Let $ABC$ be a triangle with $AB < AC$. A point $P$ on the circumcircle of $ABC$ (on the same side of $BC$ as $A$) is chosen in such a way that $BP = CP$. Let $BP$ and the angle bisector of $\angle BAC$ intersect at $Q$, and let the line through $Q$ and parallel to $BC$ intersect $AC$ at $R$. Prove that $BR = CR$.

Find all functions $f: \mathbb{N} \mapsto \mathbb{N}$ so that for any positive integer $n$ and finite sequence of positive integers $a_0, \dots, a_n$, whenever the polynomial $a_0+a_1x+\dots+a_nx^n$ has at least one integer root, so does \[f(a_0)+f(a_1)x+\dots+f(a_n)x^n.\] [i]Proposed by Sutanay Bhattacharya[/i]

Let $ABC$ be a triangle with $\angle A= 105^\circ$ and $\angle B= 45^\circ$. Let $L$ be a point on side $BC$ such that $AL$ is the bisector of angle $\angle BAC$ and let $M$ be the midpoint of side $AC$. Suppose that lines $AL$ and $BM$ intersect at point $P$. Calculate the ratio $\dfrac{AP}{AL}$.

Prove that for any positive integers $ a$ and $ b$ \[ \left\vert a\sqrt{2}\minus{}b\right\vert >\frac{1}{2(a\plus{}b)}.\]

Let $d,p,q$ be fixed positive integers, and $d$ is not a perfect square. $\mathbb{N}$ is the set of all positive integers, and $S=\{m+n\sqrt{d}|m,n \in \mathbb{N}\} \cup \{0\}$. Suppose the function $f: S \to S$ satisfies the following conditions for all $x,y \in S$: (i) $f((xy)^p)=(f(x)f(y))^p$ (ii)$f((x+y)^q)=(f(x)+f(y))^q$ Find the function $f$.

Let $ S$ be the smallest subset of the integers with the property that $ 0\in S$ and for any $ x\in S$, we have $ 3x\in S$ and $ 3x \plus{} 1\in S$. Determine the number of non-negative integers in $ S$ less than $ 2008$.

A rectangular field consists of $1520$ unit squares. How many rectangles $6\times1$ at most can be cut out from this field?

A bus ticket is considered to be lucky if the sum of the first three digits equals to the sum of the last three ($6$ digits in Russian buses). Prove that the sum of all the lucky numbers is divisible by $13$.

Let $ABC$ be an equilateral triangle and $D$ a point on side $AC$. Let $E$ be a point on $BC$ such that $DE \perp BC$, $F$ on $AB$ such that $EF \perp AB$, and $G$ on $AC$ such that $FG \perp AC$. Lines $FG$ and $DE$ intersect in $P$. If $M$ is the midpoint of $BC$, show that $BP$ bisects $AM$.

$\definecolor{A}{RGB}{255,0,0}\color{A}\fbox{A6.}$ Let $ P (x)$ be a polynomial with real coefficients such that $\deg P \ge 3$ is an odd integer. Let $f : \mathbb{R}\rightarrow\mathbb{Z}$ be a function such that $$\definecolor{A}{RGB}{0,0,200}\color{A}\forall_{x\in\mathbb{R}}\ f(P(x)) = P(f(x)).$$ $\definecolor{A}{RGB}{255,150,0}\color{A}\fbox{(a)}$ Prove that the range of $f$ is finite. $\definecolor{A}{RGB}{255,150,0}\color{A}\fbox{(b)}$ Show that for any positive integer $n$, there exist $P$, $f$ that satisfies the above condition and also that the range of $f$ has cardinality $n$. [i]Proposed by [/i][b][color=#419DAB]ltf0501[/color][/b]. [color=#3D9186]#1735[/color]

The points $(0, 0)$, $(a, 5)$, and $(b, 11)$ are the vertices of an equilateral triangle. Find $ab$.

Let $a_1, a_2, a_3,\ldots$ and $b_1, b_2, b_3,\ldots$ be infinite increasing arithmetic progressions. Their terms are positive numbers. It is known that the ratio $a_k/b_k$ is an integer for all k. Is it true that this ratio does not depend on $k{}$? [i]Boris Frenkin[/i]

Let $n$ be a positive integer. Find the number of sequences $a_1,a_2,...,a_k$ of different numbers from $\{ 1,2,3,...,n\}$ with the following property: for every number $a$ of the sequence (except the first one) there exists a previous number $b$ such that their difference is $1$ (so $a-b= \pm 1$)

The corners of a fixed convex (but not necessarily regular) $n$-gon are labeled with distinct letters. If an observer stands at a point in the plane of the polygon, but outside the polygon, they see the letters in some order from left to right, and they spell a "word" (that is, a string of letters; it doesn't need to be a word in any language). For example, in the diagram below (where $n=4$), an observer at point $X$ would read "$BAMO$," while an observer at point $Y$ would read "$MOAB$." [center]Diagram to be added soon[/center] Determine, as a formula in terms of $n$, the maximum number of distinct $n$-letter words which may be read in this manner from a single $n$-gon. Do not count words in which some letter is missing because it is directly behind another letter from the viewer's position.

