There are $2020$ points on the coordinate plane {$A_i = (x_i, y_i):i = 1, ..., 2020$}, satisfying $$0=x_1<x_2<...<x_{2020}$$ $$0=y_{2020}<y_{2019}<...<y_1$$ Let $O=(0, 0)$ be the origin, $OA_1A_2...A_{2020}$ forms a polygon $C$. Now, you want to blacken the polygon $C$. Every time you can choose a point $(x,y)$ with $x, y > 0$, and blacken the area {$(x', y'): 0\leq x' \leq x, 0\leq y' \leq y$}. However, you have to pay $xy$ dollars for doing so. Prove that you could blacken the whole polygon $C$ by using $4|C|$ dollars. Here, $|C|$ stands for the area of the polygon $C$. [i]Proposed by me[/i]

In a convex quadrilateral $ABCD$ with $\angle ABC = \angle ADC = 135^\circ$, points $M$ and $N$ are taken on the rays $AB$ and $AD$ respectively such that $\angle MCD = \angle NCB = 90^\circ$. The circumcircles of triangles $AMN$ and $ABD$ intersect at $A$ and $K$. Prove that $AK \perp KC.$

Given an odd integer $n>3$, let $k$ and $t$ be the smallest positive integers such that both $kn+1$ and $tn$ are squares. Prove that $n$ is prime if and only if both $k$ and $t$ are greater than $\frac{n}{4}$

Points $D$ and $E$ are chosen on the exterior of $\vartriangle ABC$ such that $\angle ADC = \angle BEC = 90^o$. If $\angle ACB = 40^o$, $AD = 7$, $CD = 24$, $CE = 15$, and $BE = 20$, what is the measure of $\angle ABC $ in,degrees?

Let $\mathbb N$ denote the set of all positive integers. Find all real numbers $c$ for which there exists a function $f:\mathbb N\to \mathbb N$ satisfying: [list] [*] for any $x,a\in\mathbb N$, the quantity $\frac{f(x+a)-f(x)}{a}$ is an integer if and only if $a=1$; [*] for all $x\in \mathbb N$, we have $|f(x)-cx|<2023$. [/list] [i]Proposed by Sutanay Bhattacharya[/i]

A pair of positive integers $(m,n)$ is called [i]compatible[/i] if $m \ge \tfrac{1}{2} n + 7$ and $n \ge \tfrac{1}{2} m + 7$. A positive integer $k \ge 1$ is called [i]lonely[/i] if $(k,\ell)$ is not compatible for any integer $\ell \ge 1$. Find the sum of all lonely integers. [i]Proposed by Evan Chen[/i]

Given a scalene triangle $ABC$. Incircle of $\triangle{ABC{}}$ touches the sides $AB$ and $BC$ at points $C_1$ and $A_1$ respectively, and excircle of $\triangle{ABC}$ (on side $AC$) touches $AB$ and $BC$ at points $ C_2$ and $A_2$ respectively. $BN$ is bisector of $\angle{ABC}$ ($N$ lies on $BC$). Lines $A_1C_1$ and $A_2C_2$ intersects the line $AC$ at points $K_1$ and $K_2$ respectively. Let circumcircles of $\triangle{BK_1N}$ and $\triangle{BK_2N}$ intersect circumcircle of a $\triangle{ABC}$ at points $P_1$ and $P_2$ respectively. Prove that $AP_1$=$CP_2$

Along a round table are arranged $11$ cards with the names ( all distinct ) of the $11$ members of the $16^{th}$ JBMO Problem Selection Committee . The cards are arranged in a regular polygon manner . Assume that in the first meeting of the Committee none of its $11$ members sits in front of the card with his name . Is it possible to rotate the table by some angle so that at the end at least two members sit in front of the card with their names ?

Consider the integers $a_1, a_2, a_3, a_4, b_1, b_2, b_3, b_4$ with $a_k \ne b_k$ for all $k = 1, 2, 3, 4$. If $\{a_1, b_1\} + \{a_2, b_2\} = \{a_3, b_3\} + \{a_4, b_4\}$, show that the number $|(a_1 - b_1)(a_2 - b_2)(a_3 - b_3)(a_4 - b_4)|$ is a square. Note. For any sets $A$ and $B$, we denote $A + B = \{x + y | x \in A, y \in B\}$.

Consider equation I: $x+y+z=46$ where $x, y,$ and $z$ are positive integers, and equation II: $x+y+z+w=46$, where $x, y, z,$ and $w$ are positive integers. Then $ \textbf{(A)}\ \text{I can be solved in consecutive integers} \qquad$ $\textbf{(B)}\ \text{I can be solved in consecutive even integers} \qquad$ $\textbf{(C)}\ \text{II can be solved in consecutive integers} \qquad$ $\textbf{(D)}\ \text{II can be solved in consecutive even integers} \qquad$ $\textbf{(E)}\ \text{II can be solved in consecutive odd integers} $

Find all polynomials $p$ with real coefficients such that for all real $x$ , $xp(x)p(1-x)+x^3 +100 \ge 0$.

