The three-digit number $2a3$ is added to the number $326$ to give the three-digit number $5b9$. If $5b9$ is divisible by 9, then $a+b$ equals $ \text{(A)}\ 2\qquad\text{(B)}\ 4\qquad\text{(C)}\ 6\qquad\text{(D)}\ 8\qquad\text{(E)}\ 9$

Given are three circles $(O_1), (O_2), (O_3)$, pairwise intersecting each other, that is, every single circle meets the other two circles at two distinct points. Let $(X_1)$ be the circle externally tangent to $(O_1)$ and internally tangent to the circles $(O_2), (O_3),$ circles $(X_2), (X_3)$ are defined in the same manner. Let $(Y_1)$ be the circle internally tangent to $(O_1)$ and externally tangent to the circles $(O_2), (O_3)$, the circles $(Y_2), (Y_3)$ are defined in the same way. Let $(Z_1), (Z_2)$ be two circles internally tangent to all three circles $(O_1), (O_2), (O_3)$. Prove that the four lines $X_1Y_1, X_2Y_2, X_3Y_3, Z_1Z_2$ are concurrent. Nguyễn Văn Linh

Let $T_1$ be a triangle having $a, b, c$ as lengths of its sides and let $T_2$ be another triangle having $u, v,w$ as lengths of its sides. If $P,Q$ are the areas of the two triangles, prove that \[16PQ \leq a^2(-u^2 + v^2 + w^2) + b^2(u^2 - v^2 + w^2) + c^2(u^2 + v^2 - w^2).\] When does equality hold?

An L-shaped figure composed of $4$ unit squares (such as shown in the picture) we call L-dominoes. [img]https://cdn.artofproblemsolving.com/attachments/b/2/064b7c7de496f981cd937cbb7392efc1066420.png[/img] Determine the maximum number of L-dominoes that can be placed on a board of dimensions $n \times n$, where $n$ is natural number, so that no two dominoes overlap and it is possible get from the upper left to the lower right corner of the board by moving only across those squares that are not covered by dominoes. (By moving, we move from someone of the square on it the neighboring square, i.e. the square with which it shares the page). Note: L-Dominoes can be rotated as well as flipped, giving an symmetrical figure wrt axis compared to the one shown in the picture.

Points $A, B, C, D,$ and $F$ lie on a sphere with radius $\sqrt{10}$ such that lines $AD, BE,$ and $CF$ are concurrent at point $P$ inside the sphere and are pairwise perpendicular. If $PA=\sqrt{6}, PB=\sqrt{10},$ and $PC=\sqrt{15},$ what is the volume of tetrahedron $DEFP$?

Let $f\left(x,y\right)=x^2\left(\left(x+2y\right)^2-y^2+x-1\right)$. If $f\left(a,b+c\right)=f\left(b,c+a\right)=f\left(c,a+b\right)$ for distinct numbers $a,b,c$, what are all possible values of $a+b+c$? [i]2019 CCA Math Bonanza Individual Round #12[/i]

$(MON 4)$ Let $p$ and $q$ be two prime numbers greater than $3.$ Prove that if their difference is $2^n$, then for any two integers $m$ and $n,$ the number $S = p^{2m+1} + q^{2m+1}$ is divisible by $3.$

Let $ABC$ an acute triangle with circumcenter $O$. Points $X$ and $Y$ are chosen such that: [list] [*]$\angle XAB = \angle YCB = 90^\circ$[/*] [*]$\angle ABC = \angle BXA = \angle BYC$[/*] [*]$X$ and $C$ are in different half-planes with respect to $AB$[/*] [*]$Y$ and $A$ are in different half-planes with respect to $BC$[/*] [/list] Prove that $O$ is the midpoint of $XY$.

Let $A,B,P$ positive reals with $P\le A+B$. (a) Choose reals $\theta_1,\theta_2$ with $A\cos\theta_1 + B\cos\theta_2=P$ and prove that \[ A\sin\theta_1 + B\sin\theta_2 \le \sqrt{(A+B-P)(A+B+P)} \] (b) Prove equality is attained when $\theta_1=\theta_2=\arccos\left(\dfrac{P}{A+B}\right)$. (c) Take $A=\dfrac{1}{2}xy, B=\dfrac{1}{2}wz$ and $P=\dfrac14 \left(x^2+y^2-z^2-w^2\right)$ with $0<x\le y\le x+z+w$, $z,w>0$ and $z^2+w^2<x^2+y^2$. Show that we can translate (a) and (b) into the following theorem: from all quadrilaterals with (ordered) sidelenghts $(x,y,z,w)$, the cyclical one has the greatest area.

There are 100 saucers in a circle. Two people take turns putting marmalade of various colors in empty saucers. The first person can choose one or three empty saucers and fill each of them with marmalade of arbitrary color. The second one can choose one empty saucer and fill it with marmalade of arbitrary color. There should not be two adjacent saucers with marmalade of the same color. The game ends when all the saucers are filled. The loser is the last player to introduce a new color of marmalade into the game. Who has a winning strategy?

Let $m,\ n$ be positive integer such that $2\leq m<n$. (1) Prove the inequality as follows. \[\frac{n+1-m}{m(n+1)}<\frac{1}{m^2}+\frac{1}{(m+1)^2}+\cdots +\frac{1}{(n-1)^2}+\frac{1}{n^2}<\frac{n+1-m}{n(m-1)}\] (2) Prove the inequality as follows. \[\frac 32\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq 2\] (3) Prove the inequality which is made precisely in comparison with the inequality in (2) as follows. \[\frac {29}{18}\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq \frac{61}{36}\]

Show that , for any positive integer $n$ , the sum of $8n+4$ consecutive positive integers cannot be a perfect square .

