Found problems: 85335
2022 Czech-Polish-Slovak Junior Match, 2
The number $2022$ is written on the board. In each step, we replace one of the $2$ digits with the number $2022$.
For example $$2022 \Rightarrow 2020222 \Rightarrow 2020220222 \Rightarrow ...$$
After how many steps can a number divisible by $22$ be written on the board? Specify all options.
2024 Serbia National Math Olympiad, 1
Find all positive integers $n$, such that if their divisors are $1=d_1<d_2<\ldots<d_k=n$ for $k \geq 4$, then the numbers $d_2-d_1, d_3-d_2, \ldots, d_k-d_{k-1}$ form a geometric progression in some order.
2014 IMS, 6
Let $A=[a_{ij}]_{n \times n}$ be a $n \times n$ matrix whose elements are all numbers which belong to set $\{1,2,\cdots ,n\}$. Prove that by swapping the columns of $A$ with each other we can produce matrix $B=[b_{ij}]_{n \times n}$ such that $K(B) \le n$ where $K(B)$ is the number of elements of set $\{(i,j) ; b_{ij} =j\}$.
1995 Tuymaada Olympiad, 7
Find a continuous function $f(x)$ satisfying the identity $f(x)-f(ax)=x^n-x^m$, where $n,m\in N , 0<a<1$
1987 All Soviet Union Mathematical Olympiad, 457
Some points with the integer coordinates are marked on the coordinate plane. Given a set of nonzero vectors. It is known, that if you apply the beginnings of those vectors to the arbitrary marked point, than there will be more marked ends of the vectors, than not marked. Prove that there is infinite number of marked points.
2020 Peru Iberoamerican Team Selection Test, P7
The numbers $1, 2,\ldots ,50$ are written on a blackboard. Ana performs the following operations: she chooses any three numbers $a, b$ and $c$ from the board and replaces them with their sum $a + b + c$ and writes the number $(a + b) (b + c) (c + a)$ in the notebook. Ana performs these operations until there are only two numbers left on the board ($24$ operations in total). Then, she calculates the sum of the numbers written down in her notebook. Let $M$ and $m$ be the maximum and minimum possible of the sums obtained by Ana.
Find the value of $\frac{M}{m}$.
2025 Greece National Olympiad, 2
Let $ABC$ be an acute triangle and $D$ be a point of side $ BC$. Consider points $E,Z$ on line $AD$ such that $EB \perp AB$ and $ZC \perp AC$, and points $H, T $ on line $BC$ such that $EH \parallel AC$ and $ZT \parallel AB$. Circumcircle of triangle $BHE$ intersects for second time line $AB$ at point $M$ ($M \ne B$) and circumcircle of triangle $CTZ$ intersects for second time line $AC$ at point $N$ ($N \ne C$). Prove that lines $MH$, $NT$ and $AD$ concur.
2023 Bangladesh Mathematical Olympiad, P10
Let all possible $2023$-degree real polynomials: $P(x)=x^{2023}+a_1x^{2022}+a_2x^{2021}+\cdots+a_{2022}x+a_{2023}$,
where $P(0)+P(1)=0$, and the polynomial has 2023 real roots $r_1, r_2,\cdots r_{2023}$ [not necessarily distinct] so that $0\leq r_1,r_2,\cdots r_{2023}\leq1$. What is the maximum value of $r_1 \cdot r_2 \cdots r_{2023}?$
2006 Switzerland Team Selection Test, 2
Let $D$ be inside $\triangle ABC$ and $E$ on $AD$ different to $D$. Let $\omega_1$ and $\omega_2$ be the circumscribed circles of $\triangle BDE$ and $\triangle CDE$ respectively. $\omega_1$ and $\omega_2$ intersect $BC$ in the interior points $F$ and $G$ respectively. Let $X$ be the intersection between $DG$ and $AB$ and $Y$ the intersection between $DF$ and $AC$. Show that $XY$ is $\|$ to $BC$.
2007 Germany Team Selection Test, 1
Let $ n > 1, n \in \mathbb{Z}$ and $ B \equal{}\{1,2,\ldots, 2^n\}.$ A subset $ A$ of $ B$ is called weird if it contains exactly one of the distinct elements $ x,y \in B$ such that the sum of $ x$ and $ y$ is a power of two. How many weird subsets does $ B$ have?
2018 IMO Shortlist, G4
A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, and $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of the triangle $A_1B_1C_1$. The lines $A_1T$, $B_1T$, and $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, and $C_2$, respectively. Prove that the lines $AA_2$, $BB_2$, and $CC_2$ are concurrent on $\Omega$.
[i]Proposed by Mongolia[/i]
1999 Brazil Team Selection Test, Problem 3
Let $BD$ and $CE$ be the bisectors of the interior angles $\angle B$ and $\angle C$, respectively ($D\in AC$, $E\in AB$). Consider the circumcircle of $ABC$ with center $O$ and the excircle corresponding to the side $BC$ with center $I_a$. These two circles intersect at points $P$ and $Q$.
(a) Prove that $PQ$ is parallel to $DE$.
(b) Prove that $I_aO$ is perpendicular to $DE$.