Let $1<t<2$ be a real number. Prove that for all sufficiently large positive integers like $d$, there is a monic polynomial $P(x)$ of degree $d$, such that all of its coefficients are either $+1$ or $-1$ and $$\left|P(t)-2019\right| <1.$$ [i]Proposed by Navid Safaei[/i]

An $n\times n$ matrix $A$ with integer entries is called [i]representative[/i] if, for any integer vector $\mathbf{v}$, there is a finite sequence $0=\mathbf{v}_0,\mathbf{v}_1,\dots,\mathbf{v}_{\ell}=\mathbf{v}$ of integer vectors such that for each $0\leq i <\ell$, either $\mathbf{v}_{i+1}=A\mathbf{v}_{i}$ or $\mathbf{v}_{i+1}-\mathbf{v}_i$ is an element of the standard basis (i.e. one of its entries is $1$, the rest are all equal to $0$). Show that $A$ is not representative if and only if $A^T$ has a real eigenvector with all non-negative entries and non-negative eigenvalue.

For $ n \in \mathbb{N}$, let $s(n)$ denote the sum of all positive divisors of $n$. Show that for any $n > 1$, the product $s(n - 1)s(n)s(n + 1)$ is an even number.

Two congruent triangles are inscribed in an ellipse. Are they necessarily symmetric with respect to an axis or the center of the ellipse?

How many subsets of the set $\{1, 2, 3, \ldots, 12\}$ contain exactly one or two prime numbers?

The incircle of triangle $ABC$ meet the sides $AB, AC$ and $BC$ in $M,N$ and $P$, respectively. Prove that the orthocenter of triangle $MNP,$ the incenter and the circumcenter of triangle $ABC$ are collinear. [asy] import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ttttff = rgb(0.2,0.2,1); pen ffwwww = rgb(1,0.4,0.4); pen xdxdff = rgb(0.49,0.49,1); draw((8,17.58)--(2.84,9.26)--(20.44,9.21)--cycle); draw((8,17.58)--(2.84,9.26),ttttff+linewidth(2pt)); draw((2.84,9.26)--(20.44,9.21),ttttff+linewidth(2pt)); draw((20.44,9.21)--(8,17.58),ttttff+linewidth(2pt)); draw(circle((9.04,12.66),3.43),blue+linewidth(1.2pt)+linetype("8pt 8pt")); draw((6.04,14.42)--(8.94,9.24),ffwwww+linewidth(1.2pt)); draw((8.94,9.24)--(11.12,15.48),ffwwww+linewidth(1.2pt)); draw((11.12,15.48)--(6.04,14.42),ffwwww+linewidth(1.2pt)); draw((8.94,9.24)--(7.81,14.79)); draw((11.12,15.48)--(6.95,12.79)); draw((6.04,14.42)--(10.12,12.6)); dot((8,17.58),ds); label("$A$", (8.11,18.05),NE*lsf); dot((2.84,9.26),ds); label("$B$", (2.11,8.85), NE*lsf); dot((20.44,9.21),ds); label("$C$", (20.56,8.52), NE*lsf); dot((9.04,12.66),ds); label("$O$", (8.94,12.13), NE*lsf); dot((6.04,14.42),ds); label("$M$", (5.32,14.52), NE*lsf); dot((11.12,15.48),ds); label("$N$", (11.4,15.9), NE*lsf); dot((8.94,9.24),ds); label("$P$", (8.91,8.58), NE*lsf); dot((7.81,14.79),ds); label("$D$", (7.81,15.14),NE*lsf); dot((6.95,12.79),ds); label("$F$", (6.64,12.07),NE*lsf); dot((10.12,12.6),ds); label("$G$", (10.41,12.35),NE*lsf); dot((8.07,13.52),ds); label("$H$", (8.11,13.88),NE*lsf); clip((-0.68,-0.96)--(-0.68,25.47)--(30.71,25.47)--(30.71,-0.96)--cycle); [/asy]

We have a deck of $90$ cards that are numbered from $10$ to $99$ (all two-digit numbers). How many sets of three or more different cards in this deck are there such that the number on one of them is the sum of the other numbers, and those other numbers are consecutive?

Let $n\geqslant2$ be a natural number. Show that there is no real number $c{}$ for which \[\exp\left(\frac{T+S}{2}\right)\leqslant c\cdot \frac{\exp(T)+\exp(S)}{2}\]is satisfied for any self-adjoint $n\times n$ complex matrices $T{}$ and $S{}$. (If $A{}$ and $B{}$ are self-adjoint $n\times n$ matrices, $A\leqslant B$ means that $B-A$ is positive semi-definite.)

A rectangular parallelepiped painted blue is cut into $1 \times 1\times 1$ cubes. Find the possible dimensions if the number of cubes without blue faces is equal to one-third of the total number of cubes. [b]Note:[/b] A [i]rectangular parallelepiped[/i] is a solid with $6$ faces, all of which are rectangles (or squares).

Medians $AA_1, BB_1, CC_1$ and altitudes $AA_2, BB_2, CC_2$ are drawn in triangle $ABC$ . Prove that the length of the broken line $A_1B_2C_1A_2B_1C_2A_1$ is equal to the perimeter of triangle $ABC$.

Let $m,n$ be positive integers with $m \geq n$, and let $S$ be the set of all $n$-term sequences of positive integers $(a_1, a_2, \ldots a_n)$ such that $a_1 + a_2 + \cdots + a_n = m$. Show that \[\sum_S 1^{a_1} 2^{a_2} \cdots n^{a_n} = {n \choose n} n^m - {n \choose n-1} (n-1)^m + \cdots + (-1)^{n-2} {n \choose 2} 2^m + (-1)^{n-1} {n \choose 1}.\]