For $n \ge 2,$ let $\omega(n)$ denote the number of distinct prime factors of $n.$ We set $\omega(1) = 0.$ Compute the absolute value of $$\sum_{n=1}^{160} (-1)^{\omega(n)} \left\lfloor \frac{160}{n} \right\rfloor.$$

For what values of $ m$ does the equation $ x^2 \plus{} (2m \plus{} 6)x \plus{} 4m \plus{} 12 \equal{} 0$ has two real roots, both of them greater than $ \minus{}1$.

Among all eight-digit multiples of four, are there more numbers with the digit $1$ or without the digit $1$ in their decimal representation?

Suppose that $f:[-1,1] \to \mathbb{R}$ is continuous and satisfies \[\left(\int_{-1}^1 e^xf(x) dx\right)^2 \ge \left(\int_{-1}^1 f(x) dx\right)\left(\int_{-1}^1 e^{2x}f(x) dx\right).\] Prove that there exists a point $c \in (-1,1)$ such that $f(c)=0$.

Prove that for each positive integer $a$ there exists such an integer $b>a$, for which $1+2^a+3^a$ divides $1+2^b+3^b$.

$a;b;c;d\in\mathbb{Z^+}$. Solve the equation: $$2^{a!}+2^{b!}+2^{c!}=d^3$$

Given a succession $C$ of $1001$ positive real numbers (not necessarily distinct), and given a set $K$ of distinct positive integers, the permitted operation is: select a number $k\in{K}$, then select $k$ numbers in $C$, calculate the arithmetic mean of those $k$ numbers, and replace each of those $k$ selected numbers with the mean. If $K$ is a set such that for each $C$ we can reach, by a sequence of permitted operations, a state where all the numbers are equal, determine the smallest possible value of the maximum element of $K$.

Let $ n$ be a positive integer congruent to $1$ modulo $4$. Xantippa has a bag of $n + 1$ balls numbered from $ 0$ to $n$. She draws a ball (randomly, equally distributed) from the bag and reads its number: $k$, say. She keeps the ball and then picks up another $k$ balls from the bag (randomly, equally distributed, without repossession). Finally, she adds up the numbers of all the $k + 1$ balls she picked up. What is the probability that the sum will be odd?

A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point $A$ in the figure on the right. The box has base length $w$ and height $h$. What is the area of the sheet of wrapping paper? [asy]size(270pt); defaultpen(fontsize(10pt)); filldraw(((3,3)--(-3,3)--(-3,-3)--(3,-3)--cycle),lightgrey); dot((-3,3)); label("$A$",(-3,3),NW); draw((1,3)--(-3,-1),dashed+linewidth(.5)); draw((-1,3)--(3,-1),dashed+linewidth(.5)); draw((-1,-3)--(3,1),dashed+linewidth(.5)); draw((1,-3)--(-3,1),dashed+linewidth(.5)); draw((0,2)--(2,0)--(0,-2)--(-2,0)--cycle,linewidth(.5)); draw((0,3)--(0,-3),linetype("2.5 2.5")+linewidth(.5)); draw((3,0)--(-3,0),linetype("2.5 2.5")+linewidth(.5)); label('$w$',(-1,-1),SW); label('$w$',(1,-1),SE); draw((4.5,0)--(6.5,2)--(8.5,0)--(6.5,-2)--cycle); draw((4.5,0)--(8.5,0)); draw((6.5,2)--(6.5,-2)); label("$A$",(6.5,0),NW); dot((6.5,0)); [/asy] $\textbf{(A) } 2(w+h)^2 \qquad \textbf{(B) } \frac{(w+h)^2}2 \qquad \textbf{(C) } 2w^2+4wh \qquad \textbf{(D) } 2w^2 \qquad \textbf{(E) } w^2h $

If $a@b=\dfrac{a^3-b^3}{a-b}$, for how many real values of $a$ does $a@1=0$?

In trapezoid $ABCD$ with $AB$ parallel to $CD$ show that : $\frac{|AB|^2-|BC|^2+|AC|^2}{|CD|^2-|AD|^2+|AC|^2}=\frac{|AB|}{|CD|}=\frac{|AB|^2-|AD|^2+|BD|^2}{|CD|^2-|BC|^2+|BD|^2}$

Suppose that $m=nq$, where $n$ and $q$ are positive integers. Prove that the sum of binomial coefficients \[\sum_{k=0}^{n-1}{ \gcd(n, k)q \choose \gcd(n, k)}\] is divisible by $m$.

A spinning turntable is rotating in a vertical plane with period $ 500 \, \text{ms} $. It has diameter 2 feet carries a ping-pong ball at the edge of its circumference. The ball is bolted on to the turntable but is released from its clutch at a moment in time when the ball makes a subtended angle of $\theta>0$ with the respect to the horizontal axis that crosses the center. This is illustrated in the figure. The ball flies up in the air, making a parabola and, when it comes back down, it does not hit the turntable. This can happen only if $\theta>\theta_m$. Find $\theta_m$, rounded to the nearest integer degree? [asy] filldraw(circle((0,0),1),gray(0.7)); draw((0,0)--(0.81915, 0.57358)); dot((0.81915, 0.57358)); draw((0.81915, 0.57358)--(0.475006, 1.06507)); arrow((0.417649,1.14698), dir(305), 12); draw((0,0)--(1,0),dashed); label("$\theta$", (0.2, 0.2/3), fontsize(8)); label("$r$", (0.409575,0.28679), NW, fontsize(8)); [/asy] [i]Problem proposed by Ahaan Rungta[/i]