular distance between the parallel lines is 1. Find the area of the square.



Determine the sum of all positive integers which do not appear as one of the entries in a

Pythagorean triple.



The area of a circle centered at the origin, which is inscribed in the parabola y=x2−100, can be

expressed as abπ, where a and b are coprime positive integers. What is the value of a+b?



The sum

∑k=1154(−1)k+1(2k+1k2+k)

can be expressed as ab where a and b are coprime, positive integers. What is the value of a+b?No answer given See worked solution


As θ ranges over all real values, the maximum value of sin3θcosθtan2θ+1 can be written as ab,

where a and b are coprime positive integers. What is the value of a+b?

This problem is posed by Ed M.



What is the largest possible value of 15sinx+8cosx, where x is a real number?Your answer 17 See worked solution

Find the natural number k such that 2k+234+237 is the square of an integer n.Your answer 38 See worked solution


Given that the roots of the equation x3−3x2−13x+c=0 form an arithmetic progression, find the

constant term c.Your answer 15 See worked solution


The equation 2x2−62x+k=0 has two real roots, one of which is 1 more than twice the other. Find

the value of k.Your answer 420 See worked solution


A strip of 41 squares is numbered 0,1,2,…,40 from left to right and a token is placed on the square

marked 0. Pinar rolls a pair of standard six-sided dice and moves the token right a number of

squares equal to the total of the dice roll. If Pinar rolls doubles, then she rolls the dice a second

time and moves the token in the same manner. If Pinar gets doubles again, she rolls the dice a third

time and moves the token in the same manner. If Pinar rolls doubles a third time she simply moves

the token to the square marked 36. The expected value of the square that the token ends on can be

expressed as ab where a and b are coprime positive integers. What is the value of a+b?No answer given See worked solution


ABCD is a convex cyclic quadrilateral such that AB=AD and ∠BAD=90∘.E is the foot of the

perpendicular from A to BC, and F is the foot of the perpendicular from A to DC. If AE+AF=16,

what is the area of ABCD?No answer given See worked solution


Six people are playing Secret Santa for Christmas. They will each give one gift to someone, and

each receive one gift from someone. They are not allowed to receive their own gift. How many

different ways are there to exchange gifts?



Eight light bulbs are placed on the eight lattice points (±1,±1,±1). Each light bulb can either be

turned on or off. However, the lightbulbs are unstable, and if two light bulbs with distance less

than or equal to 2 are on simultaneously, both lights explode. How many possible configurations of

on/off light bulbs exist if no explosions occur?

This problem is shared by Muhammad A. Lewis Chen proposed it for an NIMO competition.



A newly-opened restaurant has 5 menu items. If the first 4 customers each choose one menu item

uniformly at random, the probability that the 4th customer orders a previously unordered item

is ab, where a and b are relatively prime positive integers. What is a+b?

This problem is posed by Muhammad A.



Missy and Mussy are very messy sisters. Their dresser drawer consists of 43 white socks, 2 black

socks, 23 blue socks and 8 red socks. What is the minimum number of socks they must remove

from the drawer, in order to be certain that they have removed four socks of the same color?



A drawer contains 200 distinct pairs of socks, and three socks are chosen at random. The

probability that the three socks contain a pair of socks is ab, wherea and b are coprime positive

integers. What is the value of a+b?No answer given See worked solution


We say that a number x is 2-happy if twice the sum of the digits of x is greater than x. How many

positive integers are 2-happy?Your answer 10 See worked solution


Let S(N) denote the digit sum of the integer N. Let R denote the smallest integer value of NS(N),

where N is a 3-digit number. What is the largest 3-digit number N that satisfies NS(N)=R?Your answer 270 See worked solution


How many (not necessarily positive) integer values of n are there, such that n2−2013n+5 is also an

integer?Your answer 12 See worked solution


Let x,y,z, and t be positive real numbers such that x+y+z+t=1. What is the minimum value of1x+1y+4z+16t?

This problem is posed by Tuong N.



Let 3a be the highest power of 3 that divides 1000!. What is a?

Note: 1000!=1×2×3…×999×1000.Your answer 498 See worked solution


How many positive integers n less than 10000 satisfy all of the following 3 properties:

A) n is prime.

B) n is a palindrome.

C) n has an even number of digits. Your answer 1 See worked solution

Seven weeks ago June 24 to 30


How many ordered pairs of non-negative integers (a,b) are there such that a√+b√=432−−−√ ?No answer given See worked solution


A Pythagorean triple is a triple of positive integers (x,y,z) such thatx2+y2=z2. Find the sum of

all z≤100, for which there exists a Pythagorean triple (x,y,z) with exactly two of the

numbers x, y and z being prime.Your answer 79 See worked solution


How many ordered pairs of positive integers (m,n) satisfygcd(m3,n2)=22⋅32 and lcm(m2,n3)=24⋅34⋅56?

This problem is shared by Muhammad A.



The integers 1,2,…,17 are divided into 5 disjoint sets. One set has 5 elements, one set has 4

elements, two sets have 3 elements and the last set contains the 2 remaining elements. Two

players each choose a integer from 1 to 17 at random. The probability that they choose numbers

from the same set can be expressed as ab, where a and b are coprime positive integers. What is the

value of a+b?No answer given See worked solution


40 students attended the brilliant summer camp at Stanford University. k of the students at the

camp are working on math problems and the others are working on non-math problems. One of the

camp mentors noticed that whenever all the students were partitioned into 2 or more groups of

equal size, they could never have the same number of math students in every group. How many

different values could k have?No answer given See worked solution


John has 3 unique coins such that the probability of obtaining a head when the coin is flipped

is 110, 210 and 310, respectively. If he flips each of the 3 coins once, the probability that at

least 1 heads appears is p. What is the value of 1000p?



