Let $p$ be an odd prime. Show that there is at most one non-degenerate integer triangle with perimeter $4p$ and integer area. Characterize those primes for which such triangle exist.

Two distinct points $A$ and $B$ are chosen at random from 15 points equally spaced around a circle centered at $O$ such that each pair of points $A$ and $B$ has the same probability of being chosen. The probability that the perpendicular bisectors of $OA$ and $OB$ intersect strictly inside the circle can be expressed in the form $\frac{m}{n}$, where $m,n$ are relatively prime positive integers. Find $m+n$. [i]Ray Li.[/i]

For positive integers $n$, let $C(n)$ be the number of representation of $n$ as a sum of nonincreasing powers of $2$, where no power can be used more than three times. For example, $C(8)=5$ since the representations of $8$ are: $$8,4+4,4+2+2,4+2+1+1,\text{ and }2+2+2+1+1.$$Prove or disprove that there is a polynomial $P(x)$ such that $C(n)=\lfloor P(n)\rfloor$ for all positive integers $n$.

Find all pairs $(m,n)$ of integers, $m ,n \ge 2$ such that $mn - 1$ divides $n^3 - 1$.

Real numbers $x$ and $y$ satisfy the system of equations $$x^3 + 3x^2 = -3y - 1$$ $$y^3 + 3y^2 = -3x - 1.$$ What is the greatest possible value of $x$?

Find the number of pairs $(n,C)$ of positive integers such that $C\leq 100$ and $n^2+n+C$ is a perfect square.

A line parallel to the side $AC$ of a triangle $ABC$ with $\angle C = 90$ intersects side $AB$ at $M$ and side $BC$ at $N$, so that $CN/BN = AC/BC = 2/1$. The segments $CM$ and $AN$ meet at $O$. Let $K$ be a point on the segment $ON$ such that $MO+OK = KN$. The bisector of $\angle ABC$ meets the line through $K$ perpendicular to $AN$ at point $T$. Determine $\angle MTB$.

What is the minimum possible perimeter of a triangle two of whose sides are along the x- and y-axes and such that the third contains the point $(1,2)$?

Set $S=\{1,2,...,2004\}$. We denote by $d_1$ the number of subset of $S$ such that the sum of elements of the subset has remainder $7$ when divided by $32$. We denote by $d_2$ the number of subset of $S$ such that the sum of elements of the subset has remainder $14$ when divided by $16$. Compute $\frac{d_1}{d_2}$.

Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product$$n = f_1\cdot f_2\cdots f_k,$$where $k\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number $6$ can be written as $6$, $2\cdot 3$, and $3\cdot2$, so $D(6) = 3$. What is $D(96)$? $\textbf{(A) } 112 \qquad\textbf{(B) } 128 \qquad\textbf{(C) } 144 \qquad\textbf{(D) } 172 \qquad\textbf{(E) } 184$

The graph of the equation $y = ax^2 + bx + c$ is shown in the diagram. Which of the following must be positive? [DIAGRAM NEEDED] $ \mathrm{(A) \ } a \qquad \mathrm{(B) \ } ab^2 \qquad \mathrm {(C) \ } b - c \qquad \mathrm{(D) \ } bc \qquad \mathrm{(E) \ } c - a$

Find all pairs of distinct primes $(p,q)$ such that $p$ and $q$ are both prime factors of $p^3+q^2+1$, and are the only such prime factors. [i]Proposed by Takeda Shigenori[/i]

Andryusha has $100$ stones of different weight and he can distinguish the stones by appearance, but does not know their weight. Every evening, Andryusha can put exactly $10$ stones on the table and at night the brownie will order them in increasing weight. But, if the drum also lives in the house then surely he will in the morning change the places of some $2$ stones.Andryusha knows all about this but does not know if there is a drum in his house. Can he find out?

Let $S=\{1,2,3,4,5,6\}.$ Find the number of bijective functions $f:S\rightarrow S$ for which there exist exactly $6$ bijective functions $g:S\rightarrow S$ such that $f(g(x))=g(f(x))$ for all $x\in S$. [i]Proposed by Nathan Ramesh

Find all functions $f$ from reals to reals which satisfy $f (f(x) + f(y)) = f(x^2) + 2x^2 f(y) + (f(y))^2$ for all real numbers $x$ and $y$.

Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.

Let $n$ be a positive integer. Suppose that we have disks of radii $1, 2, . . . , n.$ Of each size there are two disks: a transparent one and an opaque one. In every disk there is a small hole in the centre, with which we can stack the disks using a vertical stick. We want to make stacks of disks that satisfy the following conditions: $i)$ Of each size exactly one disk lies in the stack. $ii)$ If we look at the stack from directly above, we can see the edges of all of the $n$ disks in the stack. (So if there is an opaque disk in the stack,no smaller disks may lie beneath it.) Determine the number of distinct stacks of disks satisfying these conditions. (Two stacks are distinct if they do not use the same set of disks, or, if they do use the same set of disks and the orders in which the disks occur are different.)

There is the tournament for boys and girls. Every person played exactly one match with every other person, there were no draws. It turned out that every person had lost at least one game. Furthermore every boy lost different number of matches that every other boy. Prove that there is a girl, who won a match with at least one boy.

Let $AH_A, BH_B, CH_C$ be the altitudes of triangle $ABC$. Prove that if $\frac{H_BC}{AC} = \frac{H_CA}{AB}$, then the line symmetric to $BC$ with respect to line $H_BH_C$ is tangent to the circumscribed circle of triangle $H_BH_CA$. [i](Proposed by Mykhailo Bondarenko)[/i]

Let $a$ and $b$ be integers (not necessarily positive). Prove that $a^3+5b^3 \neq 2016$.

For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an $L$-shape formed by three connected unit squares. For which values of $n$ is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?

Let $r_n$ be the $n$th smallest positive solution to $\tan x=x$, where the argument of tangent is in radians. Prove that \[ 0<r_{n+1}-r_n-\pi<\frac{1}{(n^2+n)\pi} \] for $n\geq 1$.

For all positive integers $a > 1$, there are divisors of $2021a$ that are not divisors of $2021$. If there are twelve unshared divisors, including $2021a$, which of the following answer choices could be a possible value of $a$? $\textbf{(A) }9\qquad\textbf{(B) }10\qquad\textbf{(C) }16\qquad\textbf{(D) }18\qquad\textbf{(E) }19$

Prove that for all positive reals $a,b,c$, \[\frac{1}{a+\frac{1}{b}+1}+\frac{1}{b+\frac{1}{c}+1}+\frac{1}{c+\frac{1}{a}+1}\ge \frac{3}{\sqrt[3]{abc}+\frac{1}{\sqrt[3]{abc}}+1}. \][i]Proposed by David Stoner[/i]

Let $z$ be a complex number that satisfies the equation \[\frac{z-4}{z^2-5z+1} + \frac{2z-4}{2z^2-5z+1} + \frac{z-2}{z^2-3z+1} = \frac{3}{z}.\] Over all possible values of $z$, find the sum of the values of \[\left| \frac{1}{z^2-5z+1} + \frac{1}{2z^2-5z+1} + \frac{1}{z^2-3z+1} \right|.\] [i]Proposed by Justin Hsieh[/i]