For an ordered $10$-tuple of nonnegative integers $a_1,a_2,\ldots, a_{10}$, we denote \[f(a_1,a_2,\ldots,a_{10})=\left(\prod_{i=1}^{10} {\binom{20-(a_1+a_2+\cdots+a_{i-1})}{a_i}}\right) \cdot \left(\sum_{i=1}^{10} {\binom{18+i}{19}}a_i\right).\] When $i=1$, we take $a_1+a_2+\cdots+a_{i-1}$ to be $0$. Let $N$ be the average of $f(a_1,a_2,\ldots,a_{10})$ over all $10$-tuples of nonnegative integers $a_1,a_2,\ldots, a_{10}$ satisfying \[a_1+a_2+\cdots+a_{10}=20.\] Compute the number of positive integer divisors of $N$. [i]2021 CCA Math Bonanza Individual Round #14[/i]

Evaluate $ \sum_{k\equal{}1}^n \int_0^{\pi} (\sin x\minus{}\cos kx)^2dx.$

Two symbols $A$ and $B$ obey the rule $ABBB = B$. Given a word $x_1x_2\ldots x_{3n+1}$ consisting of $n$ letters $A$ and $2n+1$ letters $B$, show that there is a unique cyclic permutation of this word which reduces to $B$.

An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4.$ One vertex of the triangle is $(0,1),$ one altitude is contained in the $y$-axis, and the length of each side is $\sqrt{\frac mn},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Determine the number of functions $f: \mathbb{N} \to \mathbb{N}$ and $g: \mathbb{N} \to \mathbb{N}$ such that for all $n \in \mathbb{N}$: \[ f(g(n)) = n + 2015, \] \[ g(f(n)) = n^2 + 2015. \]

Show that a path on a rectangular grid which starts at the northwest corner, goes through each point on the grid exactly once, and ends at the southeast corner divides the grid into two equal halves: (a) those regions opening north or east; and (b) those regions opening south or west. [img]https://cdn.artofproblemsolving.com/attachments/b/e/aa20c9f9bc44bd1e5a9b9e86d49debf0f821b7.png[/img] (The figure above shows a path meeting the conditions of the problem on a $5 \times 8$ grid. The shaded regions are those opening north or east while the rest open south or west.)

Let be $f:\left[0,1\right]\rightarrow\left[0,1\right]$ a continuous and bijective function,such that : $f\left(0\right)=0$.Then the following inequality holds: $\left(\alpha+2\right)\cdotp\int_{0}^{1}x^{\alpha}\left(f\left(x\right)+f^{-1}\left(x\right)\right)\leq2,\forall\alpha\geq0 $

For any set $A=\{a_1,a_2,\cdots,a_m\}$, let $P(A)=a_1a_2\cdots a_m$. Let $n={2010\choose99}$, and let $A_1, A_2,\cdots,A_n$ be all $99$-element subsets of $\{1,2,\cdots,2010\}$. Prove that $2010|\sum^{n}_{i=1}P(A_i)$.

Find the area of the region of the $xy$-plane defined by the inequality $|x|+|y|+|x+y| \le 1$.

Find all real-valued functions on the positive integers such that $f(x + 1019) = f(x)$ for all $x$, and $f(xy) = f(x) f(y)$ for all $x,y$.

Tamara has three rows of two 6-feet by 2-feet flower beds in her garden. The beds are separated and also surrounded by 1-foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet? [asy] unitsize(0.7cm); path p1 = (0,0)--(15,0)--(15,10)--(0,10)--cycle; fill(p1,lightgray); draw(p1); for (int i = 1; i <= 8; i += 7) { for (int j = 1; j <= 7; j += 3 ) { path p2 = (i,j)--(i+6,j)--(i+6,j+2)--(i,j+2)--cycle; draw(p2); fill(p2,white); } } draw((0,8)--(1,8),Arrows); label("1",(0.5,8),S); draw((7,8)--(8,8),Arrows); label("1",(7.5,8),S); draw((14,8)--(15,8),Arrows); label("1",(14.5,8),S); draw((11,0)--(11,1),Arrows); label("1",(11,0.5),W); draw((11,3)--(11,4),Arrows); label("1",(11,3.5),W); draw((11,6)--(11,7),Arrows); label("1",(11,6.5),W); draw((11,9)--(11,10),Arrows); label("1",(11,9.5),W); label("6",(4,1),N); label("2",(1,2),E); [/asy] $\textbf{(A) }72 \qquad \textbf{(B) }78 \qquad \textbf{(C) }90 \qquad \textbf{(D) }120 \qquad \textbf{(E) }150 $

Show that any arithmetic progression where the first term and the common difference are non-zero natural numbers contains an infinite number of composite terms. *A number is composite if it is not prime.

