Compute the number of sequences of integers $(a_1,\ldots,a_{200})$ such that the following conditions hold. [list] [*] $0\leq a_1<a_2<\cdots<a_{200}\leq 202.$ [*] There exists a positive integer $N$ with the following property: for every index $i\in\{1,\ldots,200\}$ there exists an index $j\in\{1,\ldots,200\}$ such that $a_i+a_j-N$ is divisible by $203$. [/list]

Determine all functions $f : \mathbb R \to \mathbb R$ such that for all real numbers $x$ and $y$, $$f(x^2 + xy) = f(x)f(y) + yf(x) + xf(x+y).$$ [i]Proposed by Walther Janous, Austria[/i]

In each square of a $10\times 10$ board, there is an arrow pointing upwards, downwards, left, or right. Prove that it is possible to remove $50$ arrows from the board, such that no two remaining arrows point at each other.

Find all real values of the real parameter $a$ such that the equation \[ 2x^{2}-6ax+4a^{2}-2a-2+\log_{2}(2x^{2}+2x-6ax+4a^{2})= \] \[ =\log_{2}(x^{2}+2x-3ax+2a^{2}+a+1). \] has a unique solution.

For some positive integer $n$, the sum of all odd positive integers between $n^2-n$ and $n^2+n$ is a number between $9000$ and $10000$, inclusive. Compute $n$. [i]2020 CCA Math Bonanza Lightning Round #3.1[/i]

Mereduth has many red boxes and many blue boxes. Coloon has placed five green boxes in a row on the ground, and Mereduth wants to arrange some number of her boxes on top of his row. Assume that each box must be placed so that it straddles two lower boxes. Including the one with no boxes, how many arrangements can Mereduth make?

$ n > 2 $ points are chosen on a circle and each of them is connected to every other by a segment. Is it possible to draw all of these segments in one sequence, i.e. so that the end of the first segment is the beginning of the second, the end of the second - the beginning of the third, etc., and so that the end of the last segment is the beginning of the first?

Let be a positive real number $ r, $ a natural number $ n, $ and a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ satisfying $ f(rxy)=(f(x)f(y))^n, $ for any real numbers $ x,y. $ [b]a)[/b] Give three distinct examples of what $ f $ could be if $ n=1. $ [b]b)[/b] For a fixed $ n\ge 2, $ find all possibilities of what $ f $ could be. [i]Bogdan Blaga[/i]

Let $S$ be a finite set and $P$ the set of all subsets of $S$. Show that one can label the elements of $P$ as $A_i$ such that (1) $A_1 =\emptyset$. (2) For each $n\geq1 $ we either have $A_{n-1}\subset A_{n}$ and $|A_{n} \setminus A_{n-1}|=1$ or $A_{n}\subset A_{n-1}$ and $|A_{n-1} \setminus A_{n}|=1.$

If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]

Let $G$ be a simple, connected and finite grafo. A hunter and an invisible rabbit play in the graph $G$. The rabbit is initially in a vertex $w_0$. In the $k$-th turn (for $k \geq 0$) the hunter picks freely a vertex $v_k$. If $v_k = w_k$, the rabbit is capture and the game ends. If not, the rabbit moves invisibly by an edge of $w_k$ to $w_{k+1}$ ($w_k$ and $w_{k+1}$ are adjacent and therefore distinct) and the game continues. The hunter knows these rules and the graph $G$. After the $k-$th turn he knows that $w_k \not = v_k$, but he gets no more information. Characterize the graphs $G$ such that the hunter has an strategy that guaranties that he can capture the rabbit in at most $N$ turns for some positive integer $N$. Here $N$ must depend only on $G$ and the strategy should work independently of the initial position and trajectory of the rabbit.

How many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers?

How many $10$-digit positive integers can be written by using four $0$s, five $1$s, and one $2$? $ \textbf{(A)}\ 1260 \qquad\textbf{(B)}\ 1134 \qquad\textbf{(C)}\ 756 \qquad\textbf{(D)}\ 630 \qquad\textbf{(E)}\ \text{None of above} $

Compute $$\frac{2019! \cdot 2^{2019}}{(2020^2-2018^2)(2020^2-2016^2)\dots(2020^2-2^2)}.$$ [i]Proposed by Ada Tsui[/i]

Define functions $f$ and $g$ as nonconstant, differentiable, real-valued functions on $R$. If $f(x+y)=f(x)f(y)-g(x)g(y)$, $g(x+y)=f(x)g(y)+g(x)f(y)$, and $f'(0)=0$, prove that $\left(f(x)\right)^2+\left(g(x)\right)^2=1$ for all $x$.

Let $a_{m,n}$ denote the coefficient of $x^n$ in the expansion $(1+x+x^2)^n$. Prove the inequality for all integers $k\ge 0$ : \[ 0\le \sum_{\ell=0}^{\left\lfloor{\frac{2k}{3}}\right\rfloor} (-1)^{\ell} a_{k-\ell,\ell}\le 1 \]

$ABCD$ is a parallelogram. Take $E$ on the line $AB$ so that $BE = BC$ and $B$ lies between $A$ and $E$. Let the line through $C$ perpendicular to $BD$ and the line through $E$ perpendicular to $AB$ meet at $F$. Show that $\angle DAF = \angle BAF$.

Let $180^o < \theta_1 < \theta_2 <...< \theta_n = 360^o$. For $i = 1,2,..., n$, $P_i = (\cos \theta_i^o, \sin \theta_i^o)$ is a point on the circle $C$ with centre $(0,0)$ and radius $1$. Let $P$ be any point on the upper half of $C$. Find the coordinates of $P$ such that the sum of areas $[PP_1P_2] + [PP_2P_3] + ...+ [PP_{n-1}P_n]$ attains its maximum.

Find the lowest odd positive integer with an odd number of divisors and is divisible by $d^2$ and $a+b+c+d+e+f$, where $a, b, c, d, e, f$ are consecutive prime numbers.

Does there exist a positive integer, $x$, such that $(x+2)^{2023}-x^{2023}$ has exactly $2023^{2023}$ factors? [i]Proposed by Wong Jer Ren[/i]

$V$ is a real vector space and $ f, f_{i}: V \rightarrow \mathbb{R} $ are linear for $i = 1, 2, ... , k.$ Also $f $ is zero at all points for which all of $ f_{i }$ are zero. Show that $ f $ is a linear combination of the $f_{i}$.

If $n$ is an integer, then find all values of $n$ for which $\sqrt{n}+\sqrt{n+2005}$ is an integer as well.

\[\int_0^1 \frac{1-x}{x^{5/2}+x^{3/2}+x^{1/2}}\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Suppose $f(x)$ is a function satisfying $\left | f(m+n)-f(m) \right | \leq \frac{n}{m}$ for all positive integers $m$,$n$. Show that for all positive integers $k$: \[\sum_{i=1}^{k}\left |f(2^k)-f(2^i) \right |\leq \frac{k(k-1)}{2}\].

In the following triangular lattice distance of two vertices is length of the shortest path between them. Let $ A_{1},A_{2},\dots,A_{n}$ be constant vertices of the lattice. We want to find a vertex in the lattice whose sum of distances from vertices is minimum. We start from an arbitrary vertex. At each step we check all six neighbors and if sum of distances from vertices of one of the neighbors is less than sum of distances from vertices at the moment we go to that neighbor. If we have more than one choice we choose arbitrarily, as seen in the attached picture. Obviusly the algorithm finishes a) Prove that when we can not make any move we have reached to the problem's answer. b) Does this algorithm reach to answer for each connected graph?