Prove that there exists a sequence of positive integers $a_1,a_2,a_3, ...$ such that $a_1^2+a_2^2+...+a_n^2$ is a perfect square for all positive integers $n$.

For every positive integer $n{}$, consider the numbers $a_1=n^2-10n+23, a_2=n^2-9n+31, a_3=n^2-12n+46.$ a) Prove that $a_1+a_2+a_3$ is even. b) Find all positive integers $n$ for which $a_1, a_2$ and $a_3$ are primes.

Prove that a positive integer $A$ is a perfect square if and only if, for all positive integers $n$, at least one of the numbers $(A + 1)^2 - A, (A + 2)^2 - A, (A + 3)^2 - A,.., (A + n)^2- A$ is a multiple of $n$.

A positive integer $n$ is called "strong" if there exists a positive integer $x$ such that $x^{nx} + 1$ is divisible by $2^n$. a. Prove that $2013$ is strong. b. If $m$ is strong, determine the smallest $y$ (in terms of $m$) such that $y^{my} + 1$ is divisible by $2^m$.

Find all positive integer triples $(x,y,z)$ such that $x<y<z$, $\gcd (x,y)=6$, $\gcd (y,z)=10$, $\gcd (x,z)=8$, and lcm$(x,y,z)=2400$. Note that the problems of the TST are not arranged in difficulty (Problem 1 of day 1 was probably the most difficult!)

Find all positive integers $n$ for which $2^n + 2021n$ is a perfect square.

Show that there are no 2-tuples $ (x,y)$ of positive integers satisfying the equation $ (x+1) (x+2)\cdots (x+2014)= (y+1) (y+2)\cdots (y+4028).$

Let's call the positive integer $x$ interesting, if there exists integer $y$ such that the following equation holds: $(x + y)^y = (x - y)^x.$ Suppose we list all interesting integers in increasing order. An interesting integer is considered very interesting if it is not relatively prime with any other interesting integer preceding it. Find the second very interesting integer. [i]Note: It is assumed that the first interesting integer is not very interesting.[/i] [i]Proposed by Zurab Aghdgomelashvili, Georgia[/i]

Jenn randomly chooses a number $J$ from $1, 2, 3,\ldots, 19, 20$. Bela then randomly chooses a number $B$ from $1, 2, 3,\ldots, 19, 20$ distinct from $J$. The value of $B - J$ is at least $2$ with a probability that can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

For each positive integer $k$ denote $C(k)$ to be sum of its distinct prime divisors. For example $C(1)=0,C(2)=2,C(45)=8$. Find all positive integers $n$ for which $C(2^n+1)=C(n)$.

For given positive integer $n$ find all quartets $(x_1,x_2,x_3,x_4)$ such that $x_1^2+x_2^2+x_3^2+x_4^2=4^n$

Call a pair of integers $a$ and $b$ square makers , if $ab+1$ is a perfect square. Determine for which $n$ is it possible to divide the set $\{1,2, \dots , 2n\}$ into $n$ pairs of square makers.

