Prove that the polynomial$$P(x)=(x-1)(x-2)(x-3)-1$$is irreducible in $\mathbb{Z}[x].$

$f$ is a polynomial of degree $n$ with integer coefficients and $f(x)=x^2+1$ for $x=1,2,\cdot ,n$. What are the possible values for $f(0)$?

Let $f:\mathbb R\to\mathbb R$ satisfy \[|f(x+y)-f(x-y)-y|\leq y^2\] For all $(x,y)\in\mathbb R^2.$ Show that $f(x)=\frac x2+c$ where $c$ is a constant.

A $2003\times 2004$ rectangle consists of unit squares. We consider rhombi formed by four diagonals of unit squares. What maximum number of such rhombi can be arranged in this rectangle so that no two of them have any common points except vertices? [i]Proposed by A. Golovanov[/i]

An infinite geometric sequence $a_1,a_2,a_3,\dots$ satisfies $a_1=1$ and \[\dfrac{1}{a_1a_2}+\dfrac{1}{a_2a_3}+\dfrac{1}{a_3a_4}\cdots=\dfrac{1}{2}.\] The sum of all possible values of $a_2$ can be expressed as $m+\sqrt{n}$, where $m$ and $n$ are integers and $n$ is not a positive perfect square. Find $100m+n$. [i]Individual #7[/i]

Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that for any positive integers $m, n$ the number $$(f(m))^2+ 2mf(n) + f(n^2)$$ is the square of an integer. [i]Proposed by Fedir Yudin[/i]

Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f : \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that, for all $m, n \in \mathbb{Z}_{>0 }$: $$f(mf(n)) + f(n) | mn + f(f(n)).$$

Secco and Ramon are drunk in the real line over the integer points $a$ and $b$, respectively. Our real line is a little bit special, though: the interval $(-\infty, 0)$ is covered by a sea of lava. Being aware of this fact, and also because they are drunk, they decided to play the following game: initially they choose an integer number $k>1$ using a fair dice as large as desired, and therefore they start the game. In the first round, each player writes the point $h$ for which it wants to go. After that, they throw a coin: if the result is heads, they go to the desired points; otherwise, they go to the points $2g - h$, where $g$ is the point where each of the players were in the precedent round (that is, in the first round $g = a$ for Secco and $g = b$ for Ramon). They repeat this procedure in the other rounds, and the game finishes when some of the player is over a point exactly $k$ times bigger than the other (if both of the player end up in the point $0$, the game finishes as well). Determine, in values of $k$, the initial values $a$ and $b$ such that Secco and Ramon has a winning strategy to finish the game alive. [i]Observation: If any of the players fall in the lave, he dies and both of them lose the game[/i]

$D$ is midpoint of $AC$ for $\triangle ABC$. Bisectors of $\angle ACB,\angle ABD$ are perpendicular. Find max value for $\angle BAC$ [i](S. Berlov)[/i]

Let $P$ be a set of $n$ points and $S$ a set of $l$ segments. It is known that: $(i)$ No four points of $P$ are coplanar. $(ii)$ Any segment from $S$ has its endpoints at $P$. $(iii)$ There is a point, say $g$, in $P$ that is the endpoint of a maximal number of segments from $S$ and that is not a vertex of a tetrahedron having all its edges in $S$. Prove that $l \leq \frac{n^2}{3}$

$m$ and $n$ are positive integers. Set $A=\{ 1, 2, \cdots, n \}$. Let set $B_{n}^{m}=\{ (a_1, a_2 \cdots, a_m) \mid a_i \in A, i= 1, 2, \cdots, m \}$ satisfying: (1) $|a_i - a_{i+1}| \neq n-1$, $i=1,2, \cdots, m-1$; and (2) at least three of $a_1, a_2, \cdots, a_m$ ($m \geq 3$) are pairwise distince. Find $|B_n^m|$ and $|B_6^3|$.

