In each square of an $11\times 11$ board, we are to write one of the numbers $-1$, $0$, or $1$ in such a way that the sum of the numbers in each column is nonnegative and the sum of the numbers in each row is nonpositive. What is the smallest number of zeros that can be written on the board? Justify your answer.

A jacket was originally priced $\textdollar 100$. The price was reduced by $10\%$ three times and increased by $10\%$ four times in some order. To the nearest cent, what was the final price?

For $n \ge 1$ call a finite sequence $(a_1, a_2 \ldots a_n)$ of positive integers [i]progressive[/i] if $a_i < a_{i+1}$ and $a_i$ divides $a_{i+1}$ for all $1 \le i \le n-1$. Find the number of progressive sequences such that the sum of the terms in the sequence is equal to $360$.

Find all the real $a,b,c$ such that the equality $$|ax+by+cz| + |bx+cy+az| + |cx+ay+bz| = |x|+|y|+|z|$$ is valid for all the real $x,y,z$.

Let $x=\sqrt{37-20\sqrt{3}}$. Find the value of $\frac{x^4-9x^3+5x^2-7x+68}{x^2-10x+19}$

Define $ \phi$ the positive real root of $ x^2 \minus{} x \minus{} 1$ and let $ a,b,c,d$ be positive real numbers such that $ (a \plus{} 2b)^2 \equal{} 4c^2 \plus{} 1$. Show that $ \displaystyle 2d^2 \plus{} a^2\left(\phi \minus{} \frac {1}{2}\right) \plus{} b^2\left(\frac {1}{\phi \minus{} 1} \plus{} 2\right) \plus{} 2 \ge 4(c \minus{} d) \plus{} 2\sqrt {d^2 \plus{} 2d}$ and find all cases of equality. [i](A.Naskov)[/i]

Given a positive integer $n$, an $n$-gun is a $2n$-mino that is formed by putting a $1 \times n$ grid and an $n \times 1$ grid side by side so that one of the corner unit squares of the first grid is next to one of the corner unit squares of the second grid. Find the minimum possible $k$ such that it is possible to color the infinite planar grid with $k$ colors such that any $n$-gun cannot cover two different squares with the same color. [i]Itf0501[/i]

Let $N = 123456789101112\dots4344$ be the $79$-digit number obtained that is formed by writing the integers from $1$ to $44$ in order, one after the other. What is the remainder when $N$ is divided by $45$? $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 44$

On a straight line lie $100$ points and another point outside the line. Which is the biggest the number of isosceles triangles can be formed from the vertices of these $101$ points?

You place $n^2$ indistinguishable pieces on an $n\times n$ chessboard, where $n=2^{90}\approx 1.27\times10^{27}$. Of these pieces, $n$ of them are slightly lighter than usual, while the rest are all the same standard weight, but you are unable to discern this simply by feeling the pieces.\\ However, beneath each row and column of the chessboard, you have set up a scale, which, when turned on, will tell you [i]only[/i] whether the average weight of all the pieces on that row or column is the standard weight, or lighter than standard. On a given step, you are allowed to rearrange every piece on the chessboard, and then turn on all the scales simultaneously, and record their outputs, before turning them all off again. (Note that you can only turn on the scales if all $n^2$ pieces are placed in different squares on the board.) Find an algorithm that, in at most $k$ steps, will always allow you to rearrange the pieces in such a way that every row and column measures lighter than standard on the final step. An algorithm that completes in at most $k$ steps will be awarded: 1 pt for $k>10^{55}$ 10 pts for $k=10^{55}$ 30 pts for $k=10^{30}$ 50 pts for $k=10^{28}$ 65 pts for $k=10^{20}$ 80 pts for $k=10^5$ 90 pts for $k=2021$ 100 pts for $k=500$

Given $2018 \times 4$ grids and tint them with red and blue. So that each row and each column has the same number of red and blue grids, respectively. Suppose there're $M$ ways to tint the grids with the mentioned requirement. Determine $M \pmod {2018}$.

In the below diagram, the three rectangles are similar. Find the area of rectangle $ABCD$. [asy] size(6cm); draw((0,0)--(10,0)--(10,8.16496580928)--(0,8.16496580928)--(0,0)); draw((2,0)--(6,3.26598632371)--(2,8.16496580928)--(-2,4.89897949)--(2,0)); draw((10,3.26598632371)--(6,3.26598632371)--(6,0)); label("$A$",(0,8.16496580928), NW); label("$D$",(0,0), SW); label("$C$",(10,0), SE); label("$B$",(10,8.16496580928), NE); label("$E$",(2,8.16496580928), N); label("$F$",(2,0), S); label("$3$",(1,0),S); label("$3$",(1,8.16496580928),N); label("$12$",(6,8.16496580928),N); [/asy] $\textbf{(A) }75\sqrt{3}\qquad\textbf{(B) }120\sqrt{2}\qquad\textbf{(C) }100\sqrt{3}\qquad\textbf{(D) }180\qquad\textbf{(E) }75\sqrt{6}$

In the following list of numbers (given in their binary representations), each number appears an even number of times, except for one number that appears exactly three times. Find the number that appears exactly three times. Leave the answer in its binary representation. \begin{tabular}{cccccc} 010111 & 000001 & 100000 & 011000 & 110101 & 100001 \\ 010100 & 011111 & 111001 & 010001 & 010100 & 101100 \\ 010001 & 011011 & 011111 & 011011 & 100000 & 000001 \\ 110011 & 001000 & 111101 & 100001 & 101100 & 110011 \\ 111111 & 011000 & 001000 & 101000 & 111111 & 101000 \\ 010111 & 100011 & 111001 & 100011 & 110101 & 011111 \\ 100000 & 010100 & 010001 & 101100 & 010111 & 011011 \\ 011000 & 111101 & 111111 & 100001 & 101000 & 100011 \\ 011011 & 010111 & 110011 & 111111 & 000001 & 010001 \\ 101000 & 111001 & 010100 & 110101 & 011000 & 110101 \\ 001000 & 000001 & 100000 & 111101 & 100011 & 001000 \\ 111001 & 110011 & 100001 & 011111 & 101100 \end{tabular}

