Assume that the set of all positive integers is decomposed into $ r$ (disjoint) subsets $ A_1 \cup A_2 \cup \ldots \cup A_r \equal{} \mathbb{N}.$ Prove that one of them, say $ A_i,$ has the following property: There exists a positive $ m$ such that for any $ k$ one can find numbers $ a_1, a_2, \ldots, a_k$ in $ A_i$ with $ 0 < a_{j \plus{} 1} \minus{} a_j \leq m,$ $ (1 \leq j \leq k \minus{} 1)$.

For which inequality, there exists a line such that the region defined by the inequality and the line intersect in exactly two distinct points? $\textbf{(A)}\ x^2+y^2\leq 1 \qquad\textbf{(B)}\ |x+y|+|x-y| \leq 1 \qquad\textbf{(C)}\ |x|^3+|y|^3 \leq 1 \\ \textbf{(D)}\ |x|+|y| \leq 1 \qquad\textbf{(E)}\ |x|^{1/2} + |y|^{1/2} \leq 1$

Given triangle $ABC$ the point $J$ is the centre of the excircle opposite the vertex $A.$ This excircle is tangent to the side $BC$ at $M$, and to the lines $AB$ and $AC$ at $K$ and $L$, respectively. The lines $LM$ and $BJ$ meet at $F$, and the lines $KM$ and $CJ$ meet at $G.$ Let $S$ be the point of intersection of the lines $AF$ and $BC$, and let $T$ be the point of intersection of the lines $AG$ and $BC.$ Prove that $M$ is the midpoint of $ST.$ (The [i]excircle[/i] of $ABC$ opposite the vertex $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.) [i]Proposed by Evangelos Psychas, Greece[/i]

Let the triangle $ABC$ with $BC$ the smallest side. Let $P$ on ($AB$) such that angle $PCB$ equals angle $BAC$. and $Q$ on side ($AC$) such that angle $QBC$ equals angle $BAC$. Show that the line passing through the circumenters of triangles $ABC$ and $APQ$ is perpendicular on $BC$.

Let $p_n(k)$ be the number of permutations of the set $\{1,2,3,\ldots,n\}$ which have exactly $k$ fixed points. Prove that $\sum_{k=0}^nk p_n(k)=n!$.[i](IMO Problem 1)[/i] [b][i]Original formulation [/i][/b] Let $S$ be a set of $n$ elements. We denote the number of all permutations of $S$ that have exactly $k$ fixed points by $p_n(k).$ Prove: (a) $\sum_{k=0}^{n} kp_n(k)=n! \ ;$ (b) $\sum_{k=0}^{n} (k-1)^2 p_n(k) =n! $ [i]Proposed by Germany, FR[/i]

For any integer $n\geq1$, we consider a set $P_{2n}$ of $2n$ points placed equidistantly on a circle. A [i]perfect matching[/i] on this point set is comprised of $n$ (straight-line) segments whose endpoints constitute $P_{2n}$. Let $\mathcal{M}_{n}$ denote the set of all non-crossing perfect matchings on $P_{2n}$. A perfect matching $M\in \mathcal{M}_{n}$ is said to be [i]centrally symmetric[/i], if it is invariant under point reflection at the circle center. Determine, as a function of $n$, the number of centrally symmetric perfect matchings within $\mathcal{M}_{n}$. [i]Proposed by Mirko Petrusevski[/i]

