A rectangle of dimensions $2024 \times 2023$ is filled with pieces of the following types: [asy] size(200); // Figure (A) draw((0,0)--(4,0)--(4,1)--(0,1)--cycle); draw((1,0)--(1,1)); draw((2,0)--(2,1)); draw((3,0)--(3,1)); // Figure (B) draw((6,0)--(8,0)--(8,2)--(6,2)--cycle); draw((7,0)--(7,2)); draw((6,1)--(8,1)); // Figure (C) draw((10,0)--(12,0)--(12,1)--(11,1)--(11,2)--(9,2)--(9,1)--(10,1)--cycle); draw((10,0)--(10,1)); draw((11,0)--(11,1)); draw((10,1)--(11,1)); draw((9,1)--(9,2)); draw((10,1)--(10,2)); draw((11,0)--(12,0)); draw((10,1)--(12,1)); // Labeling label("(A)", (2, -0.5)); label("(B)", (7, -0.5)); label("(C)", (10.5, -0.5)); [/asy] Each piece can be turned arround, and each square has side length $1$. Is it possible to use exactly 2023 pieces of type $(A)$?

A circle $k$ is given with a non-diameter chord $AC$. On the tangent at point $A$ select point $X \ne A$ and mark $D$ the intersection of the circle $k$ with the interior of the line $XC$ (if any). Let $B$ a point in circle $k$ such that quadrilateral $ABCD$ is a trapezoid . Determine the set of intersections of lines $BC$ and $AD$ belonging to all such trapezoids.

Let $f(x)$ be a continuous real-valued function defined on the interval $[0, 1]$. Show that \[\int_0^1\int_0^1|f(x)+f(y)|dx \; dy \ge \int_0^1 |f(x)|dx\]

Let $\text{s}_b(k)$ denote the sum of the digits of $k$ in base $b$. Compute \[\text{s}_{101}(33)+\text{s}_{101}(66)+\text{s}_{101}(99)+\cdots+\text{s}_{101}(3333).\] [i]2021 CCA Math Bonanza Individual Round #10[/i]

Let $a$ and $k$ be positive integers. Prove that for every positive integer $d$ there exists a positive integer $n$ such that $d$ divides $ka^n + n.$

Let $n_1 = \overline{abcabc}$ and $n_2= \overline{d00d}$ be numbers represented in the decimal system, with $a\ne 0$ and $d \ne 0$. a) Prove that $\sqrt{n_1}$ cannot be an integer. b) Find all positive integers $n_1$ and $n_2$ such that $\sqrt{n_1+n_2}$ is an integer number. c) From all the pairs $(n_1,n_2)$ such that $\sqrt{n_1 n_2}$ is an integer find those for which $\sqrt{n_1 n_2}$ has the greatest possible value

Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.) [asy]import three; import math; size(180); defaultpen(linewidth(.8pt)); currentprojection=orthographic(2,0.2,1); triple A=(0,0,1); triple B=(sqrt(2)/2,sqrt(2)/2,0); triple C=(sqrt(2)/2,-sqrt(2)/2,0); triple D=(-sqrt(2)/2,-sqrt(2)/2,0); triple E=(-sqrt(2)/2,sqrt(2)/2,0); triple F=(0,0,-1); draw(A--B--E--cycle); draw(A--C--D--cycle); draw(F--C--B--cycle); draw(F--D--E--cycle,dotted+linewidth(0.7));[/asy]$ \textbf{(A)}\ 210 \qquad \textbf{(B)}\ 560 \qquad \textbf{(C)}\ 840 \qquad \textbf{(D)}\ 1260 \qquad \textbf{(E)}\ 1680$

Consider the parallelogram $ ABCD, $ whose diagonals intersect at $ O. $ The bisector of the angle $ \angle DAC $ and that of $ \angle DBC $ intersect each other at $ T. $ Moreover, $ \overrightarrow{TD} +\overrightarrow{TC} =\overrightarrow{TO} . $ Find the angles of the triangle $ ABT. $

Let $D$ be the closed unit disk in the plane, and let $z_1,z_2,\ldots,z_n$ be fixed points in $D$. Prove that there exists a point $z$ in $D$ such that the sum of the distances from $z$ to each of the $n$ points is greater or equal than $n$.

The points $A_{1},A_{2},B_{1},B_{2},C_{1},C_{2}$ are on the sides $AB$, $BC$ and $AC$ of an acute triangle $ABC$ such that $AA_{1} = A_{1}A_{2} = A_{2}B = \frac{1}{3} AB$, $BB_{1} = B_{1}B_{2} = B_{2}C = \frac{1}{3}BC$ and $CC_{1} = C_{1}C_{2} = C_{2}A = \frac{1}{3} AC$. Let $k_{A}, k_{B}$ and $k_{C}$ be the circumcircles of the triangles $AA_{1}C_{2}$, $BB_{1}A_{2}$ and $CC_{1}B_{2}$ respectively. Furthermore, let $a_{B}$ and $a_{C}$ be the tangents to $k_{A}$ at $A_{1}$ and $C_{2}$, $b_{C}$ and $b_{A}$ the tangents to $k_{B}$ at $B_{1}$ and $A_{2}$ and $c_{A}$ and $c_{B}$ the tangents to $k_{C}$ at $C_{1}$ and $B_{2}$. Show that the perpendicular lines from the intersection points of $a_{B}$ and $b_{A}$, $b_{C}$ and $c_{B}$, $c_{A}$ and $a_{C}$ to $AB$, $BC$ and $CA$ respectively are concurrent.

