Determine all pairs of positive integers $(m, n)$ for which there exists a bijective function \[f : \mathbb{Z}_m \times \mathbb{Z}_n \to \mathbb{Z}_m \times \mathbb{Z}_n\]such that the vectors $f(\mathbf{v}) + \mathbf{v}$, as $\mathbf{v}$ runs through all of $\mathbb{Z}_m \times \mathbb{Z}_n$, are pairwise distinct. (For any integers $a$ and $b$, the vectors $[a, b], [a + m, b]$ and $[a, b + n]$ are treated as equal.) [i]Poland, Wojciech Nadara[/i]

Pete has an instrument which can locate the midpoint of a line segment, and also the point which divides the line segment into two segments whose lengths are in a ratio of $n : (n + 1)$, where $n$ is any positive integer. Pete claims that with this instrument, he can locate the point which divides a line segment into two segments whose lengths are at any given rational ratio. Is Pete right?

Find the smallest positive integer $N$ such that each of the $101$ intervals $$[N^2, (N+1)^2), [(N+1)^2, (N+2)^2), \cdots, [(N+100)^2, (N+101)^2)$$ contains at least one multiple of $1001.$ [i]Proposed by Kyle Lee[/i]

If the numbers $\dfrac{4}{5},81\%$ and $0.801$ are arranged from smallest to largest, the correct order is $\textbf{(A)}\ \dfrac{4}{5},81\%,0.801 \qquad \textbf{(B)}\ 81\%,0.801,\dfrac{4}{5} \qquad \textbf{(C)}\ 0.801,\dfrac{4}{5},81\% \qquad \textbf{(D)}\ 81\%,\dfrac{4}{5},0.801 \qquad \textbf{(E)}\ \dfrac{4}{5},0.801,81\%$

In triangular pyramid $S-ABC$, any two of $SA,SB,SC$ are perpendicular. $M$ is the centre of gravity of $\triangle ABC$. $D$ is the midpoint of $AB$, line $DP//SC$. Prove: [b](a)[/b] $DP$ and $SM$ intersect. [b](b)[/b] $DP\cap SM=D'$, then $D'$ is the center of circumsphere of $S-ABC$.

A set $A$ contains exactly $n$ integers, each of which is greater than $1$ and every of their prime factors is less than $10$. Determine the smallest $n$ such that $A$ must contain at least two distinct elements $a$ and $b$ such that $ab$ is the square of an integer.

Let $\mathbb{N}$ and $\mathbb{N}^*$ be the sets containing the natural numbers/positive integers respectively. We define a binary relation on $\mathbb{N}$ by $a\acute{\in}b$ iff the $a$-th bit in the binary representation of $b$ is $1$. We define a binary relation on $\mathbb{N}^*$ by $a\tilde{\in}b$ iff $b$ is a multiple of the $a$-th prime number $p_a$. i) Prove that there is no bijection $f:\mathbb{N}\to \mathbb{N}^*$ such that $a\acute{\in}b\Leftrightarrow f(a)\tilde{\in}f(b)$. ii) Prove that there is a bijection $g:\mathbb{N}\to \mathbb{N}^*$ such that $(a\acute{\in}b \vee b\acute{\in}a)\Leftrightarrow (g(a)\tilde{\in}g(b) \vee g(b)\tilde{\in}g(a))$.

Given an acute triangle $ABC$. Point $H$ denotes the foot of the altitude drawn from $A$. Prove that $$AB + AC \ge BC cos \angle BAC + 2AH sin \angle BAC$$

Let $n$ be a fixed natural number. [b]a)[/b] Find all solutions to the following equation : \[ \sum_{k=1}^n [\frac x{2^k}]=x-1 \] [b]b)[/b] Find the number of solutions to the following equation ($m$ is a fixed natural) : \[ \sum_{k=1}^n [\frac x{2^k}]=x-m \]

Let $n \ge 1$ be a positive integer. An $n \times n$ matrix is called [i]good[/i] if each entry is a non-negative integer, the sum of entries in each row and each column is equal. A [i]permutation[/i] matrix is an $n \times n$ matrix consisting of $n$ ones and $n(n-1)$ zeroes such that each row and each column has exactly one non-zero entry. Prove that any [i]good[/i] matrix is a sum of finitely many [i]permutation[/i] matrices.

A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone? [asy] real r=(3+sqrt(5))/2; real s=sqrt(r); real Brad=r; real brad=1; real Fht = 2*s; import graph3; import solids; currentprojection=orthographic(1,0,.2); currentlight=(10,10,5); revolution sph=sphere((0,0,Fht/2),Fht/2); //draw(surface(sph),green+white+opacity(0.5)); //triple f(pair t) {return (t.x*cos(t.y),t.x*sin(t.y),t.x^(1/n)*sin(t.y/n));} triple f(pair t) { triple v0 = Brad*(cos(t.x),sin(t.x),0); triple v1 = brad*(cos(t.x),sin(t.x),0)+(0,0,Fht); return (v0 + t.y*(v1-v0)); } triple g(pair t) { return (t.y*cos(t.x),t.y*sin(t.x),0); } surface sback=surface(f,(3pi/4,0),(7pi/4,1),80,2); surface sfront=surface(f,(7pi/4,0),(11pi/4,1),80,2); surface base = surface(g,(0,0),(2pi,Brad),80,2); draw(sback,rgb(0,1,0)); draw(sfront,rgb(.3,1,.3)); draw(base,rgb(.4,1,.4)); draw(surface(sph),rgb(.3,1,.3)); [/asy] $ \textbf {(A) } \dfrac {3}{2} \qquad \textbf {(B) } \dfrac {1+\sqrt{5}}{2} \qquad \textbf {(C) } \sqrt{3} \qquad \textbf {(D) } 2 \qquad \textbf {(E) } \dfrac {3+\sqrt{5}}{2} $

