Alice and Bob are playing rock paper scissors. Alice however is cheating, so in each round, she has a $\frac35$ chance of winning, $\frac25$ chance of drawing, and $\frac25$ chance of losing. The first person to win $5$ more rounds than the other person wins the match. What is the probability Alice wins?

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

How many distinct spanning trees does the graph below have? Recall that a $\emph{spanning tree}$ of a graph $G$ is a subgraph of $G$ that is a tree and containing all the vertices of $G$. [center][img]http://i.imgur.com/NMF12pE.png[/img][/center]

Prove that there exists $c>0$ such that for any odd prime $p=2k+1$, the numbers $1^0, 2^1,3^2,\dots,k^{k-1}$ give at least $c\sqrt{p}$ distinct residues modulo $p$. [i]Proposed by M. Turevsky, I. Bogdanov[/i]

Let $ABCD$ be a parallelogram with at least an angle not equal to $90^o$ and $k$ the circumcircle of the triangle $ABC$. Let $E$ be the diametrically opposite point of $B$. Show that the circumcircle of the triangle $ADE$ and $k$ have the same radius.

Let us name a move of the chess knight horizontal if it moves two cells horizontally and one vertically, and vertical otherwise. It is required to place the knight on a cell of a ${46} \times {46}$ board and alternate horizontal and vertical moves. Prove that if each cell is visited not more than once then the number of moves does not exceed 2024. Alexandr Gribalko

Prove that for each prime $p>2$ there exists exactly one positive integer $n$, such that $n^2+np$ is a perfect square.

Find all real numbers $k$ for which the inequality $(1+t)^k (1-t)^{1-k} \leq 1$ is true for every real number $t \in (-1, 1)$.

Consider a convex $n$-gon in the plane with $n$ being odd. Prove that if one may find a point in the plane from which all the sides of the $n$-gon are viewed at equal angles, then this point is unique. (We say that segment $AB$ is viewed at angle $\gamma$ from point $O$ iff $\angle AOB =\gamma$ .)

Solve the system of equations $$\begin{cases} x^2+x+y=8\\ y^2+2xy+z=168\\ z^2+2yz+2xz=12480 \end{cases}$$

Let $ABC$ be an acute triangle, $B,C$ fixed, $A$ moves on the big arc $BC$ of $(ABC)$. Let $O$ be the circumcenter of $(ABC)$ $(B,O,C$ are not collinear, $AB \ne AC)$, $(I)$ is the incircle of triangle $ABC$. $(I)$ tangents to $BC$ at $D$. Let $I_a$ be the $A$-excenter of triangle $ABC$. $I_aD$ cuts $OI$ at $L$. Let $E$ lies on $(I)$ such that $DE \parallel AI$. a) $LE$ cuts $AI$ at $F$. Prove that $AF=AI$. b) Let $M$ lies on the circle $(J)$ go through $I_a,B,C$ such that $I_aM \parallel AD$. $MD$ cuts $(J)$ again at $N$. Prove that the midpoint $T$ of $MN$ lies on a fixed circle.

For any real numbers $a$ and $b>0$, define an [i]extension[/i] of an interval $[a-b,a+b] \subseteq \mathbb{R}$ be $[a-2b, a+2b]$. We say that $P_1, P_2, \ldots, P_k$ covers the set $X$ if $X \subseteq P_1 \cup P_2 \cup \ldots \cup P_k$. Prove that there exists an integer $M$ with the following property: for every finite subset $A \subseteq \mathbb{R}$, there exists a subset $B \subseteq A$ with at most $M$ numbers, so that for every $100$ closed intervals that covers $B$, their extensions covers $A$.