[b]p1.[/b] Triangle $ABC$ has side lengths $AB = 3^2$ and $BC = 4^2$. Given that $\angle ABC$ is a right angle, determine the length of $AC$. [b]p2.[/b] Suppose $m$ and $n$ are integers such that $m^2+n^2 = 65$. Find the largest possible value of $m-n$. [b]p3.[/b] Six middle school students are sitting in a circle, facing inwards, and doing math problems. There is a stack of nine math problems. A random student picks up the stack and, beginning with himself and proceeding clockwise around the circle, gives one problem to each student in order until the pile is exhausted. Aditya falls asleep and is therefore not the student who picks up the pile, although he still receives problem(s) in turn. If every other student is equally likely to have picked up the stack of problems and Vishwesh is sitting directly to Aditya’s left, what is the probability that Vishwesh receives exactly two problems? [b]p4.[/b] Paul bakes a pizza in $15$ minutes if he places it $2$ feet from the fire. The time the pizza takes to bake is directly proportional to the distance it is from the fire and the rate at which the pizza bakes is constant whenever the distance isn’t changed. Paul puts a pizza $2$ feet from the fire at $10:30$. Later, he makes another pizza, puts it $2$ feet away from the fire, and moves the first pizza to a distance of $3$ feet away from the fire instantly. If both pizzas finish baking at the same time, at what time are they both done? [b]p5.[/b] You have $n$ coins that are each worth a distinct, positive integer amount of cents. To hitch a ride with Charon, you must pay some unspecified integer amount between $10$ and $20$ cents inclusive, and Charon wants exact change paid with exactly two coins. What is the least possible value of $n$ such that you can be certain of appeasing Charon? [b]p6.[/b] Let $a, b$, and $c$ be positive integers such that $gcd(a, b)$, $gcd(b, c)$ and $gcd(c, a)$ are all greater than $1$, but $gcd(a, b, c) = 1$. Find the minimum possible value of $a + b + c$. [b]p7.[/b] Let $ABC$ be a triangle inscribed in a circle with $AB = 7$, $AC = 9$, and $BC = 8$. Suppose $D$ is the midpoint of minor arc $BC$ and that $X$ is the intersection of $\overline{AD}$ and $\overline{BC}$. Find the length of $\overline{BX}$. [b]p8.[/b] What are the last two digits of the simplified value of $1! + 3! + 5! + · · · + 2009! + 2011!$ ? [b]p9.[/b] How many terms are in the simplified expansion of $(L + M + T)^{10}$ ? [b]p10.[/b] Ben draws a circle of radius five at the origin, and draws a circle with radius $5$ centered at $(15, 0)$. What are all possible slopes for a line tangent to both of the circles? PS. You had better use hide for answers.

Find all integer $n$ such that the equation $2x^2 + 5xy + 2y^2 = n$ has integer solution for $x$ and $y$.

Let $n$ be an even positive integer and let $a, b$ be two relatively prime positive integers. Find $a$ and $b$ such that $a + b$ is a divisor of $a^n + b^n$.

Show that infinitely many positive integers cannot be written as a sum of three fourth powers of integers.

Find all triples $(a, b, c)$ of positive integers such that $a^2 + b^2 = n\cdot lcm(a, b) + n^2$ $b^2 + c^2 = n \cdot lcm(b, c) + n^2$ $c^2 + a^2 = n \cdot lcm(c, a) + n^2$ for some positive integer $n$.

Show that there is a natural number $ n $ that satisfies the following inequalities: $$ \sqrt{3} -\frac{1}{10}<\{ n\sqrt 3\} +\{ (n+1)\sqrt 3 \} <\sqrt 3. $$

Let $c$ and $n > c$ be positive integers. Mary's teacher writes $n$ positive integers on a blackboard. Is it true that for all $n$ and $c$ Mary can always label the numbers written by the teacher by $a_1,\ldots, a_n$ in such an order that the cyclic product $(a_1-a_2)\cdot(a_2-a_3)\cdots(a_{n-1}-a_n)\cdot(a_n-a_1)$ would be congruent to either $0$ or $c$ modulo $n$?

The product of any three of these four natural numbers is a perfect square. Prove that these numbers themselves are perfect squares.

Determine all natural numbers $ \overline{ab} $ with $ a<b $ which are equal with the sum of all the natural numbers between $ a $ and $ b, $ inclusively.

A collection of $n > 2$ numbers is called [i]crowded [/i] if each of them is less than their sum divided by $n - 1$ . Let $\{a, b, c, ,...\}$ be a crowded collection of $n$ numbers whose sum equals $S$. Prove that: (a) each of the numbers is positive, (b) we always have $a + b > c$, (c) we always have $a + b \ge \frac{S}{n-1}$ . (Regina Schleifer)

Let $a_1=1,$ $a_2=2,$ $a_n=2a_{n-1}+a_{n-2},$ $n=3,4,\cdots.$ Prove that for any integer $n\geq5,$ $a_n$ has at least one prime factor $p,$ such that $p\equiv 1\pmod{4}.$

Find all prime numbers $ p $ for which $ 1 + p\cdot 2^{p} $ is a perfect square.

Prove that for any prime $p\ge5$, the number $$\sum_{0<k<\frac{2p}3}\binom pk$$is divisible by $p^2$.