Let $u, v$ be arbitrary positive real numbers. Prove that \[\min{(u, \frac{100}{v}, v+\frac{2023}{u})} \leq \sqrt{2123}.\]

Consider an equilateral triangle with side length $9$. Each side is divided into $3$ equal segments by $2$ points, for a total of $6$ points. Compute the area of the circle passing through these$ 6$ points. [img]https://cdn.artofproblemsolving.com/attachments/7/b/1860a3ff86a0e4b93a4891861300dcb09adad4.png[/img]

Let $p$ be a positive integer and $q, z$ be real numbers with $0\le q\le 1$ and $q^{p+1}\le z\le 1$. Prove that \[\prod_{k=1}^p \left|\frac{z - q^k}{z + q^k}\right| \le\prod_{k=1}^p \left|\frac{1 - q^k}{1 + q^k}\right|.\]

Suppose that $x,y$ are distinct positive reals, and $n>1$ is a positive integer. If \[x^n-y^n=x^{n+1}-y^{n+1},\] then show that \[1<x+y<\frac{2n}{n+1}.\]

It takes $3$ rabbits $5$ hours to dig $9$ holes. It takes $5$ beavers $36$ minutes to build $2$ dams. At this rate, how many more minutes does it take $1$ rabbit to dig $1$ hole than it takes $1$ beaver to build $1$ dam? [i]2016 CCA Math Bonanza Team #1[/i]

In the questions below: $G$ is a finite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group. Show that every element of $S_{n}$ is a product of $2$-cycles.

On account of the annual fluctuation of temperature the ground at the town of Ν freezes to a depth of 2 metres. To what depth would it freeze on account of a daily fluctuation of the same amplitude?

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$. Prove that there exists a point $J$ such that for any point $X$ inside $ABC$ if $AX,BX,CX$ intersect $\omega$ in $A_1,B_1,C_1$ and $A_2,B_2,C_2$ be reflections of $A_1,B_1,C_1$ in midpoints of $BC,AC,AB$ respectively then $A_2,B_2,C_2,J$ lie on a circle.

A $4\times4$ grid has its 16 cells colored arbitrarily in three colors. A [i]swap[/i] is an exchange between the colors of two cells. Prove or disprove that it always takes at most three swaps to produce a line of symmetry, regardless of the grid's initial coloring. [i]Proposed by Matthew Babbitt[/i]

Alice and Bob are painting a house. Alice can paint a house in $20$ hours by herself. Bob can paint a house in $40$ hours by himself. Both people start at the same time, paint at their own constant rate, and work together to paint one house. When the house is fully painted, what fraction of the house was painted by Alice?

Let $ABCD$ be a square such that $A=(0,0)$ and $B=(1,1)$. $P(\frac{2}{7},\frac{1}{4})$ is a point inside the square. An ant starts walking from $P$, touches $3$ sides of the square and comes back to the point $P$. The least possible distance traveled by the ant can be expressed as $\frac{\sqrt{a}}{b}$, where $a$ and $b$ are integers and $a$ not divisible by any square number other than $1$. What is the value of $(a+b)$?

In right $ \triangle ABC$ with legs $ 5$ and $ 12$, arcs of circles are drawn, one with center $ A$ and radius $ 12$, the other with center $ B$ and radius $ 5$. They intersect the hypotenuse at $ M$ and $ N$. Then, $ MN$ has length: [asy]size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(12,7), C=(12,0), M=12*dir(A--B), N=B+B.y*dir(B--A); real r=degrees(B); draw(A--B--C--cycle^^Arc(A,12,0,r)^^Arc(B,B.y,180+r,270)); pair point=incenter(A,B,C); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$M$", M, dir(point--M)); label("$N$", N, dir(point--N)); label("$12$", (6,0), S); label("$5$", (12,3.5), E);[/asy] $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ \frac {13}{5} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ \frac {24}{5}$

Initially, a non-constant polynomial $S(x)$ with real coefficients is written down on a board. Whenever the board contains a polynomial $P(x)$, not necessarily alone, one can write down on the board any polynomial of the form $P(C + x)$ or $C + P(x)$ where $C$ is a real constant. Moreover, if the board contains two (not necessarily distinct) polynomials $P(x)$ and $Q(x)$, one can write $P(Q(x))$ and $P(x) + Q(x)$ down on the board. No polynomial is ever erased from the board. Given two sets of real numbers, $A = \{ a_1, a_2, \dots, a_n \}$ and $B = \{ b_1, \dots, b_n \}$, a polynomial $f(x)$ with real coefficients is $(A,B)$-[i]nice[/i] if $f(A) = B$, where $f(A) = \{ f(a_i) : i = 1, 2, \dots, n \}$. Determine all polynomials $S(x)$ that can initially be written down on the board such that, for any two finite sets $A$ and $B$ of real numbers, with $|A| = |B|$, one can produce an $(A,B)$-[i]nice[/i] polynomial in a finite number of steps. [i]Proposed by Navid Safaei, Iran[/i]

$18.$ A subset $B$ of the set of first $100$ positive integers has the property that no two elements of $B$ sum to $125.$ What is the maximum possible number of elements in $B ?$

Consider a table of cyclic permutations ($n\ge2$) \[ \begin{matrix} 1, & 2, & \ldots, & n-1, & n \\ 2, & 3, & \ldots, & n, & 1, \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ n, & 1, & \ldots, & n-2, & n-1. \end{matrix} \] Then multiply each number of the first row by that number of the $k$-th row that is in the same column. Sum all these products and denote $s_k$ the result (e.g. $s_2=1\cdot2+2\cdot3+\cdots+(n-1)\cdot n+n\cdot1$). a) Find a recursive relation for $s_k$ in terms of $s_{k-1}$ and determine the explicit formula for $s_k$. b) Determine both an index $k$ and the value of $s_k$ such that the sum $s_k$ is minimal.