The digits 1,2,3,4,5,6 are arranged to form a 6-digit number. The probability that the number

formed is divisible by 4 can be expressed as ab where a and bare coprime positive integers. What is

the value of a+b?Your answer 19 See worked solution


Suppose that 10 dice are rolled. Each die is a regular 6-sided die with

numbers 1 through 6 labelled on the sides. How many different distinct sums of all 10 numbers are

possible?Your answer 51 See worked solution


A sequence {ai}is defined by the recurrence relation an=40−4an−1 with a0=−4. There exists real

valued constants r,s and t such that ai=r⋅si+tfor all non-negative integers i. Determine r2+s2+t2.No answer given See worked solution


The solution to the equation

1+1−x4−x2−−−−−−√−−−−−−−−−−−√=x

can be written in the form ab where a and b are coprime positive integers. Find a+b.Your answer 9 See worked solution


Three positive integers are in geometric progression, and have a sum of 19 and a product of 216.

What is the least common multiple (LCM) of the three integers?

This problem is posed by Shrajan V.



What is the sum of all possible real values of x, such that there exists a real value y which satisfies

the equation (x+y−40)2+(x−y−18)2=0?Your answer 29 See worked solution


Find the minimum possible value of

2cos4θ+8cos3θ+32cos2θ+56cosθ+100



In an urn, there are several colored balls, with equal numbers of each color. We add 14 balls which

are all of the same new color, that is different from those in the urn. It is calculated that the

probability of drawing, without replacement, two balls of the same color before the 14 balls are

added, is equal to the probability of drawing, without replacement, two balls of the same color after

the 14balls are added. How many balls are there in the urn initially?



A 999-digit number starts with 9. Every 2 consecutive digits is divisible by 17 or 23. There are 2

possibilities for the last 3 digits. What is the sum of these 2 possibilities?



Jenny places 100 pennies on a table, with 30 showing heads and 70 showing tails. She

chooses 40 distinct pennies uniformly at random and turns them over. That is, if a chosen penny

was showing heads, she turns it to show tails; if a chosen penny was showing tails, she turns it to

show heads. After this process, what is the expected number of pennies showing heads?

This problem is shared by Muhammad A.



Γ1 is a circle with center O1 and radius R1, Γ2 is a circle with center O2 and radius R2,

and R2<R1. Γ2 has O1 on its circumference. O1O2 intersect Γ2again at A. Circles Γ1 and Γ2 intersect

at points B and C such that ∠CO1B=52∘. D is a point on the circumference of Γ1 that is not

contained within Γ2. The line DB intersects Γ2 at E. What is the measure (in degrees) of

the acute angle between lines DE and EA?Your answer 26 See worked solution


ABCD is a square. Γ1 is a circle that circumscribes ABCD (i.e. Γ1 passes through

points A,B,C and D). Γ2 is a circle that is inscribed in ABCD (i.e. Γ2is tangential to

sides AB,BC,CD and DA). If the area of Γ1 is 100, what is the area of Γ2?Your answer 50 See worked solution


Let A1B1C1D1A2B2C2D2 be a unit cube, where the vertex X2 is vertically above the vertex X1.

Let M be the center of face A2B2C2D2. Rectangular pyramid MA1B1C1D1 is cut out of the cube. The

surface area of the solid that remains after the pyramid is removed is expressed in the form a+b√,

where aand b are positive integers and b is not divisible by the square of any prime. What is a+b?




Vector v⃗ =2i+4j+7k. What is |v⃗ |2?Your answer 69 See worked solution


Find the sum of all primes p, such that p divides up, where up is the p-th Fibonacci number.No answer given See worked solution


Consider the Collatz function defined on the positive integers:

f(n){n23n+1n evenn odd

Find the smallest value of n such that f(7)(n)=5.Your answer 640 See worked solution


Let

f(n)=11√+2√+12√+3√+13√+4√…1n√+n+1−−−−√.

For how many positive integers n, in the range 1≤n≤1000, is f(n) an integer?

This problem is posed by Thaddeus A.



What is the sum of all possible positive integer values of n, such that n2+19n+130 is a perfect

square?Your answer 30 See worked solution


S=1+2(15)+3(15)2+4(15)3….

If S=ab, where a and b are coprime positive integers, what is the value of a+b?Your answer 41 See worked solution


Let S(N) denote the digit sum of the integer N. As N ranges over all 3-digit positive numbers, what

value of N would give the minimum of M=NS(N)?Your answer 199 See worked solution


Consider the function defined as

f(x)=(x2+2x+3)2+4.

As x ranges over all real values, what is the minimum value of f(x)?Your answer 8 See worked solution


The number 27+200−−−√−−−−−−−−√ can be simplified to the form a+b√, where a and bare

positive integers. Find the product ab.Your answer 10 See worked solution

Nine weeks ago June 10 to 16


Suppose f(x) is a non-constant polynomial such thatf(x3)−f(x3−2)=(f(x))2+12

for all x. Find f(5).No answer given See worked solution


The smallest possible positive value of

1−(1w+1x+1y+1z)

where w, x, y, z are odd positive integers, has the form ab, where a,b are coprime positive integers. Find a+b.Your answer 46 See worked solution


If two numbers have arithmetic mean 2700 and harmonic mean 75, find their geometric mean.

This problem is posed by Mark Ellis E.



Find the positive integer value of n such thatn3−3n3+n3−4n3+n3−5n3+…+4n3+3n3=169.

This problem is posed by Jose N.



