Prove that:$a+b+c+d \geq \frac{2}{3}(ab+bc+ca+ad+ac+bd)$ where $a;b;c;d$ are positive real numbers satisfying $2(ab+bc+cd+da+ac+bd)+abc+bcd+cda+dab=16$

You are tossing an unbiased coin. The last $ 28 $ consecutive flips have all resulted in heads. Let $ x $ be the expected number of additional tosses you must make before you get $ 60 $ consecutive heads. Find the sum of all distinct prime factors in $ x $.

Let $A$ and $B$ be positive integers. Define the arithmetic sequence $a_0, a_1, a_2, ...$ by $a_n = A_n + B$. Suppose that there exists an $n\ge 0$ such that $a_n$ is a square. Let $M$ be a positive integer such that $M^2$ is the smallest square in the sequence. Prove that $M < A +\sqrt{B}$.

Let $A$ and $B$ be points on a circle $\mathcal{C}$ with center $O$ such that $\angle AOB = \dfrac {\pi}2$. Circles $\mathcal{C}_1$ and $\mathcal{C}_2$ are internally tangent to $\mathcal{C}$ at $A$ and $B$ respectively and are also externally tangent to one another. The circle $\mathcal{C}_3$ lies in the interior of $\angle AOB$ and it is tangent externally to $\mathcal{C}_1$, $\mathcal{C}_2$ at $P$ and $R$ and internally tangent to $\mathcal{C}$ at $S$. Evaluate the value of $\angle PSR$.

We define the sets $A_1,A_2,...,A_{160}$ such that $\left|A_{i} \right|=i$ for all $i=1,2,...,160$. With the elements of these sets we create new sets $M_1,M_2,...M_n$ by the following procedure: in the first step we choose some of the sets $A_1,A_2,...,A_{160}$ and we remove from each of them the same number of elements. These elements that we removed are the elements of $M_1$. In the second step we repeat the same procedure in the sets that came of the implementation of the first step and so we define $M_2$. We continue similarly until there are no more elements in $A_1,A_2,...,A_{160}$, thus defining the sets $M_1,M_2,...,M_n$. Find the minimum value of $n$.

A circle center $O$ is inscribed in the quadrilateral $ABCD$. $AB$ is parallel to and longer than $CD$ and has midpoint $M$. The line $OM$ meets $CD$ at $F$. $CD$ touches the circle at $E$. Show that $DE = CF$ iff $AB = 2CD$.

If you have an algorithm for finding all the real zeros of any cubic polynomial, how do you find the real solutions to $x = p(y), y = p(x)$, where $p$ is a cubic polynomial?

Let $a{}$ and $b{}$ be positive integers. Prove that for any real number $x{}$ \[\sum_{j=0}^a\binom{a}{j}\big(2\cos((2j-a)x)\big)^b=\sum_{j=0}^b\binom{b}{j}\big(2\cos((2j-b)x)\big)^a.\]

Let $\triangle ABC$ be an acute angle triangle and $\omega$ be its circumcircle. Let $N$ be a point on arc $AC$ not containing $B$ and $S$ be a point on line $AB$. The line tangent to $\omega$ at $N$ intersects $BC$ at $T$, $NS$ intersects $\omega$ at $K$. Assume that $\angle NTC = \angle KSB$. Prove that $CK\parallel AN \parallel TS$.

Denote by $\mathbb{Q}^+$ the set of all positive rational numbers. Determine all functions $f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+$ which satisfy the following equation for all $x, y \in \mathbb{Q}^+:$ \[f\left( f(x)^2y \right) = x^3 f(xy).\] [i]Proposed by Thomas Huber, Switzerland[/i]

Sixteen squares of an $8\times 8$ chessboard are chosen so that there are exactly lwo in each row and two in each column. Prove that eight white pawns and eight black pawns can be placed on these sixteen squares so that there is one white pawn and one black pawn in each row and in cach colunm.

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.