## Problem(s) of the Week Description

## Problems curated/created by Problem Czar, ** Edwin Xie**

****

Solve fun and interesting problems that cover a broad range of difficulties to develop your skills! Pick a problem(or problems) that are hard enough to challenge you but easy enough to solve or make progress on. Sources are cited when external sources are used. We will go over the weekly problem at the beginning of the following meeting.

Submit answers through online form on website and earn points for both correct answers and elegant solutions.

## Request a POTW topic!

Saw a cool math topic? We can make a problem about it! Request a Problem of The Week topic. If you want math problems about infinite Markov chains or a math problem about the Prisoner's Dilemma (or anything in between), you'll get them.

Current Problem(s) of the Week

1. Given that

*f*is continuous, find all f such that f(x+y) = f(x) + f(y) +1.2. Let x be a real number such that (sin(x))^10+(cos(x))^10 = 11/36. Find (sin(x))^12+(cos(x))^12.[AIME]

3. Question Below[BMT]:

*Provide answers/solutions to all the questions that you solve*

**in one form submission**

## Past Problems of the Week

1. Find the number of pairs (A,B) of distinct subsets of {1,2,3,4,5,6,7,8}, such that A is a proper subset of B.[SMT]

2. Let the altitude of 4ABC from A intersect the circumcircle of 4ABC at D. Let E be a point on line AD such that E is not A and AD = DE. If AB = 13, BC = 14, and AC = 15, what is the area of quadrilateral BDCE?[SMT]

3. A square is inscribed in a semicircle of radius 1. Then, another square is inscribed such that two of the vertices lie on the semicircle, and the side opposite to those vertices lies on the side of the previous square. If this process is continued forever, what is the sum of the areas of the squares?

1. Let x, y be real numbers such that x+y = 2, and x^4+y^4 = 1234. Find xy.[SMT]

2. The bases of a right hexagonal prism are regular hexagons of side length s > 0, and the prism has height h. The prism contains some water, and when it is placed on a ﬂat surface with a hexagonal face on the bottom, the water has depth s√3/4 . The water depth doesn’t change when the prism is turned so that a rectangular face is on the bottom. Compute h/s.[SMT]

3. Edward has a 3×3 tic-tac-toe board and wishes to color the squares using 3 colors. How many ways can he color the board such that there is at least one row whose squares have the same color and at least one column whose squares have the same color? A coloring does not have to contain all three colors and Edward cannot rotate or reﬂect his board.[SMT]

11/23/19

1. If f(x)=xf(x/2) for all real x, what are the possible functions f(x)?

2. Square ABCD has side length 4, and E is a point on segment BC such that CE = 1. Let C1 be the circle tangent to segments AB,BE, and EA, and C2 be the circle tangent to segments CD,DA, and AE. What is the sum of the radii of circles C1 and C2? [CHMMC]

3. (Problem below)[AMC 12]

11/9/19

1. Let S(x) denote the sum of the digits of a positive integer x. Find the maximum possible value of S(x + 2019) − S(x).[HMMT Guts]

2. Find the sum of the infinite sequence 1, cos(a)/2,cos(2a)/4,cos(3a)/8,cos(4a)/16,... where cos a = 1/5.

3. Find the value of the following infinite sum.[HMMT Individual Round]

10/25

1. Let S(n) be the sum of the digits of the integer n. If S(n) = 2018, what is the smallest possible value of S(n+1)?[SMT]

2. Let a, b, c be positive real numbers such that a+b+c = 10 and ab+bc+ca = 25. Let m = min{ab,bc,ca}. Find the largest possible value of m.[HMMT]

3. Problem Below: A biased coin has a (6+2sqrt(3))/12 chance of landing heads, and a (6-2sqrt(3))/12 chance of landing tails. What is the probability that the number of times the coin lands heads after being flipped 100 times is a multiple of 4? The answer can be expressed as 1/4+(1+a^b)/c*d^e where a,b,c,d,e are positive integers. Find the minimal possible value of a+b+c+d+e. [BMT]

10/18:

Theme: Proofs!(yay)

1. Prove that 2019 cannot be expressed as the sum of two perfect squares.

2. Show that 1^3+2^3+3^3+...+n^3=(1+2+3+...+n)^2

3. Show that there exists no right triangle with integer side lengths with the area of a perfect square.

10/11:

Theme: Cyclic relays and convergence

Find all sets of solutions to these questions:

1. How many nonnegative solutions does x^3+x2=0 have, where x2 is the second answer?

2.What is the remainder of the third answer modulo 9?

3.What is the cube of the first answer?

10/4:

1(Easy): Edwin's accuracy rate when solving math problems is 15%. If he has currently solved 1000 math problems, how many problems must he get right to raise his accuracy rate to at least 50%?

2(Medium): Given that x,y are real numbers satisfying x>y>0, compute the minimum value of (5x^2 -2xy+y^2)/(x^2-y^2). [Stanford Math Tournament]

3(Hard): Prove or disprove that there are an infinite number of numbers n such that n^3 is the largest value in a primitive Pythagorean triple, where a primitive Pythagorean triple is a Pythagorean triple (a,b,c) such that gcd(a,b,c) = 1.

9/27:

1(Easy): In the middle of the night, Thomas the Tank Engine was pranked and his circular wheels were replaced with triangular ones. If he manages to maintain the same speed that he had yesterday, what is the ratio of his rpm (rotations per minute) today to his rpm yesterday?