Prove that if the positive real numbers $p$ and $q$ satisfy $\frac{1}{p}+\frac{1}{q}= 1$, then a) $\frac{1}{3} \le \frac{1}{p (p + 1)} +\frac{1}{q (q + 1)} <\frac{1}{2}$ b) $\frac{1}{p (p - 1)} + \frac{1}{q (q - 1)} \ge 1$

Find all $f: R\rightarrow R$ such that (i) The set $\{\frac{f(x)}{x}| x\in R-\{0\}\}$ is finite (ii) $f(x-1-f(x)) = f(x)-1-x$ for all $x$

a) The vertex $A_1$ of the regular $12$-gon (dodecagon) $A_1A_2...A_{12}$ is marked with "$-$" and all the rest $--$ with "$+$". You are allowed to change the sign simultaneously in the $6$ vertices in succession. Prove that is impossible to obtain dodecagon with $A_2$ marked with "$-$" and the rest of the vertices $--$ with "$+$". b) Prove the same statement if it is allowed to change the signs not in six, but in four vertices in succession. c) Prove the same statement if it is allowed to change the signs in three vertices in succession.

Let $ABC$ be an acute triangle with incenter $I$ and circumcircle $\Omega$. The line passing $I$ and perpendicular to $AI$ meets $AB, AC$ at $D, E$, respectively. $A$-excircle of $\triangle{ABC}$ meets $BC$ at $T$. $AT$ meets $\Omega$ at $P$. The line passing $P$ and parallel to $BC$ meets $\Omega$ at $Q$. The intersection of $QI$ and $AT$ is $K$. Prove that $Q,D,K,E$ are concyclic.

Two buckets each have four balls; two red balls and two white balls in the first, and two red balls and two blue balls in the second. At first, a bucket is selected, then a ball in the bucket is selected, with both buckets and balls inside the selected bucket having equal probability of being chosen. Then, without replacement of the first ball, the process is repeated once more. Determine the probability that the first ball drawn being red if the second ball drawn was blue.

Kritesh manages traffic on a $45 \times 45$ grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the $45 \times 45$ grid. Kritesh realizes that it may not always be possible for all the cars to leave the grid. Therefore, before the process begins, he will remove $k$ cars from the $45 \times 45$ grid in such a way that it becomes possible for all the remaining cars to eventually exit the grid. What is the minimum value of $k$ that guarantees that Kritesh's job is possible? $\textbf{Proposed by Shining Sun. USA}$

On the sides of the triangle $ABC$, three similar isosceles triangles $ABP \ (AP = PB)$, $AQC \ (AQ = QC)$, and $BRC \ (BR = RC)$ are constructed. The first two are constructed externally to the triangle $ABC$, but the third is placed in the same half-plane determined by the line $BC$ as the triangle $ABC$. Prove that $APRQ$ is a parallelogram.

Let $ABC$ be a triangle, the point $E$ is in the segment $AC$, the point $F$ is in the segment $AB$ and $P=BE\cap CF$. Let $D$ be a point such that $AEDF$ is a parallelogram, Prove that $D$ is in the side $BC$, if and only if, the triangle $BPC$ and the quadrilateral $AEPF$ have the same area.

Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

Prove that there exist infinitely many positive integers $ n$ such that $ n!$ is not divisible by $ n^2\plus{}1$.

Given that $f(x)+2f(4-x)=x+8$, compute $f(16)$.

Let the set of all bijective functions taking positive integers to positive integers be $\mathcal B.$ Find all functions $\mathbf F:\mathcal B\to \mathbb R$ such that $$(\mathbf F(p)+\mathbf F(q))^2=\mathbf F(p \circ p)+\mathbf F(p\circ q)+\mathbf F(q\circ p)+\mathbf F(q\circ q)$$ for all $p,q \in \mathcal B.$ [i]Proposed by ckliao914[/i]

Call a fraction $\frac{a}{b}$, not necessarily in the simplest form [i]special[/i] if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions? $\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13$

Let $ABC$ be a scalene triangle, and $A_o$, $B_o,$ $C_o$ be the midpoints of $BC$, $CA$, $AB$ respectively. The bisector of angle $C$ meets $A_oCo$ and $B_oC_o$ at points $B_1$ and $A_1$ respectively. Prove that the lines $AB_1$, $BA_1$ and $A_oB_o$ concur.

$ABC$ is a triangle with $\angle A = 90^\circ$, $\angle B = 60^\circ$. The points $A_1$, $B_1$, $C_1$ on $BC$, $CA$, $AB$ respectively are such that $A_1B_1C_1$ is equilateral and the perpendiculars (to $BC$ at $A_1$, to $CA$ at $B_1$ and to $AB$ at $C_1$) meet at a point $P$ inside the triangle. Find the ratios $PA_1:PB_1:PC_1$.