There are$n$ balls numbered $1,2,\cdots,n$, respectively. They are painted with $4$ colours, red, yellow, blue, and green, according to the following rules: First, randomly line them on a circle. Then let any three clockwise consecutive balls numbered $i, j, k$, in order. 1) If $i>j>k$, then the ball $j$ is painted in red; 2) If $i<j<k$, then the ball $j$ is painted in yellow; 3) If $i<j, k<j$, then the ball $j$ is painted in blue; 4) If $i>j, k>j$, then the ball $j$ is painted in green. And now each permutation of the balls determine a painting method. We call two painting methods distinct, if there exists a ball, which is painted with two different colours in that two methods. Find out the number of all distinct painting methods.

Kelvin and Quinn are collecting trading cards; there are $6$ distinct cards that could appear in a pack. Each pack contains exactly one card, and each card is equally likely. Kelvin buys packs until he has at least one copy of every card, and then he stops buying packs. If Quinn is missing exactly one card, the probability that Kelvin has at least two copies of the card Quinn is missing is expressible as $\tfrac{m}{n}$ for coprime positive integers $m$ and $n$. Determine $m+n$.

Given an $ n\times n$ chessboard. a) Find the number of rectangles on the chessboard. b) Assume there exists an $ r\times r$ square (label $ B$) with $ r<n$ which is located on the upper left corner of the board. Define "inner border" of $ A$ as the border of $ A$ which is not the border of the chessboard. How many rectangles in $ B$ that touch exactly one inner border of $ B$?

We call a system of non-empty sets $H$ [i]entwined[/i], if for every disjoint pair of sets $A$ and $B$ in $H$ there exists $b\in B$ such that $A\cup\{b\}$ is in $H$ or there exists $a\in A$ such that $B\cup\{a\}$ is in $H.$ Let $H$ be an entwined system of sets containing all of $\{1\},\{2\},\ldots,\{n\}.$ Prove that if $n>k(k+1)/2,$ then $H$ contains a set with at least $k+1$ elements, and this is sharp for every $k,$ i.e. if $n=k(k+1),$ it is possible that every set in $H$ has at most $k$ elements.

Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties: [list=disc] [*]every term in the sequence is less than or equal to $2^{2023}$, and [*]there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\] [/list]

Let \( x \) be a real number. Determine the solution to the following equation: \[ \frac{x^2 + 1}{1} + \frac{x^2 + 2}{2} + \frac{x^2 + 3}{3} + \ldots + \frac{x^2 + 2024}{2024} = 2024 \]

Let $a_1 \geq a_2 \geq \cdots \geq a_n$ be $n$ real numbers such that $a_1^k +a_2^k + \cdots + a_n^k \geq 0$ for all positive integers $k$. Suppose that $p=\max\{|a_1|,|a_2|, \ldots,|a_n|\}$. Prove that $p=a_1$, and \[(x-a_1)(x-a_2)\cdots(x-a_n)\leq x^n-a_1^n \qquad \forall x>a_1.\]