2(Medium): Define f(n) = (n^2+n)/2 . Compute the number of positive integers n such that f(n) ≤ 1000 and f(n) is the product of two prime numbers. [BMT Discrete Round]

3(Hard): Cryo-freezing has been invented! Now, all of the scientists want to freeze themselves so they can be alive when someone invents something cool. However, they need some scientists to stay awake to invent cool things. There are n scientists named 1,2,3, ... n, and for every number 0<i<=n, scientist i invents something cool every i days. On day 0, everyone invents something cool (one of them was cryo-freezing!). Each day after that, 1 person freezes themselves (they never wake up). What is the maximum number of cool things that could have been invented after day 0?

2. Let the altitude of 4ABC from A intersect the circumcircle of 4ABC at D. Let E be a point on line AD such that E is not A and AD = DE. If AB = 13, BC = 14, and AC = 15, what is the area of quadrilateral BDCE?[SMT]

3. A square is inscribed in a semicircle of radius 1. Then, another square is inscribed such that two of the vertices lie on the semicircle, and the side opposite to those vertices lies on the side of the previous square. If this process is continued forever, what is the sum of the areas of the squares?

1. Let x, y be real numbers such that x+y = 2, and x^4+y^4 = 1234. Find xy.[SMT]

2. The bases of a right hexagonal prism are regular hexagons of side length s > 0, and the prism has height h. The prism contains some water, and when it is placed on a ﬂat surface with a hexagonal face on the bottom, the water has depth s√3/4 . The water depth doesn’t change when the prism is turned so that a rectangular face is on the bottom. Compute h/s.[SMT]

3. Edward has a 3×3 tic-tac-toe board and wishes to color the squares using 3 colors. How many ways can he color the board such that there is at least one row whose squares have the same color and at least one column whose squares have the same color? A coloring does not have to contain all three colors and Edward cannot rotate or reﬂect his board.[SMT]

11/23/19

1. If f(x)=xf(x/2) for all real x, what are the possible functions f(x)?

2. Square ABCD has side length 4, and E is a point on segment BC such that CE = 1. Let C1 be the circle tangent to segments AB,BE, and EA, and C2 be the circle tangent to segments CD,DA, and AE. What is the sum of the radii of circles C1 and C2? [CHMMC]

3. (Problem below)[AMC 12]

11/9/19

1. Let S(x) denote the sum of the digits of a positive integer x. Find the maximum possible value of S(x + 2019) − S(x).[HMMT Guts]

2. Find the sum of the infinite sequence 1, cos(a)/2,cos(2a)/4,cos(3a)/8,cos(4a)/16,... where cos a = 1/5.

3. Find the value of the following infinite sum.[HMMT Individual Round]

10/25

1. Let S(n) be the sum of the digits of the integer n. If S(n) = 2018, what is the smallest possible value of S(n+1)?[SMT]

2. Let a, b, c be positive real numbers such that a+b+c = 10 and ab+bc+ca = 25. Let m = min{ab,bc,ca}. Find the largest possible value of m.[HMMT]

3. Problem Below: A biased coin has a (6+2sqrt(3))/12 chance of landing heads, and a (6-2sqrt(3))/12 chance of landing tails. What is the probability that the number of times the coin lands heads after being flipped 100 times is a multiple of 4? The answer can be expressed as 1/4+(1+a^b)/c*d^e where a,b,c,d,e are positive integers. Find the minimal possible value of a+b+c+d+e. [BMT]

10/18:

Theme: Proofs!(yay)

1. Prove that 2019 cannot be expressed as the sum of two perfect squares.

2. Show that 1^3+2^3+3^3+...+n^3=(1+2+3+...+n)^2

3. Show that there exists no right triangle with integer side lengths with the area of a perfect square.

10/11:

Theme: Cyclic relays and convergence

Find all sets of solutions to these questions:

1. How many nonnegative solutions does x^3+x2=0 have, where x2 is the second answer?

2.What is the remainder of the third answer modulo 9?

3.What is the cube of the first answer?

10/4:

1(Easy): Edwin's accuracy rate when solving math problems is 15%. If he has currently solved 1000 math problems, how many problems must he get right to raise his accuracy rate to at least 50%?

2(Medium): Given that x,y are real numbers satisfying x>y>0, compute the minimum value of (5x^2 -2xy+y^2)/(x^2-y^2). [Stanford Math Tournament]

3(Hard): Prove or disprove that there are an infinite number of numbers n such that n^3 is the largest value in a primitive Pythagorean triple, where a primitive Pythagorean triple is a Pythagorean triple (a,b,c) such that gcd(a,b,c) = 1.

9/27:

1(Easy): In the middle of the night, Thomas the Tank Engine was pranked and his circular wheels were replaced with triangular ones. If he manages to maintain the same speed that he had yesterday, what is the ratio of his rpm (rotations per minute) today to his rpm yesterday?

2(Medium): Define f(n) = (n^2+n)/2 . Compute the number of positive integers n such that f(n) ≤ 1000 and f(n) is the product of two prime numbers. [BMT Discrete Round]

3(Hard): Cryo-freezing has been invented! Now, all of the scientists want to freeze themselves so they can be alive when someone invents something cool. However, they need some scientists to stay awake to invent cool things. There are n scientists named 1,2,3, ... n, and for every number 0<i<=n, scientist i invents something cool every i days. On day 0, everyone invents something cool (one of them was cryo-freezing!). Each day after that, 1 person freezes themselves (they never wake up). What is the maximum number of cool things that could have been invented after day 0?