[b]p1.[/b] Arrange the whole numbers $1$ through $15$ in a row so that the sum of any two adjacent numbers is a perfect square. In how many ways this can be done? [b]p2.[/b] Prove that if $p$ and $q$ are prime numbers which are greater than $3$ then $p^2 - q^2$ is divisible by $24$. [b]p3.[/b] If a polyleg has even number of legs he always tells truth. If he has an odd number of legs he always lies. Once a green polyleg told a dark-blue polyleg ”- I have $8$ legs. And you have only $6$ legs!” The offended dark-blue polyleg replied ”-It is me who has $8$ legs, and you have only $7$ legs!” A violet polyleg added ”-The dark-blue polyleg indeed has $8$ legs. But I have $9$ legs!” Then a stripped polyleg started ”None of you has $8$ legs. Only I have $8$ legs!” Which polyleg has exactly $8$ legs? [b][b]p4.[/b][/b] There is a small puncture (a point) in the wall (as shown in the figure below to the right). The housekeeper has a small flag of the following form (see the figure left). Show on the figure all the points of the wall where you can hammer in a nail such that if you hang the flag it will close up the puncture. [img]https://cdn.artofproblemsolving.com/attachments/a/f/8bb55a3fdfb0aff8e62bc4cf20a2d3436f5d90.png[/img] [b]p5.[/b] Assume $ a, b, c$ are odd integers. Show that the quadratic equation $ax^2 + bx + c = 0$ has no rational solutions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let there be a triangle $ABC$ with orthocenter $H$. Let the lengths of the heights be $h_a, h_b, h_c$ from points $A, B$ and respectively $C$, and the semi-perimeter $p$ of triangle $ABC$. It is known that $AH \cdot h_a + BH \cdot h_b + CH \cdot h_c = \frac{2}{3} \cdot p^2$. Show that $ABC$ is equilateral.

Suppose two regular pyramids with the same base $ABC$: $P-ABC$ and $Q-ABC$ are circumscribed by the same sphere. If the angle formed by one of the lateral face and the base of pyramid $P-ABC$ is $\frac{\pi}{4}$, find the tangent value of the angle formed by one of the lateral face and the base of the pyramid $Q-ABC$.

Let $S$ denote $\{1, \dots , 100\}$, and let $f$ be a permutation of $S$ such that for all $x\in S$, $f(x)\ne x$. Over all such $f$, find the maximum number of elements $j$ that satisfy $\underbrace{f(\dots(f(j))\dots)}_{\text{j times}}=j$. [i]Proposed by Hari Desikan[/i]

For each positive integer $n$, let $f(n)$ denote the smallest possible value of $$|A_1 \cup A_2 \cup \dots \cup A_n|$$ where $A_1, A_2, A_3 \dots A_n$ are sets such that $A_i \not\subseteq A_j$ and $|A_i| \neq |A_j|$ whenever $i \neq j$. Determine $f(n)$ for each positive integer $n$.

Let $a,b$ and $c$ be positive real numbers. Prove that \[ \frac{a^2b(b-c)}{a+b}+\frac{b^2c(c-a)}{b+c}+\frac{c^2a(a-b)}{c+a} \ge 0. \]

Let $p(x)$ be an $n$-degree $(n \ge 2)$ polynomial with integer coefficients. If there are infinitely many positive integers $m$, such that $p(m)$ at most $n -1$ different prime factors $f$, prove that $p(x)$ has at most $n-1$ different rational roots . [color=#f00]a help in translation is welcome[/color]

Positive integers $x_1,...,x_m$ (not necessarily distinct) are written on a blackboard. It is known that each of the numbers $F_1,...,F_{2018}$ can be represented as a sum of one or more of the numbers on the blackboard. What is the smallest possible value of $m$? (Here $F_1,...,F_{2018}$ are the first $2018$ Fibonacci numbers: $F_1=F_2=1, F_{k+1}=F_k+F_{k-1}$ for $k>1$.)

Let \(\mathbb{R}\) denote the set of real numbers. A subset \(S\subseteq\mathbb{R}\) is called [i]dense[/i] if any non-empty open interval of \(\mathbb{R}\) contains at least one element in \(S\). For a function \(f:\mathbb{R}\to\mathbb{R}\), let \(\mathcal{O}_f(x)\) denote the set \(\left\{x,f(x),f(f(x)),\ldots\right\}\). (a) Is there a function \(g:\mathbb{R}\to\mathbb{R}\), continuous everywhere in \(\mathbb{R}\) such that \(\mathcal{O}_g(x)\) is dense for all \(x\in\mathbb{R}\) for all \(x\in\mathbb{R}\)? (b) Is there a function \(h:\mathbb{R}\to\mathbb{R}\), continuous at all but a single \(x_0\in\mathbb{R}\), such that \(\mathcal{O}_h(x)\) is dense for all \(x\in\mathbb{R}\)? [i]Proposed by the ICMC Problem Committee[/i]

In acute $\vartriangle ABC$, let points $D$, $E,$ and $F$ be the feet of the altitudes of the triangle from $A$, $B$,and $C$, respectively. The area of $\vartriangle AEF$ is $1$, the area of $\vartriangle CDE$ is $2$, and the area of $\vartriangle BF D$ is $2 -\sqrt3$. What is the area of $\vartriangle DEF$?

