Let there be $131$ given distinct natural numbers, each having prime divisors not exceeding $42$. Prove that one can choose four of them whose product is a perfect square.

You wish to color every vertex, edge, face, and the interior of a cube one color each such that no two adjacent objects are the same color. Faces are adjacent if they share an edge. Edges are adjacent if they share a vertex. The interior is adjacent to all of its faces, edges, and vertices. Each face is adjacent to all of its edges and vertices. Each edge is adjacent to both of its vertices. What is the minimum number of colors required to do this?

Consider the expanded form of $\left(x+\frac{1}{2\sqrt[4]{x}}\right)^n$, put all items in number (from high power to low power). If the coefficients of the first three items are arithmetic sequence, then the number of items with an integral power is________.

Let $n\ge 3$ be a given integer. Nonzero real numbers $a_1,..., a_n$ satisfy: $\frac{-a_1-a_2+a_3+...a_n}{a_1}=\frac{a_1-a_2-a_3+a_4+...a_n}{a_2}=...=\frac{a_1+...+a_{n-2}-a_{n-1}-a_n}{a_{n-1}}=\frac{-a_1+a_2+...+a_{n-1}-a_n}{a_{n}}$ What values can be taken by the product $\frac{a_2+a_3+...a_n}{a_1}\cdot \frac{a_1+a_3+a_4+...a_n}{a_2}\cdot ...\cdot \frac{+a_1+a_2+...+a_{n-1}}{a_{n}}$ ?

A deck consists of $2^n$ cards. The deck is shuffled using the following operation: if the cards are initially in the order $a_1,a_2,a_3,a_4,...,a_{2^n-1},a_{2^n}$ then after shuffling the order becomes $a_{2^{n-1}+1},a_1,a_{2^{n-1}+2},a_2,...,a_{2^n},a_{2^{n-1}}$ . Find the smallest number of such operations after which the original order of the cards is restored. (R. Palm)

Find the total number of times the digit ‘$2$’ appears in the set of integers $\{1,2,..,1000\}$. For example, the digit ’$2$’ appears twice in the integer $229$.

Determine a right parallelepiped with minimal area, if its volume is strictly greater than $1000$, and the lengths of it sides are integer numbers.

A polynomial with integer coefficients when divided by $x^2-12x+11$ gives the remainder $990x-889$. Prove that the polynomial has no integer roots.

Diagonal sections of a regular 8-gon pyramid, which are drawn through the smallest and largest diagonals of the base, are equal. At what angle is the plane passing through the vertex, the pyramids and the smallest diagonal of the base inclined to the base? [hide=original wording]Діагональні перерізи правильної 8-кутної піраміди, які Проведені через найменшу і найбільшу діагоналі основи, рівновеликі. Під яким кутом до основи нахилена площина, що проходить через вершину, піраміди і найменшу діагональ основи?[/hide]

Find the maximal positive integer $n$, so that for any real number $x$ we have $\sin^{n}{x}+\cos^{n}{x} \geq \frac{1}{n}$.

Find the minimum value of $\int_0^{\frac{\pi}{2}} |\cos x -a|\sin x \ dx$

Natural numbers $1,2,...,500$ are written on a blackboard. Two players $A$ and $B$ consecutively make moves, $A$ starts. Each move a player chooses two numbers $n$ and $2n$ and erases them from the blackboard. If a player cannot perform a valid move, he loses. Which player can guarantee a win?

On an $8 \times 8$ chessboard, a rook stands on the bottom left corner square. We want to move it to the upper right corner, subject to the following rules: we have to move the rook exactly $9$ times, such that the length of each move is either $3$ or $4$. (It is allowed to mix the two lengths throughout the "journey".) How many ways are there to do this? In each move, the rook moves horizontally or vertically.

A [i]binary string[/i] is a sequence, each of whose terms is $0$ or $1$. A set $\mathcal{B}$ of binary strings is defined inductively according to the following rules. [list] [*]The binary string $1$ is in $\mathcal{B}$.[/*] [*]If $s_1,s_2,\dotsc ,s_n$ is in $\mathcal{B}$ with $n$ odd, then both $s_1,s_2,\dotsc ,s_n,0$ and $0,s_1,s_2,\dotsc ,s_n$ are in $\mathcal{B}$.[/*] [*]If $s_1,s_2,\dotsc ,s_n$ is in $\mathcal{B}$ with $n$ even, then both $s_1,s_2,\dotsc ,s_n,1$ and $1,s_1,s_2,\dotsc ,s_n$ are in $\mathcal{B}$.[/*] [*]No other binary strings are in $\mathcal{B}$.[/*] [/list] For each positive integer $n$, let $b_n$ be the number of binary strings in $\mathcal{B}$ of length $n$. [list=a] [*]Prove that there exist constants $c_1,c_2>0$ and $1.6<\lambda_1,\lambda_2<1.9$ such that $c_1\lambda_1^n<b_n<c_2\lambda_2^n$ for all positive integer $n$.[/*] [*]Determine $\liminf_{n\to \infty} {\sqrt[n]{b_n}}$ and $\limsup_{n\to \infty} {\sqrt[n]{b_n}}$[/*] [/list] [i]Note: The problem is open in the sense that no solution is currently known to part (b).[/i]

Let $X$ be a non-empty set of positive integers which satisfies the following: [list] [*] if $x \in X$, then $4x \in X$, [*] if $x \in X$, then $\lfloor \sqrt{x}\rfloor \in X$. [/list] Prove that $X=\mathbb{N}$.

