(a) On $[0, 1]$, let $f(x)$ have a continuous derivative satisfying $0 <f'(x) \leq1$. Also suppose that $f(0) = 0.$ Prove that $$ \left( \int_{0}^{1} f(x)\; dx \right)^{2} \geq \int_{0}^{1} f(x)^{3}\; dx.$$ (b) Show an example in which equality occurs.

[b]3.[/b] A quadrilateral $ABCD$ has the incircle $\omega$ and is such that the semi-lines $AB$ and $DC$ intersect at point $P$ and the semi-lines $AD$ and $BC$ intersect at point $Q$. The lines $AC$ and $PQ$ intersect at point $R$. Let $T$ be the point of $\omega$ closest from line $PQ$. Prove that the line $RT$ passes through the incenter of triangle $PQC$.

Let $ \varepsilon_1,\varepsilon_2,...,\varepsilon_{2n}$ be independent random variables such that $ P(\varepsilon_i\equal{}1)\equal{}P(\varepsilon_i\equal{}\minus{}1)\equal{}\frac 12$ for all $ i$, and define $ S_k\equal{}\sum_{i\equal{}1}^k \varepsilon_i, \;1\leq k \leq 2n$. Let $ N_{2n}$ denote the number of integers $ k\in [2,2n]$ such that either $ S_k>0$, or $ S_k\equal{}0$ and $ S_{k\minus{}1}>0$. Compute the variance of $ N_{2n}$.

The operation $*$ is defined by $a*b=a+b+ab$, where $a$ and $b$ are real numbers. Find the value of \[\frac{1}{2}*\bigg(\frac{1}{3}*\Big(\cdots*\big(\frac{1}{9}*(\frac{1}{10}*\frac{1}{11})\big)\Big)\bigg).\] [i]2017 CCA Math Bonanza Team Round #3[/i]

Let $ABC$ be a triangle with $AB=13$, $AC=25$, and $\tan A = \frac{3}{4}$. Denote the reflections of $B,C$ across $\overline{AC},\overline{AB}$ by $D,E$, respectively, and let $O$ be the circumcenter of triangle $ABC$. Let $P$ be a point such that $\triangle DPO\sim\triangle PEO$, and let $X$ and $Y$ be the midpoints of the major and minor arcs $\widehat{BC}$ of the circumcircle of triangle $ABC$. Find $PX \cdot PY$. [i]Proposed by Michael Kural[/i]

There are $10000$ equal tiles in the shape of an equilateral triangle. With these little triangles, regular hexagons are formed, without overlaps or gaps. If the regular hexagon that wastes the fewest triangles is formed, how many triangles are left over?

Consider a square $ABCD$ of side length $16$. Let $E,F$ be points on $CD$ such that $CE = EF = FD$. Let the line $BF$ and $AE$ meet in $M$. The area of $\bigtriangleup MAB$ is:

We have an $n \times n$ board, and in every square there is an integer. The sum of all integers on the board is $0$. We define an action on a square where the integer in the square is decreased by the number of neighbouring squares, and the number inside each of the neighbouring squares is increased by 1. Determine if there exists $n\geq 2$ such that we can turn all the integers into zeros in a finite number of actions.

Let $a,b,c$ be non-zero distinct real numbers so that there exist functions $f,g:\mathbb{R}^{+} \to \mathbb{R}$ so that: $af(xy)+bf(\frac{x}{y})=cf(x)+g(y)$ For all positive real $x$ and large enough $y$. Prove that there exists a function $h:\mathbb{R}^{+} \to \mathbb{R}$ so that: $f(xy)+f(\frac{x}{y})=2f(x)+h(y)$ For all positive real $x$ and large enough $y$.

A portion of a corner of a tiled floor is shown. If the entire floor is tiled in this way and each of the four corners looks like this one, then what fraction of the tiled floor is made of darker tiles? [asy]/* AMC8 2002 #23 Problem */ fill((0,2)--(1,3)--(2,3)--(2,4)--(3,5)--(4,4)--(4,3)--(5,3)--(6,2)--(5,1)--(4,1)--(4,0)--(2,0)--(2,1)--(1,1)--cycle, mediumgrey); fill((7,1)--(6,2)--(7,3)--(8,3)--(8,4)--(9,5)--(10,4)--(7,0)--cycle, mediumgrey); fill((3,5)--(2,6)--(2,7)--(1,7)--(0,8)--(1,9)--(2,9)--(2,10)--(3,11)--(4,10)--(4,9)--(5,9)--(6,8)--(5,7)--(4,7)--(4,6)--cycle, mediumgrey); fill((6,8)--(7,9)--(8,9)--(8,10)--(9,11)--(10,10)--(10,9)--(11,9)--(11,7)--(10,7)--(10,6)--(9,5)--(8,6)--(8,7)--(7,7)--cycle, mediumgrey); draw((0,0)--(0,11)--(11,11)); for ( int x = 1; x &lt; 11; ++x ) { draw((x,11)--(x,0), linetype("4 4")); } for ( int y = 1; y &lt; 11; ++y ) { draw((0,y)--(11,y), linetype("4 4")); } clip((0,0)--(0,11)--(11,11)--(11,5)--(4,1)--cycle);[/asy] $ \textbf{(A)}\ \frac13\qquad\textbf{(B)}\ \frac49\qquad\textbf{(C)}\ \frac12\qquad\textbf{(D)}\ \frac59\qquad\textbf{(E)}\ \frac58$

Find all pairs $(x,y)$ of real numbers such that \[16^{x^{2}+y} + 16^{x+y^{2}} = 1\]

Is it possible to arrange the numbers $1, 2, . . . , 60$ in that order, so that the sum of any two numbers between which there is one number, divisible by $2$, the sum of any two numbers between which there are two numbers divisible by $3$, . . . , the sum of any two numbers between which there is are there six numbers, divisible by $7$?

