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 $p + q$.

Let $D,E,F$ be points on the sides $BC,CA,AB$ respectively of a triangle $ABC$ such that $BD=CE=AF$ and $\angle BDF=\angle CED=\angle AFE.$ Show that $\triangle ABC$ is equilateral.

Suppose $A$, $B$, and $C$ are points in the plane with $AB=40$ and $AC=42$, and let $x$ be the length of the line segment from $A$ to the midpoint of $\overline{BC}$. Define a function $f$ by letting $f(x)$ be the area of $\triangle ABC$. Then the domain of $f$ is an open interval $(p,q)$, and the maximum value $r$ of $f(x)$ occurs at $x=s$. What is $p+q+r+s$? $ \textbf{(A) }909\qquad \textbf{(B) }910\qquad \textbf{(C) }911\qquad \textbf{(D) }912\qquad \textbf{(E) }913\qquad $

Let $ABCD$ be a convex quadrilateral and points $E$ and $F$ on sides $AB,CD$ such that \[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\] If $S$ is the area of $AEFD$ show that ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$

Let $ ABCDA’B’C’D’ $ a cube. $ M,P $ are the midpoints of $ AB, $ respectively, $ DD’. $ [b]a)[/b] Show that $ MP, A’C $ are perpendicular, but not coplanar. [b]b)[/b] Calculate the distance between the lines above.

Find all primes $p$ such that $4p^2+1$ and $6p^2+1$ are both primes.

A triangle has lattice points as vertices and contains no other lattice points. Prove that its area is $\frac{1}{2}$.

When a function $f(x)$ is differentiated $n$ times ,the function we get id denoted $f^n(x)$.If $f(x)=\dfrac {e^x}{x}$.Find the value of \[\lim_{n \to \infty} \dfrac {f^ {2n}(1)}{(2n)!}\]

Find all functions $f:\mathbb Z\to \mathbb Z$ such that for all surjective functions $g:\mathbb Z\to \mathbb Z$, $f+g$ is also surjective. (A function $g$ is surjective over $\mathbb Z$ if for all integers $y$, there exists an integer $x$ such that $g(x)=y$.) [i]Proposed by Sean Li[/i]

A chicken breeder went to check what price per chick he had charged the previous year. He found an invoice, half erased, which read: $72$ chickens sold for $*679*$ escudos” (the digits of the units and tens of thousands were illegible). What price did each chick sell for last year?

Let $a, b, c, d$ be real numbers, such that $ab(c+d)=cd(a+b)$. Prove that $\frac{a+1}{a^2+3}+\frac{b+1}{b^2+3} \geq \frac{c-1}{c^2+3}+\frac{d-1}{d^2+3}$.

Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that \[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \] Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.

Let $\overline{CH}$ be an altitude of $\triangle ABC$. Let $R$ and $S$ be the points where the circles inscribed in the triangles $ACH$ and $BCH$ are tangent to $\overline{CH}$. If $AB = 1995$, $AC = 1994$, and $BC = 1993$, then $RS$ can be expressed as $m/n$, where $m$ and $n$ are relatively prime integers. Find $m + n$

Find all $3$-digit numbers $\overline{abc}$ ($a,b \ne 0$) such that $\overline{bcd} \times  a = \overline{1a4d}$ for some integer $d$ from $1$ to $9$

A trapezium is given with parallel bases having lengths $1$ and $4$. Split it into two trapeziums by a cut, parallel to the bases, of length $3$. We now want to divide the two new trapeziums, always by means of cuts parallel to the bases, in $m$ and $n$ trapeziums, respectively, so that all the $m + n$ trapezoids obtained have the same area. Determine the minimum possible value for $m + n$ and the lengths of the cuts to be made to achieve this minimum value.