In $\Delta ABC$ points $A_1$, $B_1$, and $C_1$ are the tangential points of the excircles of $ABC$ with its sides. a) Prove that $AA_1$, $BB_1$, and $CC_1$ intersect in one point $N$. b) If $AC+BC=3AB$, prove that the center of the inscribed circle of $ABC$, its tangential point with $AB$, and the point $N$ are collinear.

A number $k$ is [i]nice [/i] in base $b$ if there exists a $k$-digit number $n$ such that $n, 2n, . . . kn$ are each some cyclic shifts of the digits of $n$ in base $b$ (for example, $2$ is [i]nice [/i] in base $5$ because $2\cdot 135 = 315$). Determine all nice numbers in base $18$.

In the Cartesian plane, let $G_1$ and $G_2$ be the graphs of the quadratic functions $f_1(x) = p_1x^2 + q_1x + r_1$ and $f_2(x) = p_2x^2 + q_2x + r_2$, where $p_1 > 0 > p_2$. The graphs $G_1$ and $G_2$ cross at distinct points $A$ and $B$. The four tangents to $G_1$ and $G_2$ at $A$ and $B$ form a convex quadrilateral which has an inscribed circle. Prove that the graphs $G_1$ and $G_2$ have the same axis of symmetry.

Let $K$ denote the set $\{a, b, c, d, e\}$. $F$ is a collection of $16$ different subsets of $K$, and it is known that any three members of $F$ have at least one element in common. Show that all $16$ members of $F$ have exactly one element in common.

Let $H$ be the hyperbolic plane, $I(H)$ be the isometry group of $H$, and $O\in H$ be a fixed starting point. Determine those continuous $\sigma\colon H\rightarrow I(H)$ mappings that satisfty the following three conditions: (a) $\sigma(O)=\mathrm{id}$, and $\sigma (X)O=X$ for all $X\in H$; (b) for every $X\in H\backslash \{ O\}$ point, the $\sigma(X)$ isometry is a paracyclic shift, i.e. every member of a system of paracycles through a common infinitely far point is left invariant; (c) for any pair $P,Q\in H$ of points there exists a point $X\in H$ such that $\sigma(X)P=Q$. Prove that the $\sigma\colon H\rightarrow I(H)$ mappings satisfying the above conditions are differentiable with the exception of a point.

Positive integer $k(\ge 8)$ is given. Prove that if there exists a pair of positive integers $(x,y)$ that satisfies the conditions below, then there exists infinitely many pairs $(x,y)$. (1) $ $ $x\mid y^2-3, y\mid x^2-2$ (2) $ $ $gcd\left(3x+\frac{2(y^2-3)}{x},2y+\frac{3(x^2-2)}{y}\right)=k$ $ $

Find all positive real solutions to: \begin{eqnarray*} (x_1^2-x_3x_5)(x_2^2-x_3x_5) &\le& 0 \\ (x_2^2-x_4x_1)(x_3^2-x_4x_1) &\le& 0 \\ (x_3^2-x_5x_2)(x_4^2-x_5x_2) &\le& 0 \\ (x_4^2-x_1x_3)(x_5^2-x_1x_3) &\le & 0 \\ (x_5^2-x_2x_4)(x_1^2-x_2x_4) &\le& 0 \\ \end{eqnarray*}

Let $a, b, c$ be positive reals with $a^{2014}+b^{2014}+c^{2014}+abc=4$. Prove that \[ \frac{a^{2013}+b^{2013}-c}{c^{2013}} + \frac{b^{2013}+c^{2013}-a}{a^{2013}} + \frac{c^{2013}+a^{2013}-b}{b^{2013}} \ge a^{2012}+b^{2012}+c^{2012}. \][i]Proposed by David Stoner[/i]

Let $ABC$ be a triangle with $AB=AC$. Also, let $D\in[BC]$ be a point such that $BC>BD>DC>0$, and let $\mathcal{C}_1,\mathcal{C}_2$ be the circumcircles of the triangles $ABD$ and $ADC$ respectively. Let $BB'$ and $CC'$ be diameters in the two circles, and let $M$ be the midpoint of $B'C'$. Prove that the area of the triangle $MBC$ is constant (i.e. it does not depend on the choice of the point $D$). [i]Greece[/i]

If $a,b,c$ are positive numbers, prove the inequality $$\frac{a^2}{(a+b)(a+c)}+\frac{b^2}{(b+c)(b+a)}+\frac{c^2}{(c+a)(c+b)}\ge\frac34.$$

Given a random string of 33 bits (0 or 1), how many (they can overlap) occurrences of two consecutive 0's would you expect? (i.e. "100101" has 1 occurrence, "0001" has 2 occurrences)

There exist $4$ positive integers $a,b,c,d$ such that $abcd \neq 1$ and each pair of them have a GCD of $1$. Two functions $f,g : \mathbb{N} \rightarrow \{0,1\}$ are multiplicative functions such that for each positive integer $n$ we have : $$f(an+b)=g(cn+d)$$ Prove that at least one of the followings hold. $i)$ for each positive integer $n$ we have $f(an+b)=g(cn+d)=0$ $ii)$ There exists a positive integer $k$ such that for all $n$ where $(n,k)=1$ we have $g(n)=f(n)=1$ (Function $f$ is multiplicative if for any natural numbers $a,b$ we have $f(ab)=f(a)f(b)$) Proposed by [i]Navid Safaii[/i]

If $m$ is a positive integer, prove that $2^m$ cannot be written as a sum of two or more consecutive natural numbers.

