Given positive integers $m$ and $n$. Let $P$ and $Q$ be two collections of $m \times n$ numbers of $0$ and $1$, arranged in $m$ rows and $n$ columns. An example of such collections for $m=3$ and $n=4$ is \[\left[ \begin{array}{cccc} 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right].\] Let those two collections satisfy the following properties: (i) On each row of $P$, from left to right, the numbers are non-increasing, (ii) On each column of $Q$, from top to bottom, the numbers are non-increasing, (iii) The sum of numbers on the row in $P$ equals to the same row in $Q$, (iv) The sum of numbers on the column in $P$ equals to the same column in $Q$. Show that the number on row $i$ and column $j$ of $P$ equals to the number on row $i$ and column $j$ of $Q$ for $i=1,2,\dots,m$ and $j=1,2,\dots,n$. [i]Proposer: Stefanus Lie[/i]

$a,b,c,x,y,z$ are positive real numbers and $bz+cy=a$, $az+cx=b$, $ay+bx=c$. Find the least value of following function $f(x,y,z)=\frac{x^2}{1+x}+\frac{y^2}{1+y}+\frac{z^2}{1+z}$

Let $ABCDEF$ be a convex equilateral hexagon such that lines $BC$, $AD$, and $EF$ are parallel. Let $H$ be the orthocenter of triangle $ABD$. If the smallest interior angle of the hexagon is $4$ degrees, determine the smallest angle of the triangle $HAD$ in degrees.

Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$. [i]Proposed by Morteza Saghafian, Iran[/i]

Annie takes a $6$ question test, with each question having two parts each worth $1$ point. On each [b]part[/b], she receives one of nine letter grades $\{\text{A,B,C,D,E,F,G,H,I}\}$ that correspond to a unique numerical score. For each [b]question[/b], she receives the sum of her numerical scores on both parts. She knows that $\text{A}$ corresponds to $1$, $\text{E}$ corresponds to $0.5$, and $\text{I}$ corresponds to $0$. When she receives her test, she realizes that she got two of each of $\text{A}$, $\text{E}$, and $\text{I}$, and she is able to determine the numerical score corresponding to all $9$ markings. If $n$ is the number of ways she can receive letter grades, what is the exponent of $2$ in the prime factorization of $n$? [i]2020 CCA Math Bonanza Individual Round #10[/i]

All diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon (not on the boundary) do two or more diagonals intersect? $\textbf{(A)} \ 49 \qquad \textbf{(B)} \ 65 \qquad \textbf{(C)} \ 70 \qquad \textbf{(D)} \ 96 \qquad \textbf{(E)} \ 128$

The smallest value of $x^2+8x$ for real values of $x$ is: $\textbf{(A) }-16.25\qquad\textbf{(B) }-16\qquad\textbf{(C) }-15\qquad\textbf{(D) }-8\qquad \textbf{(E) }\text{None of these}$

Let $n$ ($n\ge 3$) be a natural number. Denote by $f(n)$ the least natural number by which $n$ is not divisible (e.g. $f(12)=5$). If $f(n)\ge 3$, we may have $f(f(n))$ in the same way. Similarly, if $f(f(n))\ge 3$, we may have $f(f(f(n)))$, and so on. If $\underbrace{f(f(\dots f}_{k\text{ times}}(n)\dots ))=2$, we call $k$ the “[i]length[/i]” of $n$ (also we denote by $l_n$ the “[i]length[/i]” of $n$). For arbitrary natural number $n$ ($n\ge 3$), find $l_n$ with proof.

Lola, Lolo, Tiya, and Tiyo participated in a ping pong tournament. Each player competed against each of the other three players exactly twice. Shown below are the win-loss records for the players. The numbers $1$ and $0$ represent a win or loss, respectively. For example, Lola won five matches and lost the fourth match. What was Tiyo’s win-loss record? \[\begin{tabular}{c | c} Player & Result \\ \hline Lola & \texttt{111011}\\ Lolo & \texttt{101010}\\ Tiya & \texttt{010100}\\ Tiyo & \texttt{??????} \end{tabular}\] $\textbf{(A)}\ \texttt{000101} \qquad \textbf{(B)}\ \texttt{001001} \qquad \textbf{(C)}\ \texttt{010000} \qquad \textbf{(D)}\ \texttt{010101} \qquad \textbf{(E)}\ \texttt{011000}$

Let scalene triangle $ABC$ have altitudes $AD, BE, CF$ and circumcenter $O$. The circumcircles of $\triangle ABC$ and $\triangle ADO$ meet at $P \ne A$. The circumcircle of $\triangle ABC$ meets lines $PE$ at $X \ne P$ and $PF$ at $Y \ne P$. Prove that $XY \parallel BC$. [i]Proposed by Daniel Hu[/i]

$x,y,z\in\mathbb{R^+}$. If $xyz=1$, then prove the following: $$\sum\frac{x^6+2}{x^3}\geq3(\frac{x}{y}+\frac{y}{z}+\frac{z}{x})$$

What is the biggest shadow that a cube of side length $1$ can have, with the sun at its peak? Note: "The biggest shadow of a figure with the sun at its peak" is understood to be the biggest possible area of the orthogonal projection of the figure on a plane.

