Let $P(n) = (n + 1)(n + 3)(n + 5)(n + 7)(n + 9)$. What is the largest integer that is a divisor of $P(n)$ for all positive even integers $n$?

Let $n \ge 3$ be a positive integer. We call a $3 \times 3$ grid [i]beautiful[/i] if the cell located at the center is colored white and all other cells are colored black, or if it is colored black and all other cells are colored white. Determine the minimum value of $a+b$ such that there exist positive integers $a$, $b$ and a coloring of an $a \times b$ grid with black and white, so that it contains $n^2 - n$ [i]beautiful[/i] subgrids.

Let $n \geq 2$ be a positive integer. Call a sequence $a_1, a_2, \cdots , a_k$ of integers an $n$[i]-chain[/i] if $1 = a_2 < a_ 2 < \cdots < a_k =n$, $a_i$ divides $a_{i+1}$ for all $i$, $1 \leq i \leq k-1$. Let $f(n)$ be the number of $n$[i]-chains[/i] where $n \geq 2$. For example, $f(4) = 2$ corresponds to the $4$-chains $\{1,4\}$ and $\{1,2,4\}$. Prove that $f(2^m \cdot 3) = 2^{m-1} (m+2)$ for every positive integer $m$.

Let $P$ be a given point inside quadrilateral $ABCD$. Points $Q_1$ and $Q_2$ are located within $ABCD$ such that \[\angle Q_1BC=\angle ABP,\quad\angle Q_1CB=\angle DCP,\quad\angle Q_2AD=\angle BAP,\quad\angle Q_2DA=\angle CDP.\] Prove that $\overline{Q_1Q_2}\parallel\overline{AB}$ if and only if $\overline{Q_1Q_2}\parallel\overline{CD}$.

Let $P$ be the set of points in the plane and $D$ the set of lines in the plane. Determine whether there exists a bijective function $f: P \rightarrow D$ such that for any three collinear points $A$, $B$, $C$, the lines $f(A)$, $f(B)$, $f(C)$ are either parallel or concurrent. [i]Gefry Barad[/i]

A farmer noticed that, during the last year, there were exactly as many calves born as during the two preceding years together. Even better, the number of pigs born during the last year was one larger than the number of pigs born during the two preceding years together. The farmer promised that if such a trend will continue then, after some years, at least twice as many pigs as calves will be born in his cattle, even though this far this target has not yet ever been reached. Will the farmer be able to keep his promise?

Let a tetrahedron $ABCD$ be inscribed in a sphere $S$. Find the locus of points $P$ inside the sphere $S$ for which the equality \[\frac{AP}{PA_1}+\frac{BP}{PB_1}+\frac{CP}{PC_1}+\frac{DP}{PD_1}=4\] holds, where $A_1,B_1, C_1$, and $D_1$ are the intersection points of $S$ with the lines $AP,BP,CP$, and $DP$, respectively.

Three fair six-sided dice are labeled with the numbers $\{1, 2, 3, 4, 5, 6\},$ $\{1, 2, 3, 4, 5, 6\},$ and $\{1, 2, 3, 7, 8, 9\},$ respectively. All three dice are rolled. The probability that at least two of the dice have the same value is $m/n,$ where $m, n$ are relatively prime positive integers. Find $100m + n.$ [i]Proposed by Michael Tang[/i]

Let $m$ be a positive integer, and consider a $m\times m$ checkerboard consisting of unit squares. At the centre of some of these unit squares there is an ant. At time $0$, each ant starts moving with speed $1$ parallel to some edge of the checkerboard. When two ants moving in the opposite directions meet, they both turn $90^{\circ}$ clockwise and continue moving with speed $1$. When more than $2$ ants meet, or when two ants moving in perpendicular directions meet, the ants continue moving in the same direction as before they met. When an ant reaches one of the edges of the checkerboard, it falls off and will not re-appear. Considering all possible starting positions, determine the latest possible moment at which the last ant falls off the checkerboard, or prove that such a moment does not necessarily exist. [i]Proposed by Toomas Krips, Estonia[/i]

Consider the sequences with 2016 terms formed by the digits 1, 2, 3, and 4. Find the number of those sequences containing an even number of the digit 1.

Let $a_1, b_1, a_2, b_2, \dots , a_n, b_n$ be nonnegative real numbers. Prove that \[ \sum_{i, j = 1}^{n} \min\{a_ia_j, b_ib_j\} \le \sum_{i, j = 1}^{n} \min\{a_ib_j, a_jb_i\}. \]

[b]i.)[/b] Consider a circle $K$ with diameter $AB;$ with circle $L$ tangent to $AB$ and to $K$ and with a circle $M$ tangent to circle $K,$ circle $L$ and $AB.$ Calculate the ration of the area of circle $K$ to the area of circle $M.$ [b]ii.)[/b] In triangle $ABC, AB = AC$ and $\angle CAB = 80^{\circ}.$ If points $D,E$ and $F$ lie on sides $BC, AC$ and $AB,$ respectively and $CE = CD$ and $BF = BD,$ then find the size of $\angle EDF.$

Through a point $M$ inside an angle $a$ line is drawn. It cuts off this angle a triangle of the least possible area. Prove that $M$ is the midpoint of the segment on this line that the angle intercepts.