We define the binary operation ⊕ as: a⊕b=1ab for all real non-zero values aand b. We define the

binary operation ⊗ as: a⊗b=a+bab, for all real non-zero values a and b. For what integer N does the

equation 4⊗11=115⊕N hold?Your answer 44 See worked solution


The sum of squares formula is given by

12+22+32+…+n2=n(n+1)(2n+1)6.

The sum of odd squares can be expressed as

12+32+52+…+(2n−1)2=An3+Bn2+Cn+D.

The value of A can be expressed as ab, where a and b are positive coprime integers. What is the

value of a+b?Your answer 7 See worked solution


A monic polynomial f(x) of degree four satisfies f(1)=10, f(2)=20, and f(3)=30. Determinef(12)+f(−8)−19000.

This problem is posed by Garvil S.



a,b and c are positive real numbers greater than or equal to 1 satisfyingabcalgablgbclgc=100,≥10000.

What is the value of a+b+c?No answer given See worked solution


We have a list of N consecutive 3-digit numbers, each of which is not divisible by its digit sum.

What is the largest possible value of N?No answer given See worked solution


Two congruent circles Γ1 and Γ2 each have radius 213, and the center of Γ1 lies

on Γ2. Suppose Γ1 and Γ2 intersect at A and B. The line through Aperpendicular

to AB meets Γ1 and Γ2 again at C and D, respectively. Find the length of CD.



10 weeks ago June 3 to 9


For a positive integer x, let f(x) be the function which returns the number of distinct positive

factors of x. If p is a prime number, what is the minimum possible value of f(75p2)?

This problem is posed by Michael T.



Evaluate

(84+64)(164+64)(244+64)(324+64)(404+64)(484+64)(564+64)(644+64)(44+64)(124+64)(204+64)(284+64)(364+64)(444+64)(524+64)

(604+64).No answer given See worked solution


Find the largest positive integer x such that3620+322x−4x2−−−−−−−−−−−−−−√

is a real number.Your answer 90 See worked solution


Find the sum of all positive integers m such that 2m can be expressed as sums of four factorials (of

positive integers).No answer given See worked solution


The sum of all real numbers x such thatx=129(3−29x−1−−−−−−√2013)

can be written as ab√c, where a and c are positive coprime integers and b is a positive integer that

is not divisible by the square of a prime. Find a+b+c.

This problem is posed by Zi Song Y .



Let f(x)=3171x−3−3114x+2+357x+3−1. Let S be the sum of all real values of x that satisfy f(x)=0.

What is the value of 1S?Your answer 57 See worked solution


How many ordered pairs of positive integers 1≤k≤n≤50 are there, such that k divides n, and(nk)!=n!k!?



x satisfies the equation (x+1x+1)(x+1x)=1. What is the value of(x20+1x20+1)(x20+1x20)?



6 friends (Andy, Bandy, Candy, Dandy, Endy and Fandy) are out to dinner. They will be seated in a

circular table (with 6 seats). How many ways are there to seat them?



Calvin and Dan are playing a game of chance. Calvin tosses 8 fair coins, and wins if he obtains at

least 4 heads. The probability that Calvin wins can be expressed as ab, where a,b are coprime

positive integers. What is the value of a+b?No answer given See worked solution


How many three digits numbers N=abc¯¯¯¯¯ are there such that a≤b and c≤b?Your answer 945 See worked solution


Three squares, all with side lengths 16, are placed so that two share a common side and the third

is centered on top of the bottom two as in the figure below. The radius of the smallest circle

containing all three squares is r. Determine r2.



The integers from 1 through 10 (inclusive) are divided into three groups, each containing at least

one number. These groups satisfy the additional property that if x is in a group and 2x≤10,

then 2x is in the same group. How many different ways are there to create the groups?No answer given See worked solution


Three points are chosen uniformly at random from the perimeter of circle. The probability that the

triangle formed by these is acute can be expressed as abwhere a and b are coprime positive

integers. What is the value of a+b?No answer given See worked solution


Let X be the random variable denoting the number of heads out of 100 independent, fair coin

tosses. What is the variance of X?No answer given See worked solution


P is a point outside of circle Γ. The tangent from P to Γ touches at A. A line

from P intersects Γ at B and C such that ∠ACP=120∘. If AC=16 and AP=19, then the radius

of Γ can be expressed as ab√c, where b is an integer not divisible by the square of any prime

and a,c are coprime positive integers. What is a+b+c?No answer given See worked solution


Given that sin(2θ)=23, the value ofsin6θ+cos6θ

can be written as ab with a and b coprime positive integers. Find a+b.

This problem is posed by Ryan P.



There are four complex fourth roots to the number 4−43√i. These can be expressed in polar form

asz1=r1(cosθ1+isinθ1)

z2=r2(cosθ2+isinθ2)

z3=r3(cosθ3+isinθ3)

z4=r4(cosθ4+isinθ4),

where ri is a real number and 0∘≤θi<360∘. What is the value of θ1+θ2+θ3+θ4 (in degrees)?No answer given See worked solution


Suppose a,b, and c are positive integers such thata+b+c+ab+bc+ca+abc=1000.

Find a+b+c.

This problem is posed by Vishnoo P.



If a+b+c=0 and a2+b2+c2=22, what is a4+b4+c4?No answer given See worked solution


Given x such that 9x−4x=665 and 3x−2x=19, evaluate 3x+2x.