The $2011$th prime number is $17483$ and the next prime is $17489$. Does there exist a sequence of $2011^{2011}$ consecutive positive integers that contain exactly $2011$ prime numbers?

The ratio between the circumradius and the inradius of a given triangle is $7 : 2$. If the length of two sides of the triangle are $3$ and $7$, and the length of the remaining side is also an integer, what is the length of the remaining side?

There are $2^n$ airports, numbered with binary strings of length $n{}$. Any two stations whose numbers differ in exactly one digit are connected by a flight that has a price (which is the same in both directions). The sum of the prices of all $n{}$ flights leaving any station does not exceed 1. Prove that one can travel between any two airports by paying no more than 1.

Does there exist an infinite set $ M$ of straight lines on the coordinate plane such that (i) no two lines are parallel, and (ii) for any integer point there is a line from $ M$ containing it?

If $ V \equal{} gt \plus{} V_0$ and $ S \equal{} \frac {1}{2}gt^2 \plus{} V_0t$, then $ t$ equals: $ \textbf{(A)}\ \frac {2S}{V \plus{} V_0} \qquad\textbf{(B)}\ \frac {2S}{V \minus{} V_0} \qquad\textbf{(C)}\ \frac {2S}{V_0 \minus{} V} \qquad\textbf{(D)}\ \frac {2S}{V} \qquad\textbf{(E)}\ 2S \minus{} V$

There are $12$ acrobats who are assigned a distinct number ($1, 2, \cdots , 12$) respectively. Half of them stand around forming a circle (called circle A); the rest form another circle (called circle B) by standing on the shoulders of every two adjacent acrobats in circle A respectively. Then circle A and circle B make up a formation. We call a formation a “[i]tower[/i]” if the number of any acrobat in circle B is equal to the sum of the numbers of the two acrobats whom he stands on. How many heterogeneous [i]towers[/i] are there? (Note: two [i]towers[/i] are homogeneous if either they are symmetrical or one may become the other one by rotation. We present an example of $8$ acrobats (see attachment). Numbers inside the circle represent the circle A; numbers outside the circle represent the circle B. All these three formations are “[i]towers[/i]”, however they are homogeneous [i]towers[/i].)

For a set $S$ of $n$ points, let $d_1 > d_2 >... > d_k > ...$ be the distances between the points. A function $f_k: S \to N$ is called a [i]coloring function[/i] if, for any pair $M,N$ of points in $S$ with $MN \le d_k$ , it takes the value $f_k(M)+ f_k(N)$ at some point. Prove that for each $m \in N$ there are positive integers $n,k$ and a set $S$ of $n$ points such that every coloring function $f_k$ of $S$ satisfies $| f_k(S)| \le m$

Express \[ \sum_{k=0}^n (-1)^k (n-k)!(n+k)! \] in closed form.

Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that $$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$ for all positive integers $n$. Show that $a_{2022}\leq 1$.

For a natural number $n{},$ determine the number of ordered pairs $(S,T)$ of subsets of $\{1,2,\ldots,n\}$ for which $s>|T|$ for any element $s\in S$ and $t>|S|$ for any element $t\in T.$

Given a set of 102 elements. Is it possible to choose 102 17-element subsets so that the intersection of any two subsets contains no more than 3 elements?

Let $p$ be a prime number, $n$ a natural number which is not divisible by $p$, and $\mathbb{K}$ is a finite field, with $char(K) = p, |K| = p^n, 1_{\mathbb{K}}$ unity element and $\widehat{0} = 0_{\mathbb{K}}.$ For every $m \in \mathbb{N}^{*}$ we note $ \widehat{m} = \underbrace{1_{\mathbb{K}} + 1_{\mathbb{K}} + \ldots + 1_{\mathbb{K}}}_{m \text{ times}} $ and define the polynomial \[ f_m = \sum_{k = 0}^{m} (-1)^{m - k} \widehat{\binom{m}{k}} X^{p^k} \in \mathbb{K}[X]. \] a) Show that roots of $f_1$ are $ \left\{ \widehat{k} | k \in \{0,1,2, \ldots , p - 1 \} \right\}$. b) Let $m \in \mathbb{N}^{*}.$ Determine the set of roots from $\mathbb{K}$ of polynomial $f_{m}.$

Consider the polynomials $P\left(x\right)=16x^4+40x^3+41x^2+20x+16$ and $Q\left(x\right)=4x^2+5x+2$. If $a$ is a real number, what is the smallest possible value of $\frac{P\left(a\right)}{Q\left(a\right)}$? [i]2016 CCA Math Bonanza Team #6[/i]

Let $[a_1 , b_1 ] , \ldots, [a_n ,b_n ]$ be a collection of closed intervals such that any of these closed intervals have a point in common. Prove that there exists a point contained in every one of these intervals.