Find all isosceles triangles that cannot be cut into three isosceles triangles with the same sides.

In a triangle $ABC$, the median $AD$ divides $\angle{BAC}$ in the ratio $1:2$. Extend $AD$ to $E$ such that $EB$ is perpendicular $AB$. Given that $BE=3,BA=4$, find the integer nearest to $BC^2$.

Michael and Andrew are playing the game Bust, which is played as follows: Michael chooses a positive integer less than or equal to $99$, and writes it on the board. Andrew then makes a move, which consists of him choosing a positive integer less than or equal to $ 8$ and increasing the integer on the board by the integer he chose. Play then alternates in this manner, with each person making exactly one move, until the integer on the board becomes greater than or equal to $100$. The person who made the last move loses. Let S be the sum of all numbers for which Michael could choose initially and win with both people playing optimally. Find S.

Prove that a necessary and sufficient for the existence of a set $ S \subset \{1,2,...,n \}$ with the property that the integers $ 0,1,...,n\minus{}1$ all have an odd number of representations in the form $ x\minus{}y, x,y \in S$, is that $ (2n\minus{}1)$ has a multiple of the form $ 2.4^k\minus{}1$ [i]L. Lovasz, J. Pelikan[/i]

Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $$f(f(x)-9f(y))=(x+3y)^2f(x-3y)$$ for all $x,y\in \mathbb{R}$.

Find the set of solutions to the equation $\log_{\lfloor x\rfloor}(x^2-1)=2$

Let $S$ be a finite set of polynomials in two variables, $x$ and $y$. For $n$ a positive integer, define $ \Omega _ { n } ( S ) $ to be the collection of all expressions $ p _ { 1 } p _ { 2 } \dots p _ { k } ,$ where $p_i \in S$ and $1\leq k \leq n$. Let $d_n(S)$ indicate the maximum number of linearly independent polynomials in $ \Omega _ { n } ( S ) $. For example, $ \Omega _ { 2 } \left( \left\{ x ^ { 2 } , y \right\} \right) = \left\{ x ^ { 2 } , y , x ^ { 2 } y , x ^ { 4 } , y ^ { 2 } \right\} $ and $d _ { 2 } \left( \left\{ x ^ { 2 } , y \right\} \right) = 5 $ (a) Find $ d _ { 2 } ( \{ 1 , x , x + 1 , y \} ) $. (b) Find a closed formula in $n$ for $ d _ { n } ( \{ 1 , x , y \} ) $. (c) Calculate the least upper bound over all such sets of $ \overline{\text{lim}} _ { n \rightarrow \infty } \frac { \log d _ { n } ( S ) } { \log n } $ ($ \overline{\text{lim}} _ { n \rightarrow \infty } a _ { n } = \lim _ { n \rightarrow \infty } ( \sup \left\{ a _ { n } , a _ { n + 1 } , \ldots \right\} $, where sup means supremum or least upper bound.)

Let $a$ and $b$ be two non-zero rational numbers such that the equation $ax^2+by^2=0$ has a non-zero solution in rational numbers . Prove that for any rational number $t$ , there is a solution of the equation $ax^2+by^2=t$.

Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations: \begin{align*} abc&=70,\\ cde&=71,\\ efg&=72. \end{align*}

Show that for $n \geq 5$, the integers $1, 2, \ldots n$ can be split into two groups so that the sum of the integers in one group equals the product of the integers in the other group.

Let $ABC$ be a triangle with $AB=72,AC=98,BC=110$, and circumcircle $\Gamma$, and let $M$ be the midpoint of arc $BC$ not containing $A$ on $\Gamma$. Let $A'$ be the reflection of $A$ over $BC$, and suppose $MB$ meets $AC$ at $D$, while $MC$ meets $AB$ at $E$. If $MA'$ meets $DE$ at $F$, find the distance from $F$ to the center of $\Gamma$. [i]Proposed by Michael Kural[/i]

The value of $$\frac{\log_35\log_25}{\log_35+\log_25}$$ can be expressed as $a\log_bc$, where $a$, $b$, and $c$ are positive integers, and $a+b$ is as small as possible. Find $a+2b+3c$.

Find all functions $f : \mathbb Z \to \mathbb Z$ such that \[f (n |m|) + f (n(|m| +2)) = 2f (n(|m| +1)) \qquad \forall m,n \in \mathbb Z.\] [b]Note.[/b] $|x|$ denotes the absolute value of the integer $x.$

A scalene acute triangle has angles whose measures (in degrees) are whole numbers. What is the smallest possible measure of one of the angles, in degrees?