Juliet has attempted 213 problems on Brilliant and solved 210 of them correctly. Her friend Romeo

has just joined Brilliant, and has attempted 4problems and solved 2 correctly. From now on, Juliet

and Romeo will attempt all the same new problems. Find the minimum number of problems they

must attempt such that it is possible that Romeo's ratio of correct solutions to attempted problems

will be strictly greater than Juliet's.Your answer 413 See worked solution


What is the sum of all integer values of x such that(x2−17x+71)(x2−34x+240)=1?



What is the expected number of heads that will appear in 52 fair coin tosses?

Note: A fair coin has probability 12 of landing heads, and probability 12 of landing tails. It has

probability 0 of landing 'on its side'.Your answer 26 See worked solution


In basketball, a player can score points in 3 different ways: 1 point from the foul line, 2 points for

shots close to the basket, or 3 points for shots far away from the basket. If a player

scores 30 points in a game, how many different ways can this be achieved?Your answer 49 See worked solution


A full binary tree of height 4 has 15 nodes, as pictured below.

The 8 nodes at the bottom of the tree are called end nodes or leaf nodes. Two distinct end nodes

are uniformly chosen. The expected length of the shortest path between them can be expressed

as ab, where a and b are coprime positive integers. What is the value of a+b?No answer given See worked solution


Triangle ABC has area equal to 903√4 and perimeter equal to 30. Also, one of its angles is equal

to 60∘. What is the product of the sides of ABC?No answer given See worked solution


Points A,B,C are given on circle Γ such that AB=BC. The tangents at A and at B intersect again at

point D. The line CD intersects Γ again at E. The line AE intersects BD at F. If AD=120, what is the

length of FB?Your answer 55 See worked solution


f(x) is a cubic polynomial such that f(n)=1n2+1 for n=1,2,3,4. If f(0)=ab, where a and b are coprime

positive integers, what is the value of a+b?No answer given See worked solution


Miles has 4 Lego people in his Lego town. A Lego person consists of exactly 1 set of legs, a torso,

and a head. All of these body parts are distinct. Miles' son comes in and separates the Lego people

into their individual pieces. How many different sets of 4 Lego people can be created from the

pieces?



In triangle ABC, points D,E,F are on sides BC,CA,AB respectively such that AD,BE,CF are angle

bisectors of triangle ABC. The lines AD,BE,CFare concurrent at I, the incenter of triangle ABC.

If ∠BAC=92∘, what is the measure (in degrees) of ∠EIF?Your answer 136 See worked solution


Let S be a set of 31 equally spaced points on a circle centered at O, and consider a uniformly

random pair of distinct points A and B (A,B∈S). The probability that the perpendicular bisectors

of OA and OB intersect strictly inside the circle can be expressed as mn, where m,n are relatively

prime positive integers. Find m+n.



12 weeks ago May 20 to 26


Suppose a,b, and c are positive integers such thatab+bc=130,

ac+bc=168,

ab+ac=228.

Find a+b+c.

This problem is posed by Kiriti M.



Find the value of x satisfying the equation3x+5−−−−√−x−2−−−−√−−−−−−−−−−−−−−√=3.



x and y are positive real numbers that satisfy logxy+logyx=174 and xy=2883√. If x+y=a+bc√,

where a, b and c are positive integers and c is not divisible by the square of any prime, what is the

value of a+b+c?Your answer 149 See worked solution


How many integers satisfy (n√−23×24−−−−−−√)2<1?Your answer 93 See worked solution


How many pairs of positive integers (a,b), where a≤b satisfy 1a+1b=150?Your answer 8 See worked solution


a, b, c, A and C are real numbers that satisfy the equation(ax2+bx+c)2=Ax4+16x3+28x2+24x+C

How many possible ordered triples of (a,b,c) are there?Your answer 4 See worked solution


Let N be the number that consists of 61 consecutive 3's, so N=333…333613′s. Let M be

the number that consists of 62 consecutive 6's, so M=6666…666626′s. What is the digit

sum of N×M?Your answer 558 See worked solution


The sequence {ak}112k=1 satisfies a1=1 and an=1337+nan−1, for all positive integers n. LetS=⌊a10a13+a11a14+a12a15+⋯+a109a112⌋.

Find the remainder when S is divided by 1000.

This problem is posed by Akshaj.



Given quadrilateral ABCD where AB is parallel to DC and AC and BDintersect at E,

with ∠BEC=52∘ and ∠ACD=25∘, what is ∠ABE (in degrees)?Your answer 27 See worked solution


A bicycle manufacturer has 4 different colors to paint the bike frame and another 7 colors to paint

the bike wheels. How many different color combinations of bicycles can this manufacturer make?Your answer 28 See worked solution


Two independent, fair 6 sided dice are rolled. The first die face shows a 3. The second die rolls

under the table, so you can not see its face. The probability that the dice under the table is 3 given

that the first dice is 3 can be written as ab, where a and b are positive coprime integers. What is

the value of a+b?Your answer 7 See worked solution


An isosceles triangle ABC has AB=AC. Points D and E are within line segment BC such

that B,D,E,C lie in order and AD=AE. If ∠DAE=34∘and ∠ACB=37∘, what is the measure

of ∠BAE (in degrees)?Your answer 70 See worked solution


Vectors u ⃗ =2i+9j+0k and v⃗ =13i+18j+0k. What is the value of |u ⃗ ×v⃗ |?Your answer 81 See worked solution


Equilateral triangle ABC has a circumcircle Γ with center O and circumradius 10. Another

circle Γ1 is drawn inside Γ such that it is tangential to radii OC andOB and circle Γ. The radius

of Γ1 can be expressed in the form ab√−c, where a,b and c are positive integers, and b is not

divisible by the square of any prime. What is the value of a+b+c?



