Alice and Bob play the following game on a $100\times 100$ grid, taking turns, with Alice starting first. Initially the grid is empty. At their turn, they choose an integer from $1$ to $100^2$ that is not written yet in any of the cells and choose an empty cell, and place it in the chosen cell. When there is no empty cell left, Alice computes the sum of the numbers in each row, and her score is the maximum of these $100$ numbers. Bob computes the sum of the numbers in each column, and his score is the maximum of these $100$ numbers. Alice wins if her score is greater than Bob's score, Bob wins if his score is greater than Alice's score, otherwise no one wins. Find if one of the players has a winning strategy, and if so which player has a winning strategy. [i]Théo Lenoir, France[/i]

Let $\zeta_1, \ldots, \zeta_n$ be (not necessarily independent) random variables with normal distribution for which $E\zeta_j=0$ and $E\zeta_j^2\le 1$ for all $1\le j\le n$. Prove that $$E\left( \max_{1\le j\le n} \zeta_j \right)\le\sqrt{2\log n}$$ (translated by Miklós Maróti)

Each of the integers from 1 to 4027 has been colored either green or red. Changing the color of a number is making it red if it was green and making it green if it was red. Two positive integers $m$ and $n$ are said to be [i]cuates[/i] if either $\frac{m}{n}$ or $\frac{n}{m}$ is a prime number. A [i]step[/i] consists in choosing two numbers that are cuates and changing the color of each of them. Show it is possible to apply a sequence of steps such that every integer from 1 to 2014 is green.

p1. Let $U_n$ be a sequence of numbers that satisfy: $U_1=1$, $U_n=1+U_1U_2U_3...U_{n-1}$ for $n=2,3,...,2020$ Prove that $\frac{1}{U_1}+\frac{1}{U_2}+...+\frac{1}{U_{2019}}<2$ p2. If $a= \left \lceil \sqrt{2020+\sqrt{2020+...+\sqrt{2020}}} \right\rceil$ , $b= \left \lfloor \sqrt{1442+\sqrt{1442+...+\sqrt{1442}}} \right \rfloor$, and $c=a-b$, then determine the value of $c$. p3. Fajar will buy a pair of koi fish in the aquarium. If he randomly picks $2$ fish, then the probability that the $2$ fish are of the same sex is $1/2$. Prove that the number of koi fish in the aquarium is a perfect square. p4. A pharmacist wants to put $155$ ml of liquid into $3$ bottles. There are 3 bottle choices, namely a. Bottle A $\bullet$ Capacity: $5$ ml $\bullet$ The price of one bottle is $10,000$ Rp $\bullet$ If you buy the next bottle, you will get a $20\%$ discount, up to the $4$th purchase or if you buy $4$ bottles, get $ 1$ free bottle A b. Bottle B $\bullet$ Capacity: $8$ ml $\bullet$ The price of one bottle is $15.000$ Rp $\bullet$ If you buy $2$ : $20\%$ discount $\bullet$ If you buy $3$ : Free $ 1$ bottle of B c. Bottle C $\bullet$ Capacity : $14$ ml $\bullet$ Buy $ 1$ : $25.000$ Rp $\bullet$ Buy $2$ : Free $ 1$ bottle of A $\bullet$ Buy $3$ : Free $ 1$ bottle of B If in one purchase, you can only buy a maximum of $4$ bottles, then look for the possibility of pharmacists putting them in bottles so that the cost is minimal (bottles do not have to be filled to capacity). p5. Two circles, let's say $L_1$ and $L_2$ have the same center, namely at point $O$. Radius of $L_1$ is $10$ cm and radius of $L_2$ is $5$ cm. The points $A, B, C, D, E, F$ lie on $L_1$ so the arcs $AB,BC,CD,DE,EF,FA$ are equal. The points $P, Q, R$ lie on $L_2$ so that the arcs $PQ,QR,RS$ are equal and $PA=PF=QB=QC=RD=RD$ . Determine the area of ​​the shaded region. [img]https://cdn.artofproblemsolving.com/attachments/b/5/0729eca97488ddfc82ab10eda02c708fecd7ae.png[/img]

Let $a, b, c$ be distinct real numbers and $x$ be a real number. Given that three numbers among $ax^2+bx+c, ax^2+cx+b, bx^2+cx+a, bx^2+ax+c, cx^2+ax+b, cx^2+bx+a$ coincide, prove that $x=1$.

Let $a,b,c,d$ be positive real numbers with $abcd=1$. Prove that $$\sqrt{\frac{a}{b+c+d^2+a^3}}+\sqrt{\frac{b}{c+d+a^2+b^3}}+\sqrt{\frac{c}{d+a+b^2+c^3}}+\sqrt{\frac{d}{a+b+c^2+d^3}} \leq 2$$

10. The sum $\sum_{k=1}^{2020} k \cos \left(\frac{4 k \pi}{4041}\right)$ can be written in the form $$ \frac{a \cos \left(\frac{p \pi}{q}\right)-b}{c \sin ^{2}\left(\frac{p \pi}{q}\right)} $$ where $a, b, c$ are relatively prime positive integers and $p, q$ are relatively prime positive integers where $p<q$. Determine $a+b+c+p+q$.

In triangle $ABC$, $D$ is the midpoint of $BC$, $E$ is the foot of the perpendicular from $A$ to $BC$, and $F$ is the foot of the perpendicular from $D$ to $AC$. Given that $BE=5$, $EC=9$, and the area of triangle $ABC$ is $84$, compute $|EF|$.