Prove that : $\left|\sum_{i=0}^n \left(1-\pi \sin \frac{i\pi}{4n}\cos \frac{i\pi}{4n}\right)\right|<1.$

Let $a_n$ be the $n$th positive integer such that when $n$ is written in base $3$, the sum of the digits of $n$ is divisible by $3$. For example, $a_1 = 5$ because $5 = 12_3$. Compute $a_{2016}$.

Source: 2018 Canadian Open Math Challenge Part C Problem 2 ----- Alice has two boxes $A$ and $B$. Initially box $a$ contains $n$ coins and box $B$ is empty. On each turn, she may either move a coin from box $a$ to box $B$, or remove $k$ coins from box $A$, where $k$ is the current number of coins in box $B$. She wins when box $A$ is empty. $\text{(a)}$ If initially box $A$ contains 6 coins, show that Alice can win in 4 turns. $\text{(b)}$ If initially box $A$ contains 31 coins, show that Alice cannot win in 10 turns. $\text{(c)}$ What is the minimum number of turns needed for Alice to win if box $A$ initially contains 2018 coins?

Let $n \geq 2$ be a positive integer. Consider the following equation: \[ \{x\}+\{2x\}+ \dots + \{nx\} = \lfloor x \rfloor + \lfloor 2x \rfloor + \dots + \lfloor 2nx \rfloor\] a) For $n=2$, solve the given equation in $\mathbb{R}$. b) Prove that, for any $n \geq 2$, the equation has at most $2$ real solutions.

An equilateral triangle is dissected into white and black triangles. It is known that all white triangles are right-angled and mutually congruent, and all black triangles are isosceles and also mutually congruent. Is it necessarily true that a) all angles of white triangles are multiples of $30^{\circ}$; (4 marks) b) all angles of black triangles are multiples of $30^{\circ}$ ? (5 marks)

Let $k\geq 4$ be an integer number. $P(x)\in\mathbb{Z}[x]$ such that $0\leq P(c)\leq k$ for all $c=0,1,...,k+1$. Prove that $P(0)=P(1)=...=P(k+1)$.

Let $a, b, c$, and $d$ be four distinct integers. Prove that $(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$ is divisible by $12$.

Find $A=x^5+y^5+z^5$ if $x+y+z=1$, $x^2+y^2+z^2=2$ and $x^3+y^3+z^3=3$.

[b]p1.[/b] What is the smallest positive integer $x$ such that $\frac{1}{x} <\sqrt{12011} - \sqrt{12006}$? [b]p2. [/b] Two soccer players run a drill on a $100$ foot by $300$ foot rectangular soccer eld. The two players start on two different corners of the rectangle separated by $100$ feet, then run parallel along the long edges of the eld, passing a soccer ball back and forth between them. Assume that the ball travels at a constant speed of $50$ feet per second, both players run at a constant speed of $30$ feet per second, and the players lead each other perfectly and pass the ball as soon as they receive it, how far has the ball travelled by the time it reaches the other end of the eld? [b]p3.[/b] A trapezoid $ABCD$ has $AB$ and $CD$ both perpendicular to $AD$ and $BC =AB + AD$. If $AB = 26$, what is $\frac{CD^2}{AD+CD}$ ? [b]p4.[/b] A hydrophobic, hungry, and lazy mouse is at $(0, 0)$, a piece of cheese at $(26, 26)$, and a circular lake of radius $5\sqrt2$ is centered at $(13, 13)$. What is the length of the shortest path that the mouse can take to reach the cheese that also does not also pass through the lake? [b]p5.[/b] Let $a, b$, and $c$ be real numbers such that $a + b + c = 0$ and $a^2 + b^2 + c^2 = 3$. If $a^5 + b^5 + c^5\ne 0$, compute $\frac{(a^3+b^3+c^3)(a^4+b^4+c^4)}{a^5+b^5+c^5}$. [b]p6. [/b] Let $S$ be the number of points with integer coordinates that lie on the line segment with endpoints $\left( 2^{2^2}, 4^{4^4}\right)$ and $\left(4^{4^4}, 0\right)$. Compute $\log_2 (S - 1)$. [b]p7.[/b] For a positive integer $n$ let $f(n)$ be the sum of the digits of $n$. Calculate $$f(f(f(2^{2006})))$$ [b]p8.[/b] If $a_1, a_2, a_3, a_4$ are roots of $x^4 - 2006x^3 + 11x + 11 = 0$, find $|a^3_1 + a^3_2 + a^3_3 + a^3_4|$. [b]p9.[/b] A triangle $ABC$ has $M$ and $N$ on sides $BC$ and $AC$, respectively, such that $AM$ and $BN$ intersect at $P$ and the areas of triangles $ANP$, $APB$, and $PMB$ are $5$, $10$, and $8$ respectively. If $R$ and $S$ are the midpoints of $MC$ and $NC$, respectively, compute the area of triangle $CRS$. [b]p10.[/b] Jack's calculator has a strange button labelled ''PS.'' If Jack's calculator is displaying the positive integer $n$, pressing PS will cause the calculator to divide $n$ by the largest power of $2$ that evenly divides $n$, and then adding 1 to the result and displaying that number. If Jack randomly chooses an integer $k$ between $ 1$ and $1023$, inclusive, and enters it on his calculator, then presses the PS button twice, what is the probability that the number that is displayed is a power of $2$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all positive integers $n$ for which all coefficients of polynomial $P(x)$ are divisible by $7,$ where \[P(x) = (x^2 + x + 1)^n - (x^2 + 1)^n - (x + 1)^n - (x^2 + x)^n + x^{2n} + x^n + 1.\]