Find all sets $(a, b, c)$ of different positive integers $a$, $b$, $c$, for which: [b]*[/b] $2a - 1$ is a multiple of $b$; [b]*[/b] $2b - 1$ is a multiple of $c$; [b]*[/b] $2c - 1$ is a multiple of $a$.

In a coordinate plane, consider the set of points with integer cooedinates at least one of which is not divisible by $4$. Prove that these points cannot be partitioned into pairs such that the distance between points in each pair equals $1$. In other words, an infinite chessboard, whose cells with both coordinates divisible by $4$ are cut out, cannot be tiled by dominoes.

$A, B$ and $C$ are points in the plane with integer coordinates. The lengths of the sides of triangle $ABC$ are integer numbers. Prove that the perimeter of the triangle is an even number.

For all valid values ​​of $a, b$, and $c$, solve the equation $$\frac{a (x-b) (x-c) }{(a-b) (a-c)} + \frac{b (x-c) (x-a)}{(b-c) (b-a)} +\frac{c (x-a) (x-b) }{(c-a ) (c-b)} = x^2$$

Find all functions $f:\mathbb{N}^* \rightarrow \mathbb{N}$ such that [list] [*] $f(x \cdot y) = f(x) + f(y)$ [*] $f(30) = 0$ [*] $f(x)=0$ always when the units digit of $x$ is $7$ [/list]

a) Prove that for every $\epsilon>0$ there is a positive integer $n$ and real numbers $\lambda_{1},\dots,\lambda_{n}$ such that $$\max_{x\in [-1,1]}|x-\sum_{k=1}^{n}\lambda_{k}x^{2k+1}|<\epsilon.$$ b) Prove that for every odd continuous function $f$ on $[-1,1]$ and for every $\epsilon>0$ there is a positive integer $n$ and real numbers $\mu_{1},\dots,\mu_{n}$ such that $$\max_{x\in [-1,1]}|f(x)-\sum_{k=1}^{n}\mu_{k}x^{2k+1}|<\epsilon.$$

Find all pairs of integers $x, y$ such that $y^5 + 2xy = x^2 + 2y^4.$ .

Let $F_r=x^r\sin{rA}+y^r\sin{rB}+z^r\sin{rC}$, where $x,y,z,A,B,C$ are real and $A+B+C$ is an integral multiple of $\pi$. Prove that if $F_1=F_2=0$, then $F_r=0$ for all positive integral $r$.

Find all positive integer values of $n$ such that the value of the $$\frac{2^{n!}-1}{2^n-1}$$ is a square of an integer.

Let $m$ be a positive integer, and consider a $m\times m$ checkerboard consisting of unit squares. At the centre of some of these unit squares there is an ant. At time $0$, each ant starts moving with speed $1$ parallel to some edge of the checkerboard. When two ants moving in the opposite directions meet, they both turn $90^{\circ}$ clockwise and continue moving with speed $1$. When more than $2$ ants meet, or when two ants moving in perpendicular directions meet, the ants continue moving in the same direction as before they met. When an ant reaches one of the edges of the checkerboard, it falls off and will not re-appear. Considering all possible starting positions, determine the latest possible moment at which the last ant falls off the checkerboard, or prove that such a moment does not necessarily exist. [i]Proposed by Toomas Krips, Estonia[/i]

In space, there are \( 13 \) points, no four of which lie in the same plane. Three of the points are colored blue, and the triangle with these points as vertices will be called a [i]blue triangle[/i]. The remaining \( 10 \) points are colored red. We say that a triangle with three red vertices is [i]attached[/i] to the blue triangle if the boundary of the red triangle intersects the blue triangle (either in its interior or on its boundary) at exactly one point. Is it possible for the number of attached triangles to be \( 33 \)?

Kylie is trying to count to $202250$. However, this would take way too long, so she decides to only write down positive integers from $1$ to $202250$, inclusive, that are divisible by $125$. How many times does she write down the digit $2$?

The four faces of a right triangular pyramid are equilateral triangles whose edge measures $3$ dm. Suppose the pyramid is hollow, resting on one of its faces at a horizontal surface (see attached figure) and that there is $2$ dm$^3$ of water inside. Determine the height that the liquid reaches inside the pyramid. [img]https://cdn.artofproblemsolving.com/attachments/0/7/6cd6e1077620371e56ed57d19fd3d05a58904e.png[/img]

Prove that for all positive integers $n$, all complex roots $r$ of the polynomial \[P(x) = (2n)x^{2n} + (2n-1)x^{2n-1} + \dots + (n+1)x^{n+1} + nx^n + (n+1)x^{n-1} + \dots + (2n-1)x + 2n\] lie on the unit circle (i.e. $|r| = 1$).