[u]Round 1[/u] [b]p1.[/b] Alex is on a week-long mining quest. Each morning, she mines at least $1$ and at most $10$ diamonds and adds them to her treasure chest (which already contains some diamonds). Every night she counts the total number of diamonds in her collection and finds that it is divisible by either $22$ or $25$. Show that she miscounted. [b]p2.[/b] Hermione set out a row of $11$ Bertie Bott’s Every Flavor Beans for Ron to try. There are $5$ chocolateflavored beans that Ron likes and $6$ beans flavored like earwax, which he finds disgusting. All beans look the same, and Hermione tells Ron that a chocolate bean always has another chocolate bean next to it. What is the smallest number of beans that Ron must taste to guarantee he finds a chocolate one? [b]p3.[/b] There are $101$ pirates on a pirate ship: the captain and $100$ crew. Each pirate, including the captain, starts with $1$ gold coin. The captain makes proposals for redistributing the coins, and the crew vote on these proposals. The captain does not vote. For every proposal, each crew member greedily votes “yes” if he gains coins as a result of the proposal, “no” if he loses coins, and passes otherwise. If strictly more crew members vote “yes” than “no,” the proposal takes effect. The captain can make any number of proposals, one after the other. What is the largest number of coins the captain can accumulate? [b]p4.[/b] There are $100$ food trucks in a circle and $10$ gnomes who sample their menus. For the first course, all the gnomes eat at different trucks. For each course after the first, gnome #$1$ moves $1$ truck left or right and eats there; gnome #$2$ moves $2$ trucks left or right and eats there; ... gnome #$10$ moves $10$ trucks left or right and eats there. All gnomes move at the same time. After some number of courses, each food truck had served at least one gnome. Show that at least one gnome ate at some food truck twice. [b]p5.[/b] The town of Lumenville has $100$ houses and is preparing for the math festival. The Tesla wiring company lays lengths of power wire in straight lines between the houses so that power flows between any two houses, possibly by passing through other houses.The Edison lighting company hangs strings of lights in straight lines between pairs of houses so that each house is connected by a string to exactly one other. Show that however the houses are arranged, the Edison company can always hang their strings of lights so that the total length of the strings is no more than the total length of the power wires the Tesla company used. [img]https://cdn.artofproblemsolving.com/attachments/9/2/763de9f4138b4dc552247e9316175036c649b6.png[/img] [u]Round 2[/u] [b]p6.[/b] What is the largest number of zeros that could appear at the end of $1^n + 2^n + 3^n + 4^n$, where n can be any positive integer? [b]p7.[/b] A tennis academy has $2023$ members. For every group of 1011 people, there is a person outside of the group who played a match against everyone in it. Show there is someone who has played against all $2022$ other members. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a quartet of positive numbers $a,b,c,d$, and is known, that $abcd=1$. Prove that $$a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+dc \ge 10$$

The parliament of the country Ar consists of two houses, upper and lower, both have the same number of people. The law says that each member must vote "Yes" or "No". One day, when all members of both houses were present and voted on an important issue, the speaker informed the press that the number of members voted "Yes" was greater by $23$ than the number of members voted "No". Prove that he made a mistake.

Let $n\geq 2$ be a positive integer and let $a_1,a_2,...,a_n\in[0,1]$ be real numbers. Find the maximum value of the smallest of the numbers: \[a_1-a_1a_2, \ a_2-a_2a_3,...,a_n-a_na_1.\]

Five points are given in the plane that each of $10$ triangles they define has area greater than $2$. Prove that there exists a triangle of area greater than $3$.

Ali who has $10$ candies eats at least one candy a day. In how many different ways can he eat all candies (according to distribution among days)? $ \textbf{(A)}\ 64 \qquad\textbf{(B)}\ 126 \qquad\textbf{(C)}\ 243 \qquad\textbf{(D)}\ 512 \qquad\textbf{(E)}\ 1025 $

Prove that for each $n\geq 2$, there is a set $S$ of $n$ integers such that $(a-b)^2$ divides $ab$ for every distinct $a,b\in S$.

Prove that $ 7^{2^{20}} + 7^{2^{19}} + 1 $ has at least $ 21 $ distinct prime divisors.

Find for which positive integers $n$ the $A_n$ alternating group has a permutation which is contained in exactly one $2$-Sylow subgroup of $A_n$. [i]Proposed by Péter Pál Pálfy[/i]

Find the last three digits in the product $1 \cdot 3\cdot 5\cdot 7 \cdot . . . \cdot 2009 \cdot 2011$.