Suppose that $x^2+px+q$ has two distinct roots $x=a$ and $x=b$. Furthermore, suppose that the positive difference between the roots of $x^2+ax+b$, the positive difference between the roots of $x^2+bx+a$, and twice the positive difference between the roots of $x^2+px+q$ are all equal. Given that $q$ can be expressed in the form $\frac{m}{m}$, where $m$ and $n$ are relatively prime positive integers, compute $m+n$. [i]2021 CCA Math Bonanza Lightning Round #4.1[/i]

Let $a_1,..., a_n$ be positive reals and let $x_1, ... , x_n$ be real numbers such that $a_1x_1 +...+ a_nx_n = 0$. Prove that $$\sum_{1\le i<j \le n} x_ix_j |a_i - a_j | \le 0.$$ When does the equality take place?

Find $\lim_{n\to\infty}\left(\frac{1}{n}\int_0^n (\sin ^ 2 \pi x)\ln (x+n)dx-\frac 12\ln n\right).$

Consider a polynomial $f(x) = x^4 +ax^3 +bx^2 +cx+d$ with integer coefficients. Prove that if $f(x)$ has exactly one real root, then it can be factored into nonconstant polynomials with rational coefficients

Solve the following equation in integers with gcd (x, y) = 1 $x^2 + y^2 = 2 z^2$

The Fibonacci numbers $F_1, F_2, F_3, \ldots$ are defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for each integer $n \ge 1$. Let $P$ be the unique polynomial of least degree for which $P(n) = F_n$ for all integers $1 \le n \le 10$. Compute the integer $m$ for which \[P(100) - \sum_{k=11}^{98} P(k) = \frac{m}{10} \dbinom{98}{9} + 144.\] [i]Proposed by Michael Tang[/i]

Find the smallest real number $M$ such that $$\sum_{k = 1}^{99}\frac{a_{k+1}}{a_k+a_{k+1}+a_{k+2}} < M$$ for all positive real numbers $a_1, a_2, \dots, a_{99}$. ($a_{100} = a_1, a_{101} = a_2$)

Triangle $ABC$ has a right angle at $C$. Let $D$ be the midpoint of side $\overline{AC}$, and let $E$ be the intersection of $\overline{AC}$ and the bisector of $\angle ABC$. The area of $\triangle ABC$ is $144$, and the area of $\triangle DBE$ is $8$. Find $AB^2$.

A block of mass $m_\text{1}$ is on top of a block of mass $m_\text{2}$. The lower block is on a horizontal surface, and a rope can pull horizontally on the lower block. The coefficient of kinetic friction for all surfaces is $\mu$. What is the resulting acceleration of the lower block if a force $F$ is applied to the rope? Assume that $F$ is sufficiently large so that the top block slips on the lower block. [asy] size(200); import roundedpath; draw((0,0)--(30,0),linewidth(3)); path A=(7,0.5)--(17,0.5)--(17,5.5)--(7,5.5)--cycle; filldraw(roundedpath(A,1),lightgray); path B=(10,6)--(15,6)--(15,9)--(10,9)--cycle; filldraw(roundedpath(B,1),lightgray); label("1",(12.5,6),1.5*N); label("2",(12,0.5),3*N); draw((17,3)--(27,3),EndArrow(size=13)); label(scale(1.2)*"$F$",(22,3),2*N); [/asy] (A) $a_\text{2}=(F-\mu g(2m_\text{1}+m_\text{2}))/m_\text{2}$ (B) $a_\text{2}=(F-\mu g(m_\text{1}+m_\text{2}))/m_\text{2}$ (C) $a_\text{2}=(F-\mu g(m_\text{1}+2m_\text{2}))/m_\text{2}$ (D) $a_\text{2}=(F+\mu g(m_\text{1}+m_\text{2}))/m_\text{2}$ (E) $a_\text{2}=(F-\mu g(m_\text{2}-m_\text{1}))/m_\text{2}$

How many non-congruent triangles have vertices at three of the eight points in the array shown below? [asy]dot((0,0)); dot((0,.5)); dot((.5,0)); dot((.5,.5)); dot((1,0)); dot((1,.5)); dot((1.5,0)); dot((1.5,.5));[/asy] $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.

$5000$ movie fans gathered at a convention. Each participant had watched at least one movie. The participants should be split into discussion groups of two kinds. In each group of the first kind, the members would discuss a movie they all watched. In each group of the second kind, each member would tell about the movie that no one else in this group had watched. Prove that the chairman can always split the participants into exactly 100 groups. (A group consisting of one person is allowed; in this case this person submits a report).