Find all $f:\mathbb{N}\to\mathbb{N}$ satisfying that for all $m,n\in\mathbb{N}$, the nonnegative integer $|f(m+n)-f(m)|$ is a divisor of $f(n)$. [i] Proposed by usjl[/i]

Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous, strictly decreasing function such that $f([0,1])\subseteq[0,1]$. [list=i] [*]For all positive integers $n{}$ prove that there exists a unique $a_n\in(0,1)$, solution of the equation $f(x)=x^n$. Moreover, if $(a_n){}$ is the sequence defined as above, prove that $\lim_{n\to\infty}a_n=1$. [*]Suppose $f$ has a continuous derivative, with $f(1)=0$ and $f'(1)<0$. For any $x\in\mathbb{R}$ we define \[F(x)=\int_x^1f(t) \ dt.\]Let $\alpha{}$ be a real number. Study the convergence of the series \[\sum_{n=1}^\infty F(a_n)^\alpha.\] [/list]

Find number of terms when we expand $(a+b+c)^{99}$ (in the general case).

At CMU, markers come in two colors: blue and orange. Zachary fills a hat randomly with three markers such that each color is chosen with equal probability, then Chase shuffles an additional orange marker into the hat. If Zachary chooses one of the markers in the hat at random and it turns out to be orange, the probability that there is a second orange marker in the hat can be expressed as simplified fraction $\tfrac{m}{n}$. Find $m+n$.

Let $ABC$ be a triangle and $\omega$ its incircle. $\omega$ touches $AC,AB$ at $E,F$, respectively. Let $P$ be a point on $EF$. Let $\omega_1=(BFP), \omega_2=(CEP)$. The parallel line through $P$ to $BC$ intersects $\omega_1,\omega_2$ at $X,Y$, respectively. Show that $BX=CY$.

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.