Solve the equation $$(x^2-x-1)^2-x^3=5$$

Oleksiy wrote several distinct positive integers on the board and calculated all their pairwise sums. It turned out that all digits from $0$ to $9$ appear among the last digits of these sums. What could be the smallest number of integers that Oleksiy wrote? [i]Proposed by Oleksiy Masalitin[/i]

Let $n\in\mathbb N_+.$ For $1\leq i,j,k\leq n,a_{ijk}\in\{ -1,1\} .$ Prove that: $\exists x_1,x_2,\cdots ,x_n,y_1,y_2,\cdots ,y_n,z_1,z_2,\cdots ,z_n\in \{-1,1\} ,$ satisfy $$\left| \sum\limits_{i=1}^n\sum\limits_{j=1}^n\sum\limits_{k=1}^na_{ijk}x_iy_jz_k\right| >\frac {n^2}3.$$ [i]Created by Yu Deng[/i]

Find the minimum positive value of $ 1*2*3*4*...*2020*2021*2022$ where you can replace $*$ as $+$ or $-$

In the diagram below, $\overline{AB}$ and $\overline{CD}$ are parallel, $\angle BXY = 45^\circ$, $\angle DZY = 25^\circ$, and $XY = YZ$. What is the degree measure of $\angle YXZ$? [asy] import graph; usepackage("amsmath"); size(6cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; draw((-2,4)--(3,4)); draw((-2,2)--(3,2)); draw((0,4)--(1,3)); draw((1,3)--(-1.14,2)); label("$ A $",(-2.13,4.6),SE*labelscalefactor); label("$ B $",(2.8,4.6),SE*labelscalefactor); label("$ C $",(-2.29,1.8),SE*labelscalefactor); label("$ D $",(2.83,1.8),SE*labelscalefactor); label("$ 45^\circ $",(0.49,3.9),SE*labelscalefactor); label("$ 25^\circ $",(-0.26,2.4),SE*labelscalefactor); label("$ Y $",(1.21,3.2),SE*labelscalefactor); label("$ X $",(-0.16,4.6),SE*labelscalefactor); label("$ Z $",(-1.28,1.8),SE*labelscalefactor); dot((-2,4),dotstyle); dot((3,4),dotstyle); dot((-2,2),dotstyle); dot((3,2),dotstyle); dot((0,4),dotstyle); dot((1,3),dotstyle); dot((-1.14,2),dotstyle); [/asy]

Show that there exist strictly increasing infinite arithmetic sequence of integers which has no numbers in common with the Fibonacci sequence.

Let $ x$ and $ y$ be positive integers. The least possible value of $ |11x^5 \minus{} 7y^3|$ is A. 1 B. 2 C. 3 D. 4 E. None of these

In an acute triangle $ABC$, let $D$, $E$, $F$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $BC$, $CA$, $AB$, respectively, and let $P$, $Q$, $R$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $EF$, $FD$, $DE$, respectively. Prove that $p\left(ABC\right)p\left(PQR\right) \ge \left(p\left(DEF\right)\right)^{2}$, where $p\left(T\right)$ denotes the perimeter of triangle $T$ . [i]Proposed by Hojoo Lee, Korea[/i]

$N$ coins are placed on a table, $N - 1$ are genuine and have the same weight, and one is fake, with a different weight. Using a two pan balance, the goal is to determine with certainty the fake coin, and whether it is lighter or heavier than a genuine coin. Whenever one can deduce that one or more coins are genuine, they will be inmediately discarded and may no longer be used in subsequent weighings. Determine all $N$ for which the goal is achievable. (There are no limits regarding how many times one may use the balance). Note: the only difference between genuine and fake coins is their weight; otherwise, they are identical.

Let $a_1, a_2, ..., a_k$ be a finite arithmetic sequence with \[ a_4+a_7+a_{10}=17 \] and \[ a_4+a_5+a_6+a_7+a_8+a_9+a_{10}+a_{11}+a_{12}+a_{13}+a_{14}=77 \] If $a_k=13$, then $k=$ $ \textbf{(A)}\ 16 \qquad\textbf{(B)}\ 18 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 22 \qquad\textbf{(E)}\ 24 $

A $6\text{-year stock}$ that goes up $30\%$ in the first year, down $30\%$ in the second, up $30\%$ in the third, down $30\%$ in the fourth, up $30\%$ in the fifth, and down $30\%$ in the sixth is equivalent to a $3\text{-year stock}$ that loses $x\%$ in each of its three years. Compute $x$.

There are $2017$ turtles in a room. Every second, two turtles are chosen uniformly at random and combined to form one super-turtle. (Super-turtles are still turtles.) The probability that after $2015$ seconds (meaning when there are only two turtles remaining) there is some turtle that has never been combined with another turtle can be written in the form $\tfrac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.