ABCD is a rectangle with AB=50 and BC=120. E is a point on CD(possibly extended) such

that AE=EC. What is the length of DE?Your answer 119 See worked solution


Petr has ten different two-player board games on his shelf. He has them numbered 1 through 10.

When he plays the game n against Diego, he has a n2100chance of winning. If Petr rolls a fair ten-

sided die to determine which game to play, the probability that he will win against Diego can be

expressed as ab wherea and b are positive, coprime integers. What is the value of a+b?Your answer 101 See worked solution



The three interior angles in a triangle are x∘, (2x+24)∘ and (3x+48)∘. What is the value of x?Your answer 18 See worked solution


What is the mean of the first 10 positive even numbers?



There are 10 cards, labeled from 1 to 10, lying face down on a table. You pick a card, and if that

card is a 10, you win $20. If the card is not 10 you win $0. What is the expected amount you would

win (in dollars)?Your answer 2 See worked solution


Rajiv has 1000 marbles. He wants to store them separately in different bags, so that each bag

contains a different number of marbles. What is the maximum number of bags that Rajiv will use?



What is the radius of the base of a cylinder with surface area equal to 420π and height equal to 11?



Suppose x and y are real numbers and 4x+3xi+yi−43=2y+2xi−x+2yi+5i. Find the product xy.Your answer 66 See worked solution


What is the largest prime factor of 58+22?Your answer 677 See worked solution


Find the number of 6-term strictly increasing geometric progressions, such that all terms are

positive integers less than 1000.Your answer 998 See worked solution


x and y are positive values satisfying x+y=14. When ln(y2+4xy+7)attains its maximum (subject to

the given constraint), what is the value of 1xy?No answer given See worked solution


Let f(x) be a polynomial such thatf(f(x))−x2=xf(x).

Find f(−100).




Determine the last three digits of

∑n=210,000,000(n7+n5).




How many positive integers n are there such that 10n≤n10?Your answer 10 See worked solution


How many positive integers less than 1020 have all their digits the same?Your answer 180 See worked solution


The probability that a positive divisor of 60 is greater than 9 can be written as ab,

where a and b are coprime positive integers. What is the value of a+b?Your answer 3 See worked solution


Vincenzo makes a sandwich using four kinds of Italian lunch meat: Genoa, prosciutto, Calabrese

and capicola and two kinds of Italian cheese: provolone and asiago. Vincenzo's sandwich has a

single layer of each type of meat and a single layer of each kind of cheese, but he also wants to

make sure that the two types of cheese are not next to each other. How many different ways can

Vincenzo arrange the meat and cheese on his sandwich?



An ant is walking along the cartesian plane. It starts at the point A=(−18,8)walks in a straight line

to a point on the x-axis, walks directly to the right for 7 units and then walks in a straight line to

the point B=(9,13). What is the shortest distance that the ant could have walked?No answer given See worked solution



θ is an angle such that 180∘<θ<270∘, and sinθ=−1213. What is the value of180(sin2θ2+cos2θ2+tan2θ2)?



The product of the digits of a positive two-digit number exceeds the sum of the digits by 39. If the

order of the digits is reversed, the number is increased by 27. Find the number.Your answer 69 See worked solution


Find the smallest positive integer n such that the equation 455x+1547y=50,000+n has a

solution (x,y) where both x and y are integers.Your answer 232 See worked solution


Suppose θ is an angle strictly between 0 and π2 such that sin5θ=sin5θ. The number tan2θ can be

uniquely written as ab√, where a and b are positive integers, and b is not divisible by the square of

a prime. What is the value of a+b?No answer given See worked solution


Suppose a and b are positive integers satisfying 1≤a≤31, 1≤b≤31 such that the

polynomial P(x)=x3−ax2+a2b3x+9a2b2 has roots r, s, and t.

Given that there exists a positive integer k such that (r+s)(s+t)(r+t)=k2, compute the maximum

possible value of ab.

This problem is posed by Samir K .



A 10 km race has 5 water stations set up around the course. What is the minimum number of

people that must run in the race in order to guarantee that 10 people stop at the same set of water

stations?



What is the sum of all positive integers N≤1000 such that N equals 13 times the digit sum of N?Your answer 468 See worked solution


Let x=2013, y=140542, z=−142555. Find(x−yz+y−zx+z−xy)⋅(xy−z+yz−x+zx−y).



Find the sum of squares of all real roots of the polynomial f(x)=x5−7x3+2x2−30x+6.Your answer 20 See worked solution


The game Upright is played by two players on an m×n board of squares and has the following

rules:

1. At the start of the game, a kangaroo game piece is placed on the bottom left square of the board.

2. Players alternate turns moving the kangaroo, and the first player moves first.3. On the player's turn, they can either move the kangaroo some number of squares to the right,

keeping it in the same row, or they can move it to the leftmost square on the row above.4. A player loses if they are unable to make a move.

If m and n can each be any number between 1 and 20 inclusive, for how many of the 400 possible

game board sizes can the second player win if both players play optimally?No answer given See worked solution


The magician, The Great Ziggyny, is planning his next world tour. He knows a total of 7 different

illusions but he only has time to perform 4 of them at any show. The Great Ziggyny does not want

any two of his shows to be identical, so they cannot contain the same 4 illusions in the same order.

What is the most number of cities The Great Ziggyny can visit and perform at on his tour?



A certain professional soccer team runs a 4-4-2 formation on the field. This means they play with 4

defenders, 4 midfielders, 2 forwards, and 1 goalkeeper on the field. If the team has 6 defenders, 5

midfielders, 3 forwards, and 2 goalkeepers, how many different groups of 11 starting players could

they have?