What is the smallest positive odd integer $n$ such that the product \[2^{1/7}2^{3/7}\cdots2^{(2n+1)/7}\] is greater than $1000$? (In the product the denominators of the exponents are all sevens, and the numerators are the successive odd integers from $1$ to $2n+1$.) $\textbf{(A) }7\qquad\textbf{(B) }9\qquad\textbf{(C) }11\qquad\textbf{(D) }17\qquad \textbf{(E) }19$

Suppose a class contains $100$ students. Let, for $1\le i\le 100$, the $i^{\text{th}}$ student have $a_i$ many friends. For $0\le j\le 99$ let us define $c_j$ to be the number of students who have strictly more than $j$ friends. Show that \begin{align*} & \sum_{i=1}^{100}a_i=\sum_{j=0}^{99}c_j \end{align*}

The sides of a pool table are $3$ and $4$ meters long.We push a ball with an angle of $45^o$ at the sides. Is it true that it returns to where it started no matter where we started it from?

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Gamma$. A line through $H$ intersects segments $AB$ and $AC$ at $E$ and $F$, respectively. Let $K$ be the circumcenter of $\triangle AEF$, and suppose line $AK$ intersects $\Gamma$ again at a point $D$. Prove that line $HK$ and the line through $D$ perpendicular to $\overline{BC}$ meet on $\Gamma$. [i]Gunmay Handa[/i]

Find all integers $c$ such that the equation $(2a+b) (2b+a) =5^c$ has integer solutions.

Bread draws a circle. He then selects four random distinct points on the circumference of the circle to form a convex quadrilateral. Kwu comes by and randomly chooses another 3 distinct points (none of which are the same as Bread's four points) on the circle to form a triangle. Find the probability that Kwu's triangle does not intersect Bread's quadrilateral, where two polygons intersect if they have at least one pair of sides intersecting. [i]Proposed by Nathan Cho[/i]

Let $a_1$, $a_2$, $\ldots$, $a_n$ be a sequence of integers with values between 2 and 1995 such that: (i) Any two of the $a_i$'s are relatively prime, (ii) Each $a_i$ is either a prime or a product of primes. Determine the smallest possible values of $n$ to make sure that the sequence will contain a prime number.

Real number $C > 1$ is given. Sequence of positive real numbers $a_1, a_2, a_3, \ldots$, in which $a_1=1$ and $a_2=2$, satisfy the conditions \[a_{mn}=a_ma_n, \] \[a_{m+n} \leq C(a_m + a_n),\] for $m, n = 1, 2, 3, \ldots$. Prove that $a_n = n$ for $n=1, 2, 3, \ldots$.

Find all sets of three primes $p, q$, and $r$ such that $p + q = r$ and $(r -p)(q - p) - 27p$ is a perfect square.

An urn contains one red ball and one blue ball. A box of extra red and blue balls lie nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color? $\textbf{(A) } \frac16 \qquad \textbf{(B) }\frac15 \qquad \textbf{(C) } \frac14 \qquad \textbf{(D) } \frac13 \qquad \textbf{(E) } \frac12$

Let $O$ be the center of insphere of a tetrahedron $ABCD$. The sum of areas of faces $ABC$ and $ABD$ equals the sum of areas of faces $CDA$ and $CDB$. Prove that $O$ and midpoints of $BC, AD, AC$ and $BD$ belong to the same plane.

For every positive integer $m$ prove the inquality $|\{\sqrt{m}\} - \frac{1}{2}| \geq \frac{1}{8(\sqrt m+1)} $ (The integer part $[x]$ of the number $x$ is the largest integer not exceeding $x$. The fractional part of the number $x$ is a number $\{x\}$ such that $[x]+\{x\}=x$.) A. Golovanov

Given $10$ points arranged in a equilateral triangular grid of side length $4$, how many ways are there to choose two distinct line segments, with endpoints on the grid, that intersect in exactly one point (not necessarily on the grid)?

There are $2n$ consecutive integers on a board. It is permitted to split them into pairs and simultaneously replace each pair by their difference (not necessarily positive) and their sum. Prove that it is impossible to obtain any $2n$ consecutive integers again. Alexandr Gribalko

Points $ K$, $ L$, $ M$, and $ N$ lie in the plane of the square $ ABCD$ so that $ AKB$, $ BLC$, $ CMD$, and $ DNA$ are equilateral triangles. If $ ABCD$ has an area of $ 16$, find the area of $ KLMN$. [asy]unitsize(2cm); defaultpen(fontsize(8)+linewidth(0.8)); pair A=(-0.5,0.5), B=(0.5,0.5), C=(0.5,-0.5), D=(-0.5,-0.5); pair K=(0,1.366), L=(1.366,0), M=(0,-1.366), N=(-1.366,0); draw(A--N--K--A--B--K--L--B--C--L--M--C--D--M--N--D--A); label("$A$",A,SE); label("$B$",B,SW); label("$C$",C,NW); label("$D$",D,NE); label("$K$",K,NNW); label("$L$",L,E); label("$M$",M,S); label("$N$",N,W);[/asy] $ \textbf{(A)}\ 32 \qquad \textbf{(B)}\ 16 \plus{} 16\sqrt {3} \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 32 \plus{} 16\sqrt {3} \qquad \textbf{(E)}\ 64$

Let $c \ge 1$ be a real number. Let $G$ be an Abelian group and let $A \subset G$ be a finite set satisfying $|A+A| \le c|A|$, where $X+Y:= \{x+y| x \in X, y \in Y\}$ and $|Z|$ denotes the cardinality of $Z$. Prove that \[|\underbrace{A+A+\dots+A}_k| \le c^k |A|\] for every positive integer $k$. [i]Proposed by Przemyslaw Mazur, Jagiellonian University.[/i]

Let $RANDOM$ be a regular hexagon with side length $1.$ Points $I$ and $T$ lie on segments $\overline{RA}$ and $\overline{DO},$ respectively, such that $MI=MT$ and $\angle{TMI}=90^\circ.$ Compute the area of triangle $MIT.$