We will say that a point of the plane $(u, v)$ lies between the parabolas $y = f(x)$ and $y = g(x)$ if $f(u) \leq v \leq g(u)$. Find the smallest real $p$ for which the following statement is true: for any segment, the ends and the midpoint of which lie between the parabolas $y = x^2$ and $y=x^2+1$, then they lie entirely between the parabolas $y=x^2$ and $y=x^2+p$.

Find all functions $f, g : Q \to Q$ satisfying the following equality $f(x + g(y)) = g(x) + 2 y + f(y)$ for all $x, y \in Q$. (I. Voronovich)

Let $-2 < x_1 < 2$ be a real number and define $x_2, x_3, \ldots$ by $x_{n+1} = x_n^2-2$ for $n \geq 1$. Assume that no $x_n$ is $0$ and define a number $A$, $0 \leq A \leq 1$ in the following way: The $n^{\text{th}}$ digit after the decimal point in the binary representation of $A$ is a $0$ if $x_1x_2\cdots x_n$ is positive and $1$ otherwise. Prove that $A = \frac{1}{\pi}\cos^{-1}\left(\frac{x_1}{2}\right)$. [i]Evan O' Dorney.[/i]

Let $a_1,a_2,\dots, a_n$ be positive integers with product $P,$ where $n$ is an odd positive integer. Prove that $$\gcd(a_1^n+P,a_2^n+P,\dots, a_n^n+P)\le 2\gcd(a_1,\dots, a_n)^n.$$ [i]Proposed by Daniel Liu[/i]

There are $n$ red sticks and $n$ blue sticks. The sticks of each colour have the same total length, and can be used to construct an $n$-gon. We wish to repaint one stick of each colour in the other colour so that the sticks of each colour can still be used to construct an $n$-gon. Is this always possible if (a) $n = 3$, (b) $n > 3$ ?

Right triangle $XY Z$ has right angle at $Y$ and $XY = 228$, $Y Z = 2004$. Angle $Y$ is trisected, and the angle trisectors intersect $XZ$ at $P$ and $Q$ so that $X$, $P$, $Q$,$Z$ lie on $XZ$ in that order. Find the value of $(PY + Y Z)(QY + XY )$.

There is a 100 x100 square table, a real number being written in each cell.$A$ and $B$ play the following game. They choose, turn by turn, some row of the table (if it has not been chosen before). When $A$ and $B$ have $50$ rows chosen each, they sum the numbers in the corresponding cells of the chosen rows, and then sum the squares of all $100$ obtained numbers and compare the results. $A$ player who has the greater result wins. Player $A$ begins. Show that $A$ can avoid a defeat.

Let [i]Revolution[/i]$(x) = x^3 +Ux^2 +Sx + A$, where $U$, $S$, and $A$ are all integers and $U +S + A +1 = 1773$. Given that [i]Revolution[/i] has exactly two distinct nonzero integer roots $G$ and $B$, find the minimum value of $|GB|$. [i]Proposed by Jacob Xu[/i] [hide=Solution] [i]Solution.[/i] $\boxed{392}$ Notice that $U + S + A + 1$ is just [i]Revolution[/i]$(1)$ so [i]Revolution[/i]$(1) = 1773$. Since $G$ and $B$ are integer roots we write [i]Revolution[/i]$(X) = (X-G)^2(X-B)$ without loss of generality. So Revolution$(1) = (1-G)^2(1-B) = 1773$. $1773$ can be factored as $32 \cdot 197$, so to minimize $|GB|$ we set $1-G = 3$ and $1-B = 197$. We get that $G = -2$ and $B = -196$ so $|GB| = \boxed{392}$. [/hide]

For any function $f:[0,1]\to [0,1]$, let $P_n (f)$ denote the number of fixed points of the function $\underbrace{f(f(\dotsc f}_{n} (x)\dotsc )$, i.e., the number of points $x\in [0,1]$ satisfying $\underbrace{f(f(\dotsc f}_{n} (x)\dotsc )=x$. Construct a piecewise linear, continuous, surjective function $f:[0,1] \to [0,1]$ such that for a suitable $2<A<3$, the sequence $\frac{P_n(f)}{A^n}$ converges. [i]Based on the 8th problem of the Miklós Schweitzer competition, 2018[/i]

If $x$ and $y$ are nonnegative real numbers with $x+y= 2$, show that $x^2y^2(x^2+y^2)\le 2$.

$M_n(\mathbb{C})$ is the vector space of all complex $n\times n$ matrices. Given a linear map $T:M_n(\mathbb{C})\to M_n(\mathbb{C})$ s.t. $\det (A)=\det(T(A))$ for every $A\in M_n(\mathbb{C})$. (1) If $T(A)$ is the zero matrix, then show that $A$ is also the zero matrix. (2) Prove that $\text{rank} (A)=\text{rank} (T(A))$ for any $A\in M_n(\mathbb{C})$.

Solve the system of equations \[\{\begin{array}{cc}x^3=2y^3+y-2\\ \text{ } \\ y^3=2z^3+z-2 \\ \text{ } \\ z^3 = 2x^3 +x -2\end{array}\]