In the sequence $ 2001, 2002, 2003, \ldots$, each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is $ 2001 \plus{} 2002 \minus{} 2003 \equal{} 2000$. What is the $ 2004^\text{th}$ term in this sequence? $ \textbf{(A)} \minus{} \! 2004 \qquad \textbf{(B)} \minus{} \! 2 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 4003 \qquad \textbf{(E)}\ 6007$

Given that $0 < a < b < c < d$, which of the following is the largest? $\textbf{(A)}\ \frac{a\plus{}b}{c\plus{}d} \qquad \textbf{(B)}\ \frac{a\plus{} d}{b\plus{} c} \qquad \textbf{(C)}\ \frac{b\plus{} c}{a\plus{}d}\qquad \textbf{(D)}\ \frac{b\plus{} d}{a\plus{} c} \qquad \textbf{(E)}\ \frac{c\plus{} d}{a\plus{}b}$

Ann and Max play a game on a $100 \times 100$ board. First, Ann writes an integer from 1 to 10 000 in each square of the board so that each number is used exactly once. Then Max chooses a square in the leftmost column and places a token on this square. He makes a number of moves in order to reach the rightmost column. In each move the token is moved to a square adjacent by side or vertex. For each visited square (including the starting one) Max pays Ann the number of coins equal to the number written in that square. Max wants to pay as little as possible, whereas Ann wants to write the numbers in such a way to maximise the amount she will receive. How much money will Max pay Ann if both players follow their best strategies? [i]Lev Shabanov[/i]

For which $n\in\{3900, 3901,\cdots, 3909\}$ can the set $\{1, 2, . . . , n\}$ be partitioned into (disjoint) triples in such a way that in each triple one of the numbers equals the sum of the other two?

Let $n$ be an arbitrary positive integer. (a) For every positive integers $a$ and $b$, show that $gcd(n^a + 1, n^b + 1) \le n^{gcd(a,b)} + 1$. (b) Show that there exist infinitely many composite pairs ($a, b)$, such that each of them is not a multiply of the other number and equality holds in (a).

Take a point $O$ inside $\Delta A B C$ such that $\angle B O C=90^{\circ}$, $\angle C A O=\angle A B O$, $\angle B A O=\angle B C O .$ Find the value of $\frac{A C}{O C}$ [list=1] [*] $\sqrt{2}$ [*] $\sqrt{\frac{3}{2}}$ [*] 2 [*] None of these [/list]

In a convex quadrilateral $ABCD$, $P$ and $Q$ are points on sides $BC$ and $DC$ such that $B\hat{A}P = D\hat{A}Q$. If the line that passes through the orthocenters of $\triangle ABP$ and $\triangle ADQ$ is perpendicular to $AC$, prove that the area of these triangles are equals.

In triangle $ABC$, points $D, E$, and $F$ are marked on sides $AC, BC$, and $AB$ respectively, so that $AD = AB$, $EC = DC$, $BF = BE$. After that, they erased everything except points $E, F$ and $D$. Reconstruct the triangle $ABC$ (no study required).

Let $\omega_1$ and $\omega_2$ be two circles intersecting at distinct points $A$ and $B.$ Point $X$ varies along $\omega_1,$ and point $Y$ is chosen on $\omega_2$ such that $AB$ bisects angle $\angle{XAY}.$ Prove that as $X$ varies along $\omega_1,$ the circumcenter of $\triangle{AXY}$ (if it exists) varies along a fixed line.

Construct a set $S$ of polynomials inductively by the rules: (i) $x\in S$; (ii) if $f(x)\in S$, then $xf(x)\in S$ and $x+(1-x)f(x)\in S$. Prove that there are no two distinct polynomials in $S$ whose graphs intersect within the region $\{0 < x < 1\}$.

Let $S$ be the incenter of triangle $ABC$. $A_1, B_1, C_1$ are the intersections of $AS, BS, CS$ with the circumcircle of triangle $ABC$ respectively. Prove that $SA_1 + SB_1 + SC_1 \geq SA + SB + SC.$

Suppose that $a, b, c, A, B, C$ are real numbers, $a \not= 0$ and $A \not= 0$, such that \[|ax^2+ bx + c| \le |Ax^2+ Bx + C|\] for all real numbers $x$. Show that \[|b^2- 4ac| \le |B^2- 4AC|\]

Rectangle $ABCD$ has $AB=6$ and $BC=3$. Point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$. What is the degree measure of $\angle AMD$? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75 $

$ABCD$ is inscribed. Bisector of angle between diagonals intersect $AB$ anc $CD$ at $X$ and $Y$. $M,N$ are midpoints of $AD,BC$. $XM=YM$ Prove, that $XN=YN$.