In the equation $ 2x^2 \minus{} hx \plus{} 2k \equal{} 0$, the sum of the roots is $ 4$ and the product of the roots is $ \minus{}3$. Then $ h$ and $ k$ have the values, respectively: $ \textbf{(A)}\ 8\text{ and }{\minus{}6} \qquad \textbf{(B)}\ 4\text{ and }{\minus{}3}\qquad \textbf{(C)}\ {\minus{}3}\text{ and }4\qquad \textbf{(D)}\ {\minus{}3}\text{ and }8\qquad \textbf{(E)}\ 8\text{ and }{\minus{}3}$

Denote by $\mathcal M_2(\mathbb R)$ the set of all $2\times2$ matrices with real entries. Prove that: a) for every $A\in\mathcal M_2(\mathbb R)$ there exist $B,C\in\mathcal M_2(\mathbb R)$ such that $A=B^2+C^2$; b) there do not exist $B,C\in\mathcal M_2(\mathbb R)$ such that $\begin{pmatrix}0&1\\1&0\end{pmatrix}=B^2+C^2$ and $BC=CB$.

Can the 'brick wall' (infinite in all directions) drawn at the picture be made of wires of length $1, 2, 3, \dots$ (each positive integral length occurs exactly once)? (Wires can be bent but should not overlap; size of a 'brick' is $1\times 2$). [asy] unitsize(0.5 cm); for(int i = 1; i <= 9; ++i) { draw((0,i)--(10,i)); } for(int i = 0; i <= 4; ++i) { for(int j = 0; j <= 4; ++j) { draw((2*i + 1,2*j)--(2*i + 1,2*j + 1)); } } for(int i = 0; i <= 3; ++i) { for(int j = 0; j <= 4; ++j) { draw((2*i + 2,2*j + 1)--(2*i + 2,2*j + 2)); } } [/asy]

Let $ABC$ be a triangle. Prove that there is a line $\ell$ (in the plane of triangle $ABC$) such that the intersection of the interior of triangle $ABC$ and the interior of its reflection $A'B'C'$ in $\ell$ has area more than $\frac23$ the area of triangle $ABC$.

Call a $4$-digit number $\overline{a b c d}$ [i]unnoticeable [/i] if $a +c = b +d$ and $\overline{a b c d} +\overline{c d a b}$ is a multiple of $7$. Find the number of unnoticeable numbers. Note: $a$, $b$, $c$, and $d$ are nonzero distinct digits. [i]Proposed by Aditya Rao[/i]

The function $f(x, y)$ is defined for all real numbers $x, y$. It satisfies $f(x,0) = ax$ (where $a$ is a non-zero constant) and if $(c, d)$ and $(h, k)$ are distinct points such that $f(c, d) = f(h, k)$, then $f(x, y)$ is constant on the line through $(c, d)$ and $(h, k)$. Show that for any real $b$, the set of points such that $f(x, y) = b$ is a straight line and that all such lines are parallel. Show that $f(x, y) = ax + by$, for some constant $b$.

Consider an $n$-by-$n$ board of unit squares for some odd positive integer $n$. We say that a collection $C$ of identical dominoes is a [i]maximal grid-aligned configuration[/i] on the board if $C$ consists of $(n^2-1)/2$ dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: $C$ then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let $k(C)$ be the number of distinct maximal grid-aligned configurations obtainable from $C$ by repeatedly sliding dominoes. Find the maximum value of $k(C)$ as a function of $n$. [i]Proposed by Holden Mui[/i]

Find all positive integers $n$ such that there exists a monic polynomial $P(x)$ of degree $n$ with integers coefficients satisfying $$P(a)P(b)\neq P(c)$$ for all integers $a,b,c$.

Find all prime numbers $p$ and $q$ such that $p^2|q^3+1$ and $q^2|p^6-1$

There are distinct points $O, A, B, K_1, . . . , K_n, L_1, . . . , L_n$ on a plane such that no three points are collinear. The open line segments $K_1L_1, . . . , K_nL_n$ are coloured red, other points on the plane are left uncoloured. An allowed path from point $O$ to point $X$ is a polygonal chain with first and last vertices at points $O$ and $X$, containing no red points. For example, for $n = 1$, and $K_1 = (-1, 0)$, $L_1 = (1, 0)$, $O = (0,-1)$, and $X = (0,1)$, $OK_1X$ and $OL_1X$ are examples of allowed paths from $O$ to $X$, there are no shorter allowed paths. Find the least positive integer n such that it is possible that the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $A$ is closer to $B$ than to $A$, and the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $B$ is closer to $A$ than to $B$.

Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledge is $\frac{1}{2}$, independently of what has happened before. What is the probability that Larry wins the game? $\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{3}{5}\qquad\textbf{(C) }\frac{2}{3}\qquad\textbf{(D) }\frac{3}{4}\qquad\textbf{(E) }\frac{4}{5}$