Define a sequence by $a_1=a_2=1, a_3=2,$ and $$a_n+a_{n-3}=a_{n-1}+a_{n-2}$$ for all $n>3.$ What is the value of $a_7?$

For $ n \in \mathbb{N}$, let $s(n)$ denote the sum of all positive divisors of $n$. Show that for any $n > 1$, the product $s(n - 1)s(n)s(n + 1)$ is an even number.

For positive integers $a,b,c$ let $ \alpha, \beta, \gamma$ be pairwise distinct positive integers such that \[ \begin{cases}{c} \displaystyle a &= \alpha + \beta + \gamma, \\ b &= \alpha \cdot \beta + \beta \cdot \gamma + \gamma \cdot \alpha, \\ c^2 &= \alpha\beta\gamma. \end{cases} \] Also, let $ \lambda$ be a real number that satisfies the condition \[\lambda^4 -2a\lambda^2 + 8c\lambda + a^2 - 4b = 0.\] Prove that $\lambda$ is an integer if and only if $\alpha, \beta, \gamma$ are all perfect squares.

Let $ \alpha$ be the positive root of the equation $ x^{2} \equal{} 1991x \plus{} 1$. For natural numbers $ m$ and $ n$ define \[ m*n \equal{} mn \plus{} \lfloor\alpha m \rfloor \lfloor \alpha n\rfloor. \] Prove that for all natural numbers $ p$, $ q$, and $ r$, \[ (p*q)*r \equal{} p*(q*r). \]

Let $\triangle ABC$ be the acute triangle such that $AB\ne AC.$ Let $V$ be the intersection of $BC$ and angle bisector of $\angle A.$ Let $D$ be the foot of altitude from $A$ to $BC.$ Let $E,F$ be the intersection of circumcircle of $\triangle AVD$ and $CA,AB$ respectively. Prove that the lines $AD, BE,CF$ is concurrent.

Initially, only the integer $44$ is written on a board. An integer a on the board can be re- placed with four pairwise different integers $a_1, a_2, a_3, a_4$ such that the arithmetic mean $\frac 14 (a_1 + a_2 + a_3 + a_4)$ of the four new integers is equal to the number $a$. In a step we simultaneously replace all the integers on the board in the above way. After $30$ steps we end up with $n = 4^{30}$ integers $b_1, b2,\ldots, b_n$ on the board. Prove that \[\frac{b_1^2 + b_2^2+b_3^2+\cdots+b_n^2}{n}\geq 2011.\]

Let $ABC$ be a triangle with $AB=3, BC=4, CA=5$. Let $M$ be the midpoint of $BC$, and $\Gamma$ be a circle through $A$ and $M$ that intersects $AB$ and $AC$ again at $D$ and $E$, respectively. Given that $AD=AE$, find the area of quadrilateral $MEAD$. [i]Individual #9[/i]

Determine whether or not there exists a natural number $N$ which satisfies the following three criteria: 1. $N$ is divisible by $2^{2023}$, but not by $2^{2024}$, 2. $N$ only has three different digits, and none of them are zero, 3. Exactly 99.9% of the digits of $N$ are odd.

In a triangle $ABC$, the incircle touches the sides $BC, CA, AB$ in the points $A',B', C'$, respectively; the excircle in the angle $A$ touches the lines containing these sides in $A_1,B_1, C_1$, and similarly, the excircles in the angles $B$ and $C$ touch these lines in $A_2,B_2, C_2$ and $A_3,B_3, C_3$. Prove that the triangle $ABC$ is right-angled if and only if one of the point triples $(A',B_3, C'),$ $ (A_3,B', C_3), (A',B', C_2), (A_2,B_2, C'), (A_2,B_1, C_2), (A_3,B_3, C_1),$ $ (A_1,B_2, C_1), (A_1,B_1, C_3)$ is collinear.