[b]p1.[/b] Evaluate $1+3+5+··· +2019$. [b]p2.[/b] Evaluate $1^2 -2^2 +3^2 -4^2 +...· +99^2 -100^2$. [b]p3. [/b]Find the sum of all solutions to $|2018+|x -2018|| = 2018$. [b]p4.[/b] The angles in a triangle form a geometric series with common ratio $\frac12$ . Find the smallest angle in the triangle. [b]p5.[/b] Compute the number of ordered pairs $(a,b,c,d)$ of positive integers $1 \le a,b,c,d \le 6$ such that $ab +cd$ is a multiple of seven. [b]p6.[/b] How many ways are there to arrange three birch trees, four maple, and five oak trees in a row if trees of the same species are considered indistinguishable. [b]p7.[/b] How many ways are there for Mr. Paul to climb a flight of 9 stairs, taking steps of either two or three at a time? [b]p8.[/b] Find the largest natural number $x$ for which $x^x$ divides $17!$ [b]p9.[/b] How many positive integers less than or equal to $2018$ have an odd number of factors? [b]p10.[/b] Square $MAIL$ and equilateral triangle $LIT$ share side $IL$ and point $T$ is on the interior of the square. What is the measure of angle $LMT$? [b]p11.[/b] The product of all divisors of $2018^3$ can be written in the form $2^a \cdot 2018^b$ for positive integers $a$ and $b$. Find $a +b$. [b]p12.[/b] Find the sum all four digit palindromes. (A number is said to be palindromic if its digits read the same forwards and backwards. [b]p13.[/b] How ways are there for an ant to travel from point $(0,0)$ to $(5,5)$ in the coordinate plane if it may only move one unit in the positive x or y directions each step, and may not pass through the point $(1, 1)$ or $(4, 4)$? [b]p14.[/b] A certain square has area $6$. A triangle is constructed such that each vertex is a point on the perimeter of the square. What is the maximum possible area of the triangle? [b]p15.[/b] Find the value of ab if positive integers $a,b$ satisfy $9a^2 -12ab +2b^2 +36b = 162$. [b]p16.[/b] $\vartriangle ABC$ is an equilateral triangle with side length $3$. Point $D$ lies on the segment $BC$ such that $BD = 1$ and $E$ lies on $AC$ such that $AE = AD$. Compute the area of $\vartriangle ADE$. [b]p17[/b]. Let $A_1, A_2,..., A_{10}$ be $10$ points evenly spaced out on a line, in that order. Points $B_1$ and $B_2$ lie on opposite sides of the perpendicular bisector of $A_1A_{10}$ and are equidistant to $l$. Lines $B_1A_1,...,B_1A_{10}$ and $B_2A_1,...· ,B_2A_{10}$ are drawn. How many triangles of any size are present? [b]p18.[/b] Let $T_n = 1+2+3··· +n$ be the $n$th triangular number. Determine the value of the infinite sum $\sum_{k\ge 1} \frac{T_k}{2^k}$. [b]p19.[/b] An infinitely large bag of coins is such that for every $0.5 < p \le 1$, there is exactly one coin in the bag with probability $p$ of landing on heads and probability $1- p$ of landing on tails. There are no other coins besides these in the bag. A coin is pulled out of the bag at random and when flipped lands on heads. Find the probability that the coin lands on heads when flipped again. [b]p20.[/b] The sequence $\{x_n\}_{n\ge 1}$ satisfies $x1 = 1$ and $(4+ x_1 + x_2 +··· + x_n)(x_1 + x_2 +··· + x_{n+1}) = 1$ for all $n \ge 1$. Compute $\left \lfloor \frac{x_{2018}}{x_{2019}} \right \rfloor$. PS. You had better use hide for answers.

Consider the sets $M = \{0,1,2,, 2019\}$ and $$A=\left\{ x\in M\,\, | \frac{x^3-x}{24} \in N\right\} $$ a) How many elements does the set $A$ have? b) Determine the smallest natural number $n$, $n \ge 2$, which has the property that any $n$-element subset of the set $A $contains two distinct elements whose difference is divisible by $40$.