Find all functions $ f:\mathbb{Q}\longrightarrow\mathbb{R} $ satisfying the equation $$ f(x+y)+f(x-y)=f(x)+f(y) +f(f(x+y)) , $$ for any rational numbers $ x,y. $ [i]Mihai Onucu Drîmbe[/i]

Let $a,b>1$ be odd positive integers. A board with $a$ rows and $b$ columns without fields $(2,1),(a-2,b)$ and $(a,b)$ is tiled with $2\times 2$ squares and $2\times 1$ dominoes (that can be rotated). Prove that the number of dominoes is at least $$\frac{3}{2}(a+b)-6.$$

Let $\{a,b,c,d,e,f,g,h,i\}$ be a permutation of $\{1,2,3,4,5,6,7,8,9\}$ such that $\gcd(c,d)=\gcd(f,g)=1$ and \[(10a+b)^{c/d}=e^{f/g}.\] Given that $h>i$, evaluate $10h+i$. [i]Proposed by James Lin[/i]

Two three-dimensional objects are said to have the same coloring if you can orient one object (by moving or turning it) so that it is indistinguishable from the other. For example, suppose we have two unit cubes sitting on a table, and the faces of one cube are all black except for the top face which is red, and the the faces of the other cube are all black except for the bottom face, which is colored red. Then these two cubes have the same coloring. In how many different ways can you color the edges of a regular tetrahedron, coloring two edges red, two edges black, and two edges green? (A regular tetrahedron has four faces that are each equilateral triangles. The figure below depicts one coloring of a tetrahedron, using thick, thin, and dashed lines to indicate three colors.)

Prove that we can color every subset with $n$ element of a set with $3n$ elements with $8$ colors . In such a way that there are no $3$ subsets $A,B,C$ with the same color where : $$|A \cap B| \le 1,|A \cap C| \le 1,|B \cap C| \le 1$$ Proposed by [i]Morteza Saghafian[/i] and [i]Amir Jafari[/i]

The expressions $ a\plus{}bc$ and $ (a\plus{}b)(a\plus{}c)$ are: $ \textbf{(A)}\ \text{always equal} \qquad \textbf{(B)}\ \text{never equal} \qquad \textbf{(C)}\ \text{equal whenever }a\plus{}b\plus{}c\equal{}1 \\ \textbf{(D)}\ \text{equal when }a\plus{}b\plus{}c\equal{}0 \qquad \textbf{(E)}\ \text{equal only when }a\equal{}b\equal{}c\equal{}0$

Let $f$ be the unique polynomial of degree at most $2026$ such that for all $n \in \{1,2, 3, \ldots, 2027\},$ $$f(n)=\begin{cases} 1 & \text{if } $n$ \text{ is a perfect square}, \\ 0 & \text{otherwise.} \end{cases}$$ Suppose that $\tfrac{a}{b}$ is the coefficient of $x^{2025}$ in $f,$ where $a$ and $b$ are integers such that $\gcd(a,b)=1.$ Compute the unique integer $r$ between $0$ and $2026$ (inclusive) such that $a-rb$ is divisible by $2027.$ (Note that $2027$ is prime.)

In the country of PUMACsboro, there are $n$ distinct cities labelled $1$ through $n$. There is a rail line going from city $i$ to city $j$ if and only if $i<j$; you can only take this rail line from city $i$ to city $j$. What is the smallest possible value of $n$ such that if each rail line's track is painted orange or black, you can always take the train between $2019$ cities on tracks that are all the same color? (This means there are some cities $c_1,c_2,\dots,c_{2019}$ such that there is a rail line going from city $c_i$ to $c_{i+1}$ for all $1\leq i\leq 2018$ and their rail lines' tracks are either all orange or all black.)

Paul is in the desert and has a pile of gypsum crystals. No matter how he divides the pile into two nonempty piles, at least one of the resulting piles has a number of crystals that, when written in base $10,$ has a sum of digits at least $7.$ Given that Paul’s initial pile has at least two crystals, compute the smallest possible number of crystals in the initial pile.

For how many unordered sets $\{a,b,c,d\}$ of positive integers, none of which exceed $168$, do there exist integers $w,x,y,z$ such that $(-1)^wa+(-1)^xb+(-1)^yc+(-1)^zd=168$? If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\left\lfloor25e^{-3\frac{|C-A|}C}\right\rfloor$.

In a country, mathematicians chose an $\alpha> 2$ and issued coins in denominations of 1 ruble, as well as $\alpha ^k$ rubles for each positive integer k. $\alpha$ was chosen so that the value of each coins, except the smallest, was irrational. Is it possible that any natural number of rubles can be formed with at most 6 of each denomination of coins?

Prove that from a set of ten distinct two-digit numbers, it is always possible to find two disjoint subsets whose members have the same sum.

Given natural number $k$ with a property "if $n$ is divisible by $k$, than the number, obtained from $n$ by reversing the order of its digits is also divisible by $k$". Prove that the $k$ is a divisor of $99$.

Let triangle $\triangle{ABC}$ have $AB=90$ and $AC=66$. Suppose that the line $IG$ is perpendicular to side $BC$, where $I$ and $G$ are the incenter and centroid, respectively. Find the length of $BC$.