[b]p1.[/b] Roll two dice. What is the probability that the sum of the rolls is prime? [b]p2. [/b]Compute the sum of the first $20$ squares. [b]p3.[/b] How many integers between $0$ and $999$ are not divisible by $7, 11$, or $13$? [b]p4.[/b] Compute the number of ways to make $50$ cents using only pennies, nickels, dimes, and quarters. [b]p5.[/b] A rectangular prism has side lengths $1, 1$, and $2$. What is the product of the lengths of all of the diagonals? [b]p6.[/b] What is the last digit of $7^{6^{5^{4^{3^{2^1}}}}}$ ? [b]p7.[/b] Given square $ABCD$ with side length $3$, we construct two regular hexagons on sides $AB$ and $CD$ such that the hexagons contain the square. What is the area of the intersection of the two hexagons? [img]https://cdn.artofproblemsolving.com/attachments/f/c/b2b010cdd0a270bc10c6e3bb3f450ba20a03e7.png[/img] [b]p8.[/b] Brooke is driving a car at a steady speed. When she passes a stopped police officer, she begins decelerating at a rate of $10$ miles per hour per minute until she reaches the speed limit of $25$ miles per hour. However, when Brooke passed the police officer, he immediately began accelerating at a rate of $20$ miles per hour per minute until he reaches the rate of $40$ miles per hour. If the police officer catches up to Brooke after 3 minutes, how fast was Brooke driving initially? [b]p9.[/b] Find the ordered pair of positive integers $(x, y)$ such that $144x - 89y = 1$ and $x$ is minimal. [b]p10.[/b] How many zeroes does the product of the positive factors of $10000$ (including $1$ and $10000$) have? [b]p11.[/b] There is a square configuration of desks. It is known that one can rearrange these desks such that it has $7$ fewer rows but $10$ more columns, with $13$ desks remaining. How many desks are there in the square configuration? [b]p12.[/b] Given that there are $168$ primes with $3$ digits or less, how many numbers between $1$ and $1000$ inclusive have a prime number of factors? [b]p13.[/b] In the diagram below, we can place the integers from $1$ to $19$ exactly once such that the sum of the entries in each row, in any direction and of any size, is the same. This is called the magic sum. It is known that such a configuration exists. Compute the magic sum. [img]https://cdn.artofproblemsolving.com/attachments/3/4/7efaa5ba5ad250e24e5ad7ef03addbf76bcfb4.png[/img] [b]p14.[/b] Let $E$ be a random point inside rectangle $ABCD$ with side lengths $AB = 2$ and $BC = 1$. What is the probability that angles $ABE$ and $CDE$ are both obtuse? [b]p15.[/b] Draw all of the diagonals of a regular $13$-gon. Given that no three diagonals meet at points other than the vertices of the $13$-gon, how many intersection points lie strictly inside the $13$-gon? [b]p16.[/b] A box of pencils costs the same as $11$ erasers and $7$ pencils. A box of erasers costs the same as $6$ erasers and a pencil. A box of empty boxes and an eraser costs the same as a pencil. Given that boxes cost a penny and each of the boxes contain an equal number of objects, how much does it costs to buy a box of pencils and a box of erasers combined? [b]p17.[/b] In the following figure, all angles are right angles and all sides have length $1$. Determine the area of the region in the same plane that is at most a distance of $1/2$ away from the perimeter of the figure. [img]https://cdn.artofproblemsolving.com/attachments/6/2/f53ae3b802618703f04f41546e3990a7d0640e.png[/img] [b]p18.[/b] Given that $468751 = 5^8 + 5^7 + 1$ is a product of two primes, find both of them. [b]p19.[/b] Your wardrobe contains two red socks, two green socks, two blue socks, and two yellow socks. It is currently dark right now, but you decide to pair up the socks randomly. What is the probability that none of the pairs are of the same color? [b]p20.[/b] Consider a cylinder with height $20$ and radius $14$. Inside the cylinder, we construct two right cones also with height $20$ and radius $14$, such that the two cones share the two bases of the cylinder respectively. What is the volume ratio of the intersection of the two cones and the union of the two cones? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For all real numbers $x$, let \[ f(x) = \frac{1}{\sqrt[2011]{1-x^{2011}}}. \] Evaluate $(f(f(\ldots(f(2011))\ldots)))^{2011}$, where $f$ is applied $2010$ times.

If $a$, $b$, $c>0$ and $abc=1$, then $$\sum \limits^{cyc} \frac{a+b+c^n}{a^{2n+3}+b^{2n+3}+ab} \leq a^{n+1}+b^{n+1}+c^{n+1}$$ for all $n\in \mathbb{N}$

In a mathematics competition, the sum of the scores of Bill and Dick equalled the sum of the scores of Ann and Carol. If the scores of Bill and Carol had been interchanged, then the sum of the scores of Ann and Carol would have exceeded the sum of the scores of the other two. Also, Dick's score exceeded the sum of the scores of Bill and Carol. Determine the order in which the four contestants finished, from highest to lowest. Assume all scores were nonnegative. $ \textbf{(A)}\ \text{Dick, Ann, Carol, Bill} \qquad\textbf{(B)}\ \text{Dick, Ann, Bill, Carol} \qquad\textbf{(C)}\ \text{Dick, Carol, Bill, Ann} \\ \qquad\textbf{(D)}\ \text{Ann, Dick, Carol, Bill} \qquad\textbf{(E)}\ \text{Ann, Dick, Bill, Carol} $

Some unit cubes are stacked atop a flat 4 by 4 square. The figures show views of the stacks from two different sides. Find the maximum and minimum number of cubes that could be in the stacks. Also give top views of a maximum arrangement and a minimum arrangement with each stack marked with its height. [asy] string s = "1010101010111111"; defaultpen(linewidth(0.7)); for(int x=0;x<4;++x) { for(int y=0;y<4;++y) { if(hex(substr(s,4*(3-y)+x,1))==1) { draw((x,y)--(x,y+1)--(x+1,y+1)--(x+1,y)--cycle); } }} label("South View",(2,4),N); s = "0101110111111111"; for(int x=0;x<4;++x) { for(int y=0;y<4;++y) { if(hex(substr(s,4*(3-y)+x,1))==1) { x=x+5; draw((x,y)--(x,y+1)--(x+1,y+1)--(x+1,y)--cycle); x=x-5; } }} label("East View",(7,4),N);[/asy]

Suppose that both $x^{3}-x$ and $x^{4}-x$ are integers for some real number $x$. Show that $x$ is an integer.

