## 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; however, the majority of questions will be original. 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

10/19:

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.

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.

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

**in one form submission**

## Past Problems of the Week

10/14:

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?

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?