2022 Greece Team Selection Test, 1
Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$
2019 Switzerland - Final Round, 3
Find all periodic sequences $x_1,x_2,\dots$ of strictly positive real numbers such that $\forall n \geq 1$ we have $$x_{n+2}=\frac{1}{2} \left( \frac{1}{x_{n+1}}+x_n \right)$$
2014 Turkey Team Selection Test, 2
$a_1=-5$, $a_2=-6$ and for all $n \geq 2$ the ${(a_n)^\infty}_{n=1}$ sequence defined as,
\[a_{n+1}=a_n+(a_1+1)(2a_2+1)(3a_3+1)\cdots((n-1)a_{n-1}+1)((n^2+n)a_n+2n+1)).\]
If a prime $p$ divides $na_n+1$ for a natural number n, prove that there is a integer $m$ such that $m^2\equiv5(modp)$
1971 AMC 12/AHSME, 32
If $s=(1+2^{-\frac{1}{32}})(1+2^{-\frac{1}{16}})(1+2^{-\frac{1}{8}})(1+2^{-\frac{1}{4}})(1+2^{-\frac{1}{2}})$, then $s$ is equal to
$\textbf{(A) }\textstyle{\frac{1}{2}}(1-2^{-\frac{1}{32}})^{-1}\qquad\textbf{(B) }(1-2^{-\frac{1}{32}})^{-1}\qquad\textbf{(C) }1-2^{-\frac{1}{32}}\qquad$
$\textbf{(D) }\textstyle{\frac{1}{2}}(1-2^{-\frac{1}{32}})\qquad \textbf{(E) }\frac{1}{2}$
2023 Yasinsky Geometry Olympiad, 4
Pick a point $C$ on a semicircle with diameter $AB$. Let $P$ and $Q$ be two points on segment $AB$ such that $AP= AC$ and $BQ= BC$. The point $O$ is the center of the circumscribed circle of triangle $CPQ$ and point $H$ is the orthocenter of triangle $CPQ$ . Prove that for all posible locations of point $C$, the line $OH$ is passing through a fixed point.
(Mykhailo Sydorenko)
2000 IberoAmerican, 3
A convex hexagon is called [i]pretty[/i] if it has four diagonals of length 1, such that their endpoints are all the vertex of the hexagon.
($a$) Given any real number $k$ with $0<k<1$ find a [i]pretty[/i] hexagon with area equal to $k$
($b$) Show that the area of any [i]pretty[/i] hexagon is less than 1.
1954 Moscow Mathematical Olympiad, 283
Consider five segments $AB_1, AB_2, AB_3, AB_4, AB_5$. From each point $B_i$ there can exit either $5$ segments or no segments at all, so that the endpoints of any two segments of the resulting graph (system of segments) do not coincide. Can the number of free endpoints of the segments thus constructed be equal to $1001$? (A free endpoint is an endpoint from which no segment begins.)
2022 Purple Comet Problems, 3
An isosceles triangle has a base with length $12$ and the altitude to the base has length $18$. Find the area of the region of points inside the triangle that are a distance of at most 3 from that altitude.
TNO 2008 Junior, 2
A cube of size $4 \times 4 \times 4$ is divided into 16 equal squares per face, with numbers from 1 to 96 randomly assigned to these squares. An operation consists of taking two squares that share a vertex, summing their numbers, and rewriting this sum in one of the squares while leaving the other blank. After performing several such operations, only one number remains. Prove that regardless of the order of operations, the final remaining number is always the same. Additionally, find this number.
2022 MIG, 7
Alice, Bob, and Charlie are each thinking of a number. Alice's number differs from Bob's number by $2$. Bob's number differs from Charlie's number by $6$. Charlie's number differs from Alice's number by $N$. What is the sum of all possible values for $N$?
$\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }12\qquad\textbf{(E) }14$
1994 Czech And Slovak Olympiad IIIA, 3
A convex $1994$-gon $M$ is given in the plane. A closed polygonal line consists of $997$ of its diagonals. Every vertex is adjacent to exactly one diagonal. Each diagonal divides $M$ into two sides, and the smaller of the numbers of edges on the two sides of $M$ is defined to be the length of the diagonal. Is it posible to have
(a) $991$ diagonals of length $3$ and $6$ of length $2$?
(b) $985$ diagonals of length $6, 4$ of length $8$, and $8$ of length $3$?
2015 All-Russian Olympiad, 5
An immortal flea jumps on whole points of the number line, beginning with $0$. The length of the first jump is $3$, the second $5$, the third $9$, and so on. The length of $k^{\text{th}}$ jump is equal to $2^k + 1$. The flea decides whether to jump left or right on its own. Is it possible that sooner or later the flee will have been on every natural point, perhaps having visited some of the points more than once?
2017 Harvard-MIT Mathematics Tournament, 10
[b]D[/b]enote $\phi=\frac{\sqrt{5}+1}{2}$ and consider the set of all finite binary strings without leading zeroes. Each string $S$ has a “base-$\phi$” value $p(S)$. For example, $p(1101)=\phi^3+\phi^2+1$. For any positive integer n, let $f(n)$ be the number of such strings S that satisfy $p(S) =\frac{\phi^{48n}-1}{\phi^{48}-1}$. The sequence of fractions $\frac{f(n+1)}{f(n)}$ approaches a real number $c$ as $n$ goes to infinity. Determine the value of $c$.