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? $\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{3}{5}\qquad\textbf{(C) }\frac{2}{3}\qquad\textbf{(D) }\frac{3}{4}\qquad\textbf{(E) }\frac{4}{5}$

In the diagram below, for each row except the bottom row, the number in each cell is determined by the sum of the two numbers beneath it. Find the sum of all cells denoted with a question mark. [asy] unitsize(2cm); path box = (-0.5,-0.2)--(-0.5,0.2)--(0.5,0.2)--(0.5,-0.2)--cycle; draw(box); label("$2$",(0,0)); draw(shift(1,0)*box); label("$?$",(1,0)); draw(shift(2,0)*box); label("$?$",(2,0)); draw(shift(3,0)*box); label("$?$",(3,0)); draw(shift(0.5,0.4)*box); label("$4$",(0.5,0.4)); draw(shift(1.5,0.4)*box); label("$?$",(1.5,0.4)); draw(shift(2.5,0.4)*box); label("$?$",(2.5,0.4)); draw(shift(1,0.8)*box); label("$5$",(1,0.8)); draw(shift(2,0.8)*box); label("$?$",(2,0.8)); draw(shift(1.5,1.2)*box); label("$9$",(1.5,1.2)); [/asy] $\textbf{(A) }6\qquad\textbf{(B) }8\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14$

The vertices $ A$ and $ B$ of an equilateral triangle $ ABC$ lie on a circle $k$ of radius $1$, and the vertex $ C$ is in the interior of the circle $ k$. A point $ D$, different from $ B$, lies on $ k$ so that $ AD\equal{}AB$. The line $ DC$ intersects $ k$ for the second time at point $ E$. Find the length of the line segment $ CE$.

The maximum number of solid regular tetrahedrons can be placed against each other so that one of their edges coincides with a given line segment in space? [hide=original wording]Hoeveel massieve regelmatige viervlakken kan men maximaal tegen mekaar plaatsen zodat ´e´en van hun ribben samenvalt met een gegeven lijnstuk in de ruimte?[/hide]

In the Cartesian coordinate system, we define a [i]Bongo-line[/i] as a sequence of integer points $\alpha = (\ldots, A_{-1}, A_0, A_1, \ldots)$ such that: - $A_iA_{i+1} = \sqrt{2}$ for every $i \in \mathbb{Z}$; - the polyline $\ldots A_{-1}A_0A_1 \ldots$ has no self-intersections. Let $\alpha = (\ldots, A_{-1}, A_0, A_1, \ldots)$ and $\beta = (\ldots, B_{-1}, B_0, B_1, \ldots)$ be two Bongo-lines such that there exists a bijection $f : \mathbb{Z} \to \mathbb{Z}$ such that $A_iA_{i+1}$ and $B_{f(i)}B_{f(i)+1}$ halve each other. Prove that all vertices of $\alpha$ and $\beta$ lie on two lines. [i]Proposed by Pavle Martinović[/i]

Let $n$ be a positive integer. Denote $M = \{(x, y)|x, y \text{ are integers }, 1 \leq x, y \leq n\}$. Define function $f$ on $M$ with the following properties: [b]a.)[/b] $f(x, y)$ takes non-negative integer value; [b] b.)[/b] $\sum^n_{y=1} f(x, y) = n - 1$ for $1 \eq x \leq n$; [b]c.)[/b] If $f(x_1, y_1)f(x2, y2) > 0$, then $(x_1 - x_2)(y_1 - y_2) \geq 0.$ Find $N(n)$, the number of functions $f$ that satisfy all the conditions. Give the explicit value of $N(4)$.

Find all pairs $(a, b)$ of real numbers with the following property: [list]Given any real numbers $c$ and $d$, if both of the equations $x^2+ax+1=c$ and $x^2+bx+1=d$ have real roots, then the equation $x^2+(a+b)x+1=cd$ has real roots.[/list]