Rectangle $ABCD$ is divided into four parts of equal area by five segments as shown in the figure, where $XY = YB + BC + CZ = ZW = WD + DA + AX$, and $PQ$ is parallel to $AB$. Find the length of $AB$ (in cm) if $BC = 19$ cm and $PQ = 87$ cm. [asy] size(250); pair A=origin, B=(96,0), C=(96,22), D=(0,22), W=(16,22), X=(20,0), Y=(80,0), Z=(76,22), P=(24,11), Q=(72,11); draw(P--X--A--D--W--P--Q--Y--B--C--Z--Q^^W--Z^^X--Y); dot(A^^B^^C^^D^^P^^Q^^W^^X^^Y^^Z); pair point=(48,11); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, NE); label("$D$", D, NW); label("$P$", P, dir(point--P)); label("$Q$", Q, dir(point--Q)); label("$W$", W, N); label("$X$", X, S); label("$Y$", Y, S); label("$Z$", Z, N);[/asy]

The number of values of $x$ satisfying the equation \[ \frac{2x^2-10x}{x^2-5x}=x-3 \]is: $\text{(A)} \ \text{zero} \qquad \text{(B)} \ \text{one} \qquad \text{(C)} \ \text{two} \qquad \text{(D)} \ \text{three} \qquad \text{(E)} \ \text{an integer greater than 3}$

A board $64$ inches long and $4$ inches high is inclined so that the long side of the board makes a $30$ degree angle with the ground. The distance from the ground to the highest point on the board can be expressed in the form $a+b\sqrt{c}$ where $a,b,c$ are positive integers and $c$ is not divisible by the square of any prime. What is $a+b+c$? [i]Author: Ray Li[/i] [hide="Clarification"]The problem is intended to be a two-dimensional problem. The board's dimensions are 64 by 4. The long side of the board makes a 30 degree angle with the ground. One corner of the board is touching the ground.[/hide]

Given a square $ABCD$ whose side length is $1$, $P$ and $Q$ are points on the sides $AB$ and $AD$. If the perimeter of $APQ$ is $2$ find the angle $PCQ$.

The integer $111111$ is the product of five prime numbers. Determine the sum of these prime numbers.

Find the relations among the angles of the triangle $ABC$ whose altitude $AH$ and median $AM$ satisfy $\angle BAH =\angle CAM$.

Let $p(x) = (x- x_1)(x- x_2)(x- x_3)$, where $x_1, x_2$ and $x_3$ are real. Show that $p(x) p''(x) \le p'(x)^2$ for all $x$.

Define the sequence $\{a_n\}_{n=1}^\infty$ as \[ a_1 = a_2 = 1,\quad a_{n+2} = 14a_{n+1} - a_n \; (n \geq 1) \] Prove that if $p$ is prime and there exists a positive integer $n$ such that $\frac{a_n}p$ is an integer, then $\frac{p-1}{12}$ is also an integer.

Solve in the real numbers the equation $ 2^{1+x} =2^{[x]} +2^{\{x\}} , $ where $ [],\{\} $ deonotes the ineger and fractional part, respectively. [i]Aurel Bârsan[/i]

The sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{1}=1, \; a_{n+1}=2a_{n}+\sqrt{3a_{n}^{2}+1}.\] Show that $a_{n}$ is an integer for every $n$.

An acute $\triangle ABC$ with circumcircle $\Gamma$ is given. $I$ and $I_a$ are the incenter and $A-$excenter of $\triangle ABC$ respectively. The line $AI$ intersects $\Gamma$ again at $D$ and $A'$ is the antipode of $A$ with respect to $\Gamma$. $X$ and $Y$ are point on $\Gamma$ such that $\angle IXD = \angle I_aYD = 90^\circ$. The tangents to $\Gamma$ at $X$ and $Y$ intersect in point $Z$. Prove that $A', D$ and $Z$ are collinear. [i]Proposed by Nikola Velov[/i]

Find all function $ f: R^\plus{} \rightarrow R^\plus{}$ such that for any $ x,y,z \in R^\plus{}$ such that $ x\plus{}y \ge z$ , $ f(x\plus{}y\minus{}z) \plus{}f(2\sqrt{xz})\plus{}f(2\sqrt{yz}) \equal{} f(x\plus{}y\plus{}z)$