Three spheres of radius 10 are placed on a table all touching each other. A fourth sphere of radius

10 is placed so that it lies on top of the other three. The distance from the bottom of the fourth

sphere to the table is h, and h2=ab, where a and b are coprime positive integers. What is the value

of a+b?No answer given See worked solution


Determine the least positive integer n for which the following condition holds: No matter how the

elements of the set of the first n positive integers, i.e. {1,2,…n}, are colored in red or blue, there

are (not necessarily distinct) integers x,y,z, and w in a set of the same color such that x+y+z=w.

This problem is shared by Sayan C.



Two points are chosen uniformly at random on the unit circle and joined to make a chord C1. This

process is repeated 17 more times to get chords C2,C3,…,C18. What is the expected number of pairs

of chords that intersect?Your answer 9 See worked solution


Let AB be the diameter of circle Γ1. In the interior of Γ1, there are circles Γ2and Γ3 that are tangent

to Γ1 at A and B, respectively. Γ2 and Γ3 are also externally tangent at the point C. This tangent line

(at C) cuts Γ1 at P and Q, with PQ=20. The area that is within Γ1 but not in Γ2 or Γ3 is equal to Mπ.

Determine M.Your answer 200 See worked solution


Let r1,r2,r3 be the roots of polynomial P(x)=x3+3x+1. Evaluate the product∏k=13(r2k+rk+1).

This problem is posed by Francisco R.



How many integers x satisfy the condition that both 2x and 3x are perfect squares?

This problem is posed by Qi Huan T .



x,y,z are positive real numbers such that x+y+z=9 and xy+9xz+25yz=9xyz. Find the sum of all

possible values of xyz.

This problem is posed by Alexander K.



ABCD is a rectangle with AC=20 and AB=2BC. What is the area of rectangle ABCD?Your answer 120 See worked solution


A right triangle has perimeter equal to 80 and hypotenuse equal to 34. What is the area of the

triangle?Your answer 240 See worked solution


The roots of the polynomial f(x)=2x3+20x2+201x+2013 are α,β and γ. What is the value of −

(α+1)(β+1)(γ+1)?Your answer 915 See worked solution


Three roots of f(x)=x4−2x3+ax2+bx+c are −5, −3 and 4. What is the value of a+b+c?

You may choose to refer to the Remainder-Factor Theorem.



We are given that log102<0.302. How many digits are there in the decimal representation of 5500?



a and b are positive integers that satisfy 18a=b3. What is the minimum possible value of a+b?Your answer 18 See worked solution


If the expression (x2+2x−1)8 is completely expanded, what is the sum of the coefficients of the

terms with even powers of x?No answer given See worked solution


Raoul wants to create a weekly schedule for going to the gym. He wants to go to the gym the same

three days each week, and he wants there to be at least one day in between each of his visits. How

many different ways can Raoul schedule his weekly gym visits?



There are 4 students in a class. A teacher wants them to each secretly choose a partner for a group

project. If everyone independently chooses a partner randomly, the probability that everyone

choses a partner who chose him/her is P. What is the value of 1P?Your answer 70 See worked solution


A point P is uniformly chosen inside a regular hexagon of side length 3. For each side of the

hexagon a line is drawn from P to the point on that side which is closest to P. The probability that

the sum of the lengths of these segments is less than or equal to 93√ can be expressed

as ab where a and b are coprime positive integers. What is the value of a+b?Your answer 1 See worked solution


ABC is an equilateral triangle with side length N. X is a point within ABC, such that the distance

from X to the vertices A,B,C are 8, 13 and 15, respectively. What is the value of N2?No answer given See worked solution


A video rental store offers 436 different movies for rent. The movies are categorized as Comedies,

Action Movies, and Dramas. A movie may be in more than one of the categories. If 234 movies are

categorized as Comedies, 97 as Action Movies, and 191 as Dramas, what's the most number of

movies that could be categorized as all three?



Γ is a circle with chord AB. P is a point outside of Γ such that PA is tangent to Γ and ∠BPA=90∘. If AB=48 and PB=8, what is the radius of Γ?Your answer 8 See worked solution

16 weeks ago April 22 to 28


x and y are real numbers that satisfy the equations x2+y2=125 and x+y=13. What is the sum of all

(distinct) possible values of x?Your answer 13 See worked solution


How many distinct integer values of N between 1 and 1000 are there, such

that N=4a+b+4c and 2N=7a+6b+7c for some positive integers a, b and c?Your answer 58 See worked solution


For how many odd positive integers n<1000 does the number of positive divisors of n divide n?No answer given See worked solution


4 distinct integers p, q, r and s are chosen from the set {1,2,3,…,16,17}. The minimum possible

value of pq+rs can be written as ab, where a and b are positive, coprime integers. What is the value

of a+b?Your answer 321 See worked solution


a,b and c are positive integers such that the simultaneous

equations (a−2b)x=1, (b−2c)x=1 and x+25=c have a positive solution for x. What is the

minimum value of a?Your answer 107 See worked solution


What is the largest 3 digit divisor of N=10243−6393−3853?Your answer 994 See worked solution


Find the smallest prime number N such that the following is true:

The largest prime factor of N−1 is A;

The largest prime factor of A−1 is B;

The largest prime factor of B−1 is 7.Your answer 709 See worked solution


In triangle ABC, we have AB=17,BC=18,CA=19. Points D,E and Fare the midpoints

of BC,CA and AB respectively. What is the perimeter of triangle DEF?Your answer 27 See worked solution