In triangle $ABC$ the midpoint of $BC$ is called $M$. Let $P$ be a variable interior point of the triangle such that $\angle CPM=\angle PAB$. Let $\Gamma$ be the circumcircle of triangle $ABP$. The line $MP$ intersects $\Gamma$ a second time at $Q$. Define $R$ as the reflection of $P$ in the tangent to $\Gamma$ at $B$. Prove that the length $|QR|$ is independent of the position of $P$ inside the triangle.

Prove that \[2010< \frac{2^{2}+1}{2^{2}-1}+\frac{3^{2}+1}{3^{2}-1}+...+\frac{2010^{2}+1}{2010^{2}-1}< 2010+\frac{1}{2}.\]

Every cell of a $2005 \times 2005$ square board is colored white or black so that every $2 \times 2$ subsquare contains an odd number of black cells. Show that among the corner cells there is an even number of black ones. How many ways are there to color the board in this manner?

$a, a_1,a_2,\dots ,a_n$ are natural numbers. We know that for any natural number $k$ which $ak+1$ is square, at least one of $a_1k+1,\dots ,a_n k+1$ is also square. Prove $a$ is one of $a_1,\dots ,a_n$ [i]Proposed by Mohsen Jamali[/i]

For which integers $n \in \{1,2,\dots,15\}$ is $n^n+1$ a prime number?

Divide the plane into $1$x$1$ squares formed by the lattice points. Let$S$ be the set-theoretic union of a finite number of such cells, and let $a$ be a positive real number less than or equal to 1/4.Show that S can be covered by a finite number of squares satisfying the following three conditions: 1) Each square in the cover is an array of $1$x$1$ cells 2) The squares in the cover have pairwise disjoint interios and 3)For each square $Q$ in the cover the ratio of the area $S \cap Q$ to the area of Q is at least $a$ and at most $a {(\lfloor a^{-1/2} \rfloor)} ^2$

Find all rational terms of sequence defined by formula $ a_n=\sqrt{\frac{9n-2}{n+1}}, n \in N $

Let $ABC$ be an acute triangle so that $2BC = AB + AC,$ with incenter $I{}.$ Let $AI{}$ meet $BC{}$ at point $A'.{}$ The perpendicular bisector of $AA'{}$ meets $BI{}$ and $CI{}$ at points $B'{}$ and $C'{}$ respectively. Let $AB'{}$ intersect $(ABC)$ at $X{}$ and let $XI{}$ intersect $AC'{}$ at $X'{}.$ Prove that $2\angle XX'A'=\angle ABC.{}$ [i](Proposed by Kang Taeyoung, South Korea)[/i]

A quick solution: Let R be the foot of the perpend. from X to BC. Let's assume Q and R are in the interior of the segms AC and BC (respectively) and P in the ext of AD. P, R, Q are colinear (Simson's thm). PQ tangent to circle XRD iff XRQ=XDR iff Pi-XCA=XDR iff XBA=XDR=XDC=ADB iff XBC+ABC=ADB=DAC+ACB iff XAC+ABC=DAC+ACD iff ABC=ACD=ACB iff AB=AC. It's the same for all the other cases.

Plane vectors form a group for addition. Show that this group has a generator system of every set $S$ that contains a Borel subset of positive linear measure of a circular arc.

Two circles of different radii are cut out of cardboard. Each circle is subdivided into $200$ equal sectors. On each circle $100$ sectors are painted white and the other $100$ are painted black. The smaller circle is then placed on top of the larger circle, so that their centers coincide. Show that one can rotate the small circle so that the sectors on the two circles line up and at least $100$ sectors on the small circle lie over sectors of the same color on the big circle.

The plan of a picture gallery is a chequered figure where each square is a room, and every room can be reached from each other by moving to rooms adjacent by side. A custodian in a room can watch all the rooms that can be reached from this room by one move of a chess rook (without leaving the gallery). What minimum number of custodians is sufficient to watch all the rooms in every gallery of $n$ rooms ($n > 1$)?

Let be two real numbers $ b>a>0, $ and a sequence $ \left( x_n \right)_{n\ge 1} $ with $ x_2>x_1>0 $ and such that $$ ax_{n+2}+bx_n\ge (a+b)x_{n+1} , $$ for any natural numbers $ n. $ Prove that $ \lim_{n\to\infty } x_n=\infty . $ [i]Dan Popescu[/i]