For a positive integer $M$, if there exist integers $a$, $b$, $c$ and $d$ so that: \[ M \leq a < b \leq c < d \leq M+49, \qquad ad=bc \] then we call $M$ a GOOD number, if not then $M$ is BAD. Please find the greatest GOOD number and the smallest BAD number.

Let $ ABC$ be a triangle. $ W_a$ is a circle with center on $ BC$ passing through $ A$ and perpendicular to circumcircle of $ ABC$. $ W_b,W_c$ are defined similarly. Prove that center of $ W_a,W_b,W_c$ are collinear.

Find all functions $f: \mathbb Q \to \mathbb Q$ such that $ f(x^{2}\plus{}y\plus{}f(xy)) \equal{} 3\plus{}(x\plus{}f(y)\minus{}2)f(x)$ for all $x,y \in \mathbb Q$.

Suppose $n$ is a natural number. In how many ways can we place numbers $1,2,....,n$ around a circle such that each number is a divisor of the sum of it's two adjacent numbers?

Let $ABC$ be an acute-angled triangle, and let $AA_1, BB_1, CC_1$ be its altitudes. Points $A', B', C'$ are chosen on the segments $AA_1, BB_1, CC_1$, respectively, so that $\angle BA'C = \angle AC'B = \angle CB'A = 90^{o}$. Let segments $AC'$ and $CA'$ intersect at $B"$; points $A", C"$ are defined similarly. Prove that hexagon $A'B"C'A"B'C"$ is circumscribed.

Three circles $\Gamma_A, \Gamma_B, \Gamma_C$ are externally tangent. The circles are centered at $A, B, C$ and have radii $4, 5, 6$ respectively. Circles $\Gamma_B$ and $\Gamma_C$ meet at $D$, circles $\Gamma_C$ and $\Gamma_A$ meet at $E$, and circles $\Gamma_A$ and $\Gamma_B$ meet at $F$. Let $GH$ be a common external tangent of $\Gamma_B$ and $\Gamma_C$ on the opposite side of $BC$ as $EF$, with $G$ on $\Gamma_B$ and $H$ on $\Gamma_C$. Lines $FG$ and $EH$ meet at $K$. Point $L$ is on $\Gamma_A$ such that $\angle DLK = 90^\circ$. Compute $\frac{LG}{LH}$.

[list=a] [*] Prove that for any positive integers $a,b,c$, there exists a positive integer $N$ such that $$(N+a^2)(N+b^2)(N+c^2)$$ is a perfect square. [*] Prove that there exist five distinct positive integers $a,b,c,d,e$ for which there exists a positive integer $N$ such that $$(N+a^2)(N+b^2)(N+c^2)(N+d^2)(N+e^2)$$ is a perfect square. [/list] [i]Luminița Popescu[/i]

Let $p$ be a prime number and let $s$ be an integer with $0 < s < p.$ Prove that there exist integers $m$ and $n$ with $0 < m < n < p$ and \[ \left \{\frac{sm}{p} \right\} < \left \{\frac{sn}{p} \right \} < \frac{s}{p} \] if and only if $s$ is not a divisor of $p-1$. Note: For $x$ a real number, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$, and let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of x.

In each cell, a strip of length $100$ is worth a chip. You can change any $2$ neighboring chips and pay $1$ rouble, and you can also swap any $2$ chips for free, between which there are exactly $4$ chips. For what is the smallest amount of rubles you can rearrange chips in reverse order?

Prove that the set of positive integers cannot be partitioned into three nonempty subsets such that, for any two integers $x$, $y$ taken from two different subsets, the number $x^{2}-xy+y^{2}$ belongs to the third subset.

Let $P$ be a point on segment $BC$. $Q, R$ are points on $AC, AB$ such that $PQ \parallel AB $ and $ PR \parallel AC$. $O, O_{1}, O_{2} $ are the circumcenters of triangle $ABC, BPR, PCQ$. The circumcircles of $BPR, PCQ $ meet at point $K (\ne P)$. Prove that $OO_{1} = KO_{2} $.

$ ABCD$ is a cyclic quadrilateral. If $ \angle B \equal{} \angle D$, $ AC\bigcap BD \equal{} \left\{E\right\}$, $ \angle BCD \equal{} 150{}^\circ$, $ \left|BE\right| \equal{} x$, $ \left|AC\right| \equal{} z$, then find $ \left|ED\right|$ in terms of $ x$ and $ z$. $\textbf{(A)}\ \frac {z \minus{} x}{\sqrt {3} } \qquad\textbf{(B)}\ \frac {z \minus{} 2x}{3} \qquad\textbf{(C)}\ \frac {z \plus{} x}{\sqrt {3} } \qquad\textbf{(D)}\ \frac {z \minus{} 2x}{2} \qquad\textbf{(E)}\ \frac {2z \minus{} 3x}{2}$

$10$ real numbers are given $a_1,a_2,\ldots ,a_{10} $, and the $45$ sums of two of these numbers are formed $a_i+a_j $, $1\leq i&lt;j\leq 10$ . It is known that not all these sums are integers. Determine the minimum value of $k$ such that it is possible that among the $45$ sums there are $k$ that are not integers and $45-k$ that are integers.