What is the sum of all numbers that occur an odd number of times in rows 0through 11 of Pascal's

triangle?Your answer 351 See worked solution


In a recent election for class president, Monika received 7 of the 10 votes and Alfred received 3 of

the 10 votes that were cast by the class. When the machine was counting the votes, it

malfunctioned and instead of giving the vote to the correct person, it gave the vote to each

candidate with probability 12 (regardless of whom the vote was cast for). The probability that the

machine gave each student the correct number of votes in the election can be expressed

as ab where a and b are positive, coprime integers. What is the value of a+b?No answer given See worked solution


P is a point in triangle ABC. The lines AP,BP, and CP intersect the sides BC,CA, and AB at

points D,E, and F, respectively. If [BDP]=10,[DPC]=16, [APB]=210, what is [APC]?Your answer 336 See worked solution


Bibi wants to send invitations to some of her friends to come to her birthday party. She has contact

information for 37 friends stored on her phone, but she only wants to invite 21 of those people to

her party. She tries to send out a message to the 21 people, but her phone malfunctions and sends

the message to three random contacts (uniformly chosen). The probability that exactly one person

who got the invitation was not supposed to be invited to the party can be expressed as ab,

where a and b are positive, coprime numbers. What is the value of a+b?Your answer 19 See worked solution


A farmer is testing a new piece of equipment he has for determining whether an animal is a cow or

a sheep. If a cow walks through the machine, 90% of the time it says it is a cow and 10% of the

time it says it is a sheep. If a sheep walks through the machine, 95% of the time it says it is a sheep

and 5% of the time it says it is a cow. The farmer has 5 cows and 36 sheep. If an animal walks

through and the machine claims it is a cow, the probability that it actually is a cow can be

expressed as ab where a and b are coprime numbers. What is a+b?Your answer 12 See worked solution


An equilateral triangle ABC has AB=203√. P is a point placed in triangle ABC and D,E and F are the

foot of the perpendiculars from P to AB, BCand AC, respectively. If PD=9 and PE=10, what is the

value of the length of PF?Your answer 11 See worked solution


Triangle ABC has integer side lengths. Rectangles BCDE,ACFG,ABHJ are constructed so

that CD=AC+AB, CF=AB+BC, and BH=(AC+BC)2. If [ABHJ]=[BCDE]+[ACFG], how many different

values can [ABC] have?Your answer 4 See worked solution


Suppose f(x) is a degree 8 polynomial such that f(2i)=12i for all integers 0≤i≤8. If f(0)=ab,

where a and b are coprime positive integers, what is the value of a+b?Your answer 3 See worked solution



4 numbers are given such that each of the 6 pairwise sums are distinct. If the 4 smallest sums

are 7,9,10 and 14, what is the largest possible product of these 4 numbers?No answer given See worked solution


The product of 2 positive integers is 1000. What is the smallest possible sum of these 2 integers?



Let σ(n) be the sum of the positive divisors of an integer n and ϕ(n) be the number of positive

integers smaller than n that are coprime to n. If p is a prime number, what is the maximum

value σ(p)ϕ(p)?No answer given See worked solution


x,y and z are positive integers such that x<y,x+y=201,z−x=200. What is the maximum value

of x+y+z?Your answer 501 See worked solution


It can be shown that for any positive integer n, the infinitely nested radical expressionn+n+n+⋯−−−−−−√−−−−−−−−−−√−−−−−−−−−−−−−−−√

equals a finite number. What is the largest positive integer n≤999 such that this expression is equal to a positive integer?Your answer 992 See worked solution


Find the sum of all 2 digit numbers N=ab¯¯¯, where a≠0, such that N divides a0b¯¯¯¯¯.Your answer 78 See worked solution


x,y and z are complex numbers that satisfy the equations x+y+z=2, xy+yz+zx=3 and xyz=4.

Given that 11−x−yz+11−y−zx+11−z−xy=ab, where a and b are coprime positive integers, what is the

value of a+b?No answer given See worked solution


If x=3√+5√, what is the value of−x2(x2−8)(x2−14)(x2−18)?



41 different balls are drawn from a bag containing balls labelled 1,2,…,300. The probability that

the number of balls drawn that are labelled with an odd number is larger than the number of balls

drawn that are labelled with an even number can be expressed as ab where a and b are coprime

positive integers. What is the value of a+b?No answer given See worked solution


In triangle ABC, we have ∠BCA=3∠CAB, BC=343 and AB=504. What is AC?Your answer 253 See worked solution


A circle has diameter AD of length 400. B and C are points on the same arc of AD such that |AB|=|

BC|=60. What is the length |CD|?Your answer 382 See worked solution


There are 100 runners, each given a distinct bib labeled 1 to 100. What is the most number of

runners that we could arrange in a circle, such that the product of the numbers on the bibs of any 2

neighboring runners, is less than 1000?



0 lines cuts the plane into at most 1 region.

A line cuts the plane into at most 2 regions.

2 lines cut the plane into at most 4 regions.

What is the most number of regions that 9 lines can cut the plane into?



A bag contains 10 red marbles, 10 green marbles, 10 yellow marbles and 10blue marbles. You

reach into the bag and grab a marble, then reach into the bag and grab a second marble. The

probability that the second marble is the same color as the first marble is ab, where a and b are

positive, coprime integers. What is the value of a+b?

Note: You do not place the first marble back into the bag.



How many 3 digit positive even integers are there, such that all the digits are distinct from each

other?



(Refer to diagram below) Two lines intersect at O at an angle of 60∘. Two circles are drawn such

that they are tangent to each other and the two lines. Aand B are the centers of the smaller and

larger, respectively. If the radius of the smaller circle is 15, what is the length of OB?




For any real number α, the parabola fα(x)=2x2+αx+3α passes through the common point (a,b).