Let $ABCD$ be a rectangle and let $E \in (BD)$ such that $m( \angle DAE) =15^o$. Let $F \in AB$ such that $EF \perp AB$. It is known that $EF=\frac12 AB$ and $AD = a$. Find the measure of the angle $\angle EAC$ and the length of the segment $(EC)$.

Let $ABCD$ be an inscribed quadrilateral in a circle $c(O,R)$ (of circle $O$ and radius $R$). With centers the vertices $A,B,C,D$, we consider the circles $C_{A},C_{B},C_{C},C_{D}$ respectively, that do not intersect to each other . Circle $C_{A}$ intersects the sides of the quadrilateral at points $A_{1} , A_{2}$ , circle $C_{B}$ intersects the sides of the quadrilateral at points $B_{1} , B_{2}$ , circle $C_{C}$ at points $C_{1} , C_{2}$ and circle $C_{D}$ at points $C_{1} , C_{2}$ . Prove that the quadrilateral defined by lines $A_{1}A_{2} , B_{1}B_{2} , C_{1}C_{2} , D_{1}D_{2}$ is cyclic.

On grid paper, mark three nodes so that in the triangle they formed, the sum of the two smallest medians equals to half-perimeter.

Let $a, b, c$ be positive reals, and let $\sqrt[2013]{\frac{3}{a^{2013}+b^{2013}+c^{2013}}}=P$. Prove that \[\prod_{\text{cyc}}\left(\frac{(2P+\frac{1}{2a+b})(2P+\frac{1}{a+2b})}{(2P+\frac{1}{a+b+c})^2}\right)\ge \prod_{\text{cyc}}\left(\frac{(P+\frac{1}{4a+b+c})(P+\frac{1}{3b+3c})}{(P+\frac{1}{3a+2b+c})(P+\frac{1}{3a+b+2c})}\right).\][i]Proposed by David Stoner[/i]

Given an acute triangle $ABC$. Points $B'$ and $C'$ are symmetrical, respectively, to vertices $B$ and $ C$ wrt straight lines $AC$ and $AB$. Let $P$ be the intersection point of the circumcircles of triangles $ABB'$ and $ACC'$, different from $A$. Prove that the center of the circumcircle of triangle $ABC$ lies on line $PA$.

Let $p,q$ be prime numbers such that $n^{3pq}-n$ is a multiple of $3pq$ for [b]all[/b] positive integers $n$. Find the least possible value of $p+q$.

Prove that there are infinitely many positive integers $m$ for which there exists consecutive odd positive integers $p_m<q_m$ such that $p_m^2+p_mq_m+q_m^2$ and $p_m^2+m\cdot p_mq_m+q_m^2$ are both perfect squares. If $m_1, m_2$ are two positive integers satisfying this condition, then we have $p_{m_1}\neq p_{m_2}$

Let $N>1$ be an integer. We are adding all remainders when we divide $N$ by all positive integers less than $N$. If this sum is less than $N$, find all possible values of $N$.

Initially, the number $1$ is written on the blackboard. At each step, the number on the blackboard is erased and another is written, which is obtained by applying any of the following operations: Operation A: Multiply the number on the board with $\frac12$. Operation B: Subtract the number on the board from $1$. For example, if the number $\frac38$ is on the board, it can be replaced by $\frac12 \frac38=\frac{3}{16}$ or by $1-\frac38=\frac58$ . Give a sequence of steps after which the number on the board is $\frac{2009}{2^{20009}}$ .

Determine the smallest positive integer $ n$ such that there exists positive integers $ a_1,a_2,\cdots,a_n$, that smaller than or equal to $ 15$ and are not necessarily distinct, such that the last four digits of the sum, \[ a_1!\plus{}a_2!\plus{}\cdots\plus{}a_n!\] Is $ 2001$.

Let $p$ be a prime number of the form $9k + 1$. Show that there exists an integer n such that $p | n^3 - 3n + 1$.

In a triangle $ABC$ with $AC\ne BC$, $M$ is the midpoint of $AB$ and $\angle A=\alpha$, $\angle B=\beta$, $\angle ACM=\varphi$ and $\angle BSM=\Psi$. Prove that $$\frac{\sin\alpha\sin\beta}{\sin(\alpha-\beta)}=\frac{\sin\varphi\sin\Psi}{\sin(\varphi-\Psi)}.$$

Prove that for all n=3,4,5.... there excist odd x,y such $2^n=x^2 + 7y^2$ .