An acute angle $A$ and a point $E$ inside it are given. Construct points $B, C$ on the sides of the angle such that $E$ is the center of the Euler circle of triangle $ABC$. (E. Diomidov)

Acute triangle $ABC$ with $AB>BC$ is inscribed in circle $\Omega$. Points $D$ and $E$, that lie on $(BC)$ and $(AB)$ are the feet of altitudes from $A$ and $C$ in triangle $ABC$, and $M$ is the midpoint of the segment $DE$. Half-line $(AM$ intersects the circle $\Omega$ for the second time in $N$. Show that the circumcenter of triangle $MDN$ lies on the line $BC$.

Edges of a complete graph with $2m$ vertices are properly colored with $2m-1$ colors. It turned out that for any two colors all the edges colored in one of these two colors can be described as union of several $4$-cycles. Prove that $m$ is a power of $2$.

Let $A_1A_2A_3A_4A_5$ be a regular pentagon inscribed in a circle with area $\tfrac{5+\sqrt{5}}{10}\pi$. For each $i=1,2,\dots,5$, points $B_i$ and $C_i$ lie on ray $\overrightarrow{A_iA_{i+1}}$ such that \[B_iA_i \cdot B_iA_{i+1} = B_iA_{i+2} \quad \text{and} \quad C_iA_i \cdot C_iA_{i+1} = C_iA_{i+2}^2\]where indices are taken modulo 5. The value of $\tfrac{[B_1B_2B_3B_4B_5]}{[C_1C_2C_3C_4C_5]}$ (where $[\mathcal P]$ denotes the area of polygon $\mathcal P$) can be expressed as $\tfrac{a+b\sqrt{5}}{c}$, where $a$, $b$, and $c$ are integers, and $c > 0$ is as small as possible. Find $100a+10b+c$. [i]Proposed by Robin Park[/i]

For a given positive integer $n\ge 2$, suppose positive integers $a_i$ where $1\le i\le n$ satisfy $a_1<a_2<\ldots <a_n$ and $\sum_{i=1}^n \frac{1}{a_i}\le 1$. Prove that, for any real number $x$, the following inequality holds \[\left(\sum_{i=1}^n\frac{1}{a_i^2+x^2}\right)^2\le\frac{1}{2}\cdot\frac{1}{a_1(a_1-1)+x^2} \] [i]Li Shenghong[/i]

Let $a$ and $b$ be two natural numbers that have an equal number $n$ of digits in their decimal expansions. The first $m$ digits (from left to right) of the numbers $a$ and $b$ are equal. Prove that if $m >\frac{n}{2},$ then $a^{\frac{1}{n}} -b^{\frac{1}{n}} <\frac{1}{n}$

Let $k<<n$ denote that $k<n$ and $k\mid n$. Let $f:\{1,2,...,2013\}\rightarrow \{1,2,...,M\}$ be such that, if $n\leq 2013$ and $k<<n$, then $f(k)<<f(n)$. What’s the least possible value of $M$?

[b]4.[/b] Let $f(x)$ be a real- or complex-value integrable function on $(0,1)$ with $\mid f(x) \mid \leq 1 $. Set $ c_k = \int_0^1 f(x) e^{-2 \pi i k x} dx $ and construct the following matrices of order $n$: $ T= (t_{pq})_{p,q=0}^{n-1}, T^{*}= (t_{pq}^{*})_{p,q =0}^{n-1} $ where $t_{pq}= c_{q-p}, t^{*}= \overline {c_{p-q}}$ . Further, consider the following hyper-matrix of order $m$: $ S= \begin{bmatrix} E & T & T^2 & \dots & T^{m-2} & T^{m-1} \\ T^{*} & E & T & \dots & T^{m-3} & T^{m-2} \\ T^{*2} & T^{*} & E & \dots & T^{m-3} & T^{m-2} \\ \dots & \dots & \dots & \dots & \dots & \dots \\ T^{*m-1} & T^{*m-2} & T^{*m-3} & \dots & T^{*} & E \end{bmatrix} $ ($S$ is a matrix of order $mn$ in the ordinary sense; E denotes the unit matrix of order $n$). Show that for any pair $(m , n) $ of positive integers, $S$ has only non-negative real eigenvalues. [b](R. 19)[/b]

We will say that a natural number is [i]acceptable[/i] if it has at most $9$ distinct prime divisors. There is a stack of $100!=1\times2\times\cdots\times100$ stones. A [i]legal move[/i] consists in removing $k$ stones from the stack, where $k$ is an acceptable number. Two players, Lucas and Nicolas, take turns making legal moves; Lucas starts the game. The one who removes the last stone wins. Determine which of the players has a winning strategy and describe this strategy.

A group of men, some of them accompanied by their wives, spent $\$1.000$ on a hotel. Each man spent $\$19$ and each woman $\$13$. Determine how many women and how many men there were.

Some figures stand in certain cells of a chess board. It is known that a figure stands on each row, and that different rows have a different number of figures. Prove that it is possible to mark $8$ figures so that on each row and column stands exactly one marked figure.