Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfy the equality, \[f(x+f(y))=f(x-f(y))+4xf(y)\] for any $x,y\in\mathbb{R}$.

Is there a finite abelian group $ G$ such that the product of the orders of all its elements is $ 2^{2009}?$

Let $ABC$ be a right triangle in $B$, with height $BT$, $T$ on the hypotenuse $AC$. Construct the equilateral triangles $BTX$ and $BTY$ so that $X$ is in the same half-plane as $A$ with respect to $BT$ and $Y$ is in the same half-plane as $C$ with respect to $BT$. Point $P$ is the intersection of $AY$ and $CX$. Show that $$PA \cdot BC = PB \cdot CA = PC \cdot AB.$$

Sherry is waiting for a train. Every minute, there is a $75\%$ chance that a train will arrive. However, she is engrossed in her game of sudoku, so even if a train arrives she has a $75\%$ chance of not noticing it (and hence missing the train). What is the probability that Sherry catches the train in the next five minutes?

Let $BH_b, CH_c$ be altitudes of an acute-angled triangle $ABC$. The line $H_bH_c$ meets the circumcircle of $ABC$ at points $X$ and $Y$. Points $P,Q$ are the reflections of $X,Y$ about $AB,AC$ respectively. Prove that $PQ \parallel BC$. [i]Proposed by Pavel Kozhevnikov[/i]

Initially, the integer $80$ is written on a blackboard. At each step, the integer $x$ on the blackboard is replaced with an integer chosen uniformly at random among [0,x−1], unless $x=0$ , in which case it is replaced by an integer chosen uniformly at random among [0,2024]. Let $P(a,b)$ be the probability that after $a$ steps, the integer on the board is $b$. Determine $$\lim_{x\to\infty}\frac{P(a,80)}{P(a,2024)}$$ (that is, the value that the function $\frac{P(a,80)}{P(a,2024)}$ approaches as $a$ goes to infinity).

Let $ k$ be a positive real number. Triangle ABC is acute and not isosceles, O is its circumcenter and AD,BE,CF are the internal bisectors. On the rays AD,BE,CF, respectively, let points L,M,N such that $ \frac {AL}{AD} \equal{} \frac {BM}{BE} \equal{} \frac {CN}{CF} \equal{} k$. Denote $ (O_1),(O_2),(O_3)$ be respectively the circle through L and touches OA at A, the circle through M and touches OB at B, the circle through N and touches OC at C. 1) Prove that when $ k \equal{} \frac{1}{2}$, three circles $ (O_1),(O_2),(O_3)$ have exactly two common points, the centroid G of triangle ABC lies on that common chord of these circles. 2) Find all values of k such that three circles $ (O_1),(O_2),(O_3)$ have exactly two common points

Given seven points in the plane, some of them are connected by segments such that: [b](i)[/b] among any three of the given points, two are connected by a segment; [b](ii)[/b] the number of segments is minimal. How many segments does a figure satisfying [b](i)[/b] and [b](ii)[/b] have? Give an example of such a figure.

Let a and b be positive integers such that $a + ab = 1443$ and $ab + b = 1444$. Find $10a + b$.

Decipher the equality $(\overline{LARN} -\overline{ACA}) : (\overline{CYP} +\overline{RUS}) = C^{Y^P} \cdot R^{U^S} $ where different symbols correspond to different digits and equal symbols correspond to equal digits. It is also supposed that all these digits are different from $0$.

Let $M$ be the set of all integers in the form of $a^2+13b^2$, where $a$ and $b$ are distinct itnegers. i) Prove that the product of any two elements of $M$ is also an element of $M$. ii) Determine, reasonably, if there exist infinite pairs of integers $(x,y)$ so that $x+y\not\in M$ but $x^{13}+y^{13}\in M$.

Let $k \geq 2, 1 < n_1 < n_2 < \ldots < n_k$ are positive integers, $a,b \in \mathbb{Z}^+$ satisfy \[ \prod^k_{i=1} \left( 1 - \frac{1}{n_i} \right) \leq \frac{a}{b} < \prod^{k-1}_{i=1} \left( 1 - \frac{1}{n_i} \right) \] Prove that: \[ \prod^k_{i=1} n_i \geq (4 \cdot a)^{2^k - 1}. \]

Let $p$ and $q$ be mutually prime natural numbers greater than $1$. Starting with the permutation $(1, 2, . . . , n)$, in one move we can switch the places of two numbers if their difference is $p$ or $q$. Prove that with such moves we can get any another permutation if and only if $n \ge p + q - 1$.

Given positive numbers $a_1,a_2,...,a_{99},a_{100}$. It is known, that $$a_1>a_0, a_2=3a_1-2a_0, a_3=3a_2-2a_1, ..., a_{100}=3a_{99}-2a_{98}$$ Prove that $$a_{100}>2^{99}.$$

Find the product of all numbers of the form $\sqrt[m]{m}-\sqrt[k]{k}$ , $m ,k$ are natural numbers satisfying the inequalities $1 \le k < m \le n$, where $n> 3$.

Call a three-term strictly increasing arithmetic sequence of integers [i]special[/i] if the sum of the squares of the three terms equals the product of the middle term and the square of the common difference. Find the sum of the third terms of all special sequences.

Find an odd natural number not exceeding $1000$ if you know that the sum of the last digits of all its divisors (including $1$ and the number itself) is $33$.

In the expansion of $ (a \plus{} b)^n$ there are $ n \plus{} 1$ dissimilar terms. The number of dissimilar terms in the expansion of $ (a \plus{} b \plus{} c)^{10}$ is: $ \textbf{(A)}\ 11\qquad \textbf{(B)}\ 33\qquad \textbf{(C)}\ 55\qquad \textbf{(D)}\ 66\qquad \textbf{(E)}\ 132$

Consider a sequence of polynomials such that $P_0(x)=2,P_1(x)=x$ and for all $n\ge1$ \[P_{n+1}(x)+P_{n-1}(x)=xP_n(x).\] a) Determine the polynomial \[Q_n(x)=P^2_n(x)-xP_n(x)P_{n-1}(x)+P^2_{n-1}(x)\] for $n=1972.$ b) Express the polynomial \[\bigl(P_{n+1}(x)-P_{n-1}(x)\bigr)^2\] in terms of $P_n(x),Q_n(x).$

(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?