What is the value of a+b?Your answer 15 See worked solution


An ordered triple of real numbers (a,b,c) is called friendly, if each number is equal to the product

of the other 2. How many (distinct ordered) friendly triples are there?Your answer 2 See worked solution


Find a positive integer n such that ⌊20n13⌋+⌈13n20⌉=2013.

This problem is proposed by Ahaan.



Given that x is a real number satisfying⌊x+11100⌋+⌊x+12100⌋+…+⌊x+90100⌋=331,

what is ⌊100x⌋?No answer given See worked solution


Raj and Vikram are two travelers in ancient India, walking along the Silk road. On the first day of

travel, Raj travels 5 yojanas, and on each successive day travels 3 yojanas more than the previous

day. Vikram started at the same place and travels the same path, but started 5 days earlier, and

travels 7 yojanas a day. On which day of Raj's travel will the two of them meet?Your answer 7 See worked solution


Let N=1!⋅2!⋅3!⋅4!…9!⋅10!. Let 2k be the largest power of 2 that divides N. What is the value of k?Your answer 38 See worked solution


If two six-sided dice are rolled, the probability that they both show the same number can be

expressed as ab where a and b are coprime positive integers. What is the value of a+b?Your answer 7 See worked solution


Triangle ABC has lengths AB=14, AC=18 and also a given angle of ∠BAC=30∘. What is the area of

triangle ABC?

You may choose to read the blog post on Sine Rule.



In triangle ABC, D is the midpoint of AC and E is the midpoint of AB. BDand CE are perpendicular

to each other and intersect at the point G. If AB=7and AC=9, what is the value of BC2?Your answer 26 See worked solution


A point (x,y) in the first quadrant is 600 units closer to the point (500,0) than to the

point (−500,0) and is a distance of 400 units away from the origin. The value of x is 2√A. Find A.Your answer 2 See worked solution


In a cookie jar, there are 6 chocolate chip cookies and 8 oatmeal cookies. Hungry Paddy takes out

the cookies one at a time and proceeds to eat them. The probability that the 7th cookie that Paddy

eats is a chocolate chip cookie is ab, where a and b are positive coprime integers. What is the value

of a+b?Your answer 10 See worked solution


A farmer is testing a new piece of equipment he has for determining whether an animal is a cow or

a sheep. If a cow walks through the machine, 90% of the time it says it is a cow and 10% of the

time it says it is a sheep. If a sheep walks through the machine, 95% of the time it says it is a sheep

and 5% of the time it says it is a cow. The farmer has 5 cows and 36 sheep. If an animal walks

through and the machine claims it is a cow, the probability that it actually is a cow can be

expressed as ab where a and b are coprime numbers. What is a+b?Your answer 12 See worked solution


Let θ=sin−1725. Consider the sequence of values defined by an=sin(nθ). They satisfy the recurrence

relationan+2=k1an+1+k0an,n∈N

for some (fixed) real numbers k1,k0. The sum k1+k0 can be written as pq, where p and q are positive coprime integers. What is the value of p+q?

This problem is proposed by Matt.



Convex quadrilateral ABCD has sides AB=BC=21, CD=15 and AD=9. Given additionally

that ∠ABC=60∘, what is the length of BD?

This problem is proposed by John.



Find the number of positive integers <1000 that can be expressed as 2k−2m,where k and m are

non-negative integers.Your answer 56 See worked solution


The function f(x)=x3+14x−14 is a monotonically increasing function, hence it is injective (one-to-

one), so its inverse function exists and is well defined. How many points of intersection are there,

between the function f(x)and its inverse f−1(x)?Your answer 2 See worked solution



What is the largest possible integer that can be chosen as one of five distinct positive integers

whose average is 10?



Let f(x)=2x2+40x+25. Given that f(x) leaves the same remainder when divided by x−a as when

divided by x+2a for a positive integer a, what is the value of a?Your answer 20 See worked solution


Given that 2.004<log10101<2.005, how many digits are there in the decimal representation

of 101101?Your answer 207 See worked solution


x,y and z are complex numbers satisfying

⎧⎩⎨x1+y1+z1x2+y2+z2x3+y3+z3=1=2=3

The value of x4+y4+z4 can be expressed as ab, where a and b are positive coprime integers. What

is the value of a+b?

This problem is proposed by Harshit.



How many ordered triples (x,y,z) of integer solutions are there to the system of equations{x2+y2+z2x2z2+y2z2=194=4225?



Let f(x)=x2−1. How many distinct real roots are there to f(f(f(x)))=0?Your answer 4 See worked solution


Yan was opening a new restaurant, so she went to the sign store to get letters to make a sign to

hang above the storefront. When she got to the sign store, the only letters they had in stock were

two copies of the letter 'X,' two copies of the letter 'E,' and one copy of the letter 'V.' Yan decided to

buy all the letters anduse all of them to make the name of her store. How many different names

can she give her store using these 5 letters?



The vertices of a cube are labeled with the numbers 0, 1, 2, 3, 4, 5, 6 and 7, such that no two

vertices are labeled with the same number. The sum of any 2numbers on an edge is prime. What is

the maximum sum of the 4 numbers on a face?Your answer 18 See worked solution


How many lattice points lie on the sphere x2+y2+z2=2013?No answer given See worked solution


A bag contains 150 balls numbered 1 through 150. Three balls are drawn and placed on the table.

The probability that the balls were drawn in increasing order can be expressed

as ab where a and b are coprime positive integers. What is the value of a+b?No answer given See worked solution

geometry and combinatorics

Documents

answer givensee

linespqandxyintersect

line segmentgp

trapezoid withadbcandab

length ofxy

lineaxintersects2again

lineayintersects1again