In a cube $ABCDA'B'C'D' $two points are considered, $M \in (CD')$ and $N \in (DA')$. Show that the $MN$ is common perpendicular to the lines $CD'$ and $DA'$ if and only if $$\frac{D'M}{D'C}=\frac{DN}{DA'} =\frac{1}{3}.$$

Different points $A$ and $D$ are on the same side of the line $BC$, with $|AB| = | BC|= |CD|$ and lines $AD$ and $BC$ are perpendicular. Let $E$ be the intersection point of lines $AD$ and $BC$. Prove that $||BE| - |CE|| < |AD| \sqrt3$

Let $M$ be the set of the integer numbers from the range $[-n, n]$. The subset $P$ of $M$ is called a [i]base subset[/i] if every number from $M$ can be expressed as a sum of some different numbers from $P$. Find the smallest natural number $k$ such that every $k$ numbers that belongs to $M$ form a base subset.

In a social network with a fixed finite setback of users, each user had a fixed set of [i]followers[/i] among the other users. Each user has an initial positive integer rating (not necessarily the same for all users). Every midnight, the rating of every user increases by the sum of the ratings that his followers had just before midnight. Let $m$ be a positive integer. A hacker, who is not a user of the social network, wants all the users to have ratings divisible by $m$. Every day, he can either choose a user and increase his rating by 1, or do nothing. Prove that the hacker can achieve his goal after some number of days. [i]Vladislav Novikov[/i]

On a unit circle with center $O$, $AB$ is an arc with the central angle $\alpha<90^\circ$. Point $H$ is the foot of the perpendicular from $A$ to $OB$, $T$ is a point on arc $AB$, and $l$ is the tangent to the circle at $T$. The line $l$ and the angle $AHB$ form a triangle $\Delta$. (a) Prove that the area of $\Delta$ is minimal when $T$ is the midpoint of arc $AB$. (b) Prove that if $S_\alpha$ is the minimal area of $\Delta$ then the function $\frac{S_\alpha}\alpha$ has a limit when $\alpha\to0$ and find this limit.

The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ meet at $O$. Let $m$ be the measure of the acute angle formed by these diagonals. A variable angle $xOy$ of measure $m$ intersects the quadrilateral by a convex quadrilateral of constant area. Prove that $ABCD$ is a square.

Jeff writes down the two-digit base-$10$ prime $\overline{ab_{10}}$. He realizes that if he misinterprets the number as the base $11$ number $\overline{ab_{11}}$ or the base $12$ number $\overline{ab_{12}}$, it is still a prime. What is the least possible value of Jeff’s number (in base $10$)? [i]Proposed byMuztaba Syed[/i]

Show that for each positive integer n, \[n!=\prod_{i=1}^n \; \text{lcm} \; \{1, 2, \ldots, \left\lfloor\frac{n}{i} \right\rfloor\}\] (Here lcm denotes the least common multiple, and $\lfloor x\rfloor$ denotes the greatest integer $\le x$.)

Rectangle $ABCD$ is given with $AB=63$ and $BC=448.$ Points $E$ and $F$ lie on $AD$ and $BC$ respectively, such that $AE=CF=84.$ The inscribed circle of triangle $BEF$ is tangent to $EF$ at point $P,$ and the inscribed circle of triangle $DEF$ is tangent to $EF$ at point $Q.$ Find $PQ.$

How many positive integers $n$ satisfy the inequality \[ \left\lceil \frac{n}{101} \right\rceil + 1 > \frac{n}{100} \, ? \] Recall that $\lceil a \rceil$ is the least integer that is greater than or equal to $a$.

Given the set $S$ whose elements are zero and the even integers, positive and negative. Of the five operations applied to any pair of elements: (1) addition (2) subtraction (3) multiplication (4) division (5) finding the arithmetic mean (average), those elements that only yield elements of $S$ are: $ \textbf{(A)}\ \text{all} \qquad\textbf{(B)}\ 1,2,3,4\qquad\textbf{(C)}\ 1,2,3,5\qquad\textbf{(D)}\ 1,2,3\qquad\textbf{(E)}\ 1,3,5 $