Found problems: 85335
2000 Belarus Team Selection Test, 4.3
Prove that for every real number $M$ there exists an infinite arithmetic progression such that:
- each term is a positive integer and the common difference is not divisible by 10
- the sum of the digits of each term (in decimal representation) exceeds $M$.
2009 International Zhautykov Olympiad, 1
On the plane, a Cartesian coordinate system is chosen. Given points $ A_1,A_2,A_3,A_4$ on the parabola $ y \equal{} x^2$, and points $ B_1,B_2,B_3,B_4$ on the parabola $ y \equal{} 2009x^2$. Points $ A_1,A_2,A_3,A_4$ are concyclic, and points $ A_i$ and $ B_i$ have equal abscissas for each $ i \equal{} 1,2,3,4$.
Prove that points $ B_1,B_2,B_3,B_4$ are also concyclic.
2011 Estonia Team Selection Test, 1
Two circles lie completely outside each other.Let $A$ be the point of intersection of internal common tangents of the circles and let $K$ be the projection of this point onto one of their external common tangents.The tangents,different from the common tangent,to the circles through point $K$ meet the circles at $M_1$ and $M_2$.Prove that the line $AK$ bisects angle $M_1 KM_2$.
2023 India IMO Training Camp, 2
Let $\mathbb R$ be the set of real numbers. We denote by $\mathcal F$ the set of all functions $f\colon\mathbb R\to\mathbb R$ such that
$$f(x + f(y)) = f(x) + f(y)$$
for every $x,y\in\mathbb R$ Find all rational numbers $q$ such that for every function $f\in\mathcal F$, there exists some $z\in\mathbb R$ satisfying $f(z)=qz$.
2019 Greece Team Selection Test, 4
Find all functions $f:(0,\infty)\mapsto\mathbb{R}$ such that $\displaystyle{(y^2+1)f(x)-yf(xy)=yf\left(\frac{x}{y}\right),}$ for every $x,y>0$.
2004 Brazil Team Selection Test, Problem 2
An integer $n\ge2$ is called [i]amicable[/i] if there exists subsets $A_1,A_2,\ldots,A_n$ of the set $\{1,2,\ldots,n\}$ such that
(i) $i\notin A_i$ for any $i=1,2,\ldots,n$,
(ii) $i\in A_j$ for any $j\notin A_i$, for any $i\ne j$
(iii) $A_i\cap A_j\ne\emptyset$ for any $i,j\in\{1,2,\ldots,n\}$
(a) Prove that $7$ is amicable.
(b) Prove that $n$ is amicable if and only if $n\ge7$.
2021 USAMO, 1
Rectangles $BCC_1B_2,$ $CAA_1C_2,$ and $ABB_1A_2$ are erected outside an acute triangle $ABC.$ Suppose that \[\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.\] Prove that lines $B_1C_2,$ $C_1A_2,$ and $A_1B_2$ are concurrent.
2004 Estonia Team Selection Test, 3
For which natural number $n$ is it possible to draw $n$ line segments between vertices of a regular $2n$-gon so that every vertex is an endpoint for exactly one segment and these segments have pairwise different lengths?
2021 Saint Petersburg Mathematical Olympiad, 4
The following functions are written on the board, $$F(x) = x^2 + \frac{12}{x^2}, G(x) = \sin(\pi x^2), H(x) = 1.$$ If functions $f,g$ are currently on the board, we may write on the board the functions $$f(x) + g(x), f(x) - g(x), f(x)g(x), cf(x)$$ (the last for any real number $c$). Can a function $h(x)$ appear on the board such that $$|h(x) - x| < \frac{1}{3}$$ for all $x \in [1,10]$ ?
2019 Singapore MO Open, 3
A robot is placed at point $P$ on the $x$-axis but different from $(0,0)$ and $(1,0)$ and can only move along the axis either to the left or to the right. Two players play the following game. Player $A$ gives a distance and $B$ gives a direction and the robot will move the indicated distance along the indicated direction. Player $A$ aims to move the robot to either $(0,0)$ or $(1,0)$. Player $B$'s aim is to stop $A$ from achieving his aim. For which $P$ can $A$ win?
2019 Adygea Teachers' Geometry Olympiad, 2
Inside the triangle $T$ there are three other triangles that do not have common points. Is it true that one can choose such a point inside $T$ and draw three rays from it so that the triangle breaks into three parts, in each of which there will be one triangle?
2011 Estonia Team Selection Test, 2
Let $n$ be a positive integer. Prove that for each factor $m$ of the number $1+2+\cdots+n$ such that $m\ge n$, the set $\{1,2,\ldots,n\}$ can be partitioned into disjoint subsets, the sum of the elements of each being equal to $m$.
[b]Edit[/b]:Typographical error fixed.
2018 IMO Shortlist, A7
Find the maximal value of
\[S = \sqrt[3]{\frac{a}{b+7}} + \sqrt[3]{\frac{b}{c+7}} + \sqrt[3]{\frac{c}{d+7}} + \sqrt[3]{\frac{d}{a+7}},\]
where $a$, $b$, $c$, $d$ are nonnegative real numbers which satisfy $a+b+c+d = 100$.
[i]Proposed by Evan Chen, Taiwan[/i]
Ukrainian TYM Qualifying - geometry, 2010.6
Find inside the triangle $ABC$, points $G$ and $H$ for which, respectively, the geometric mean and the harmonic mean of the distances to the sides of the triangle acquire maximum values. In which cases is the segment $GH$ parallel to one of the sides of the triangle? Find the length of such a segment $GH$.
2024 Miklos Schweitzer, 4
Let $\pi$ be a given permutation of the set $\{1, 2, \dots, n\}$. Determine the smallest possible value of
\[
\sum_{i=1}^n |\pi(i) - \sigma(i)|,
\]
where $\sigma$ is a permutation chosen from the set of all $n$-cycles. Express the result in terms of the number and lengths of the cycles in the disjoint cycle decomposition of $\pi$, including the fixed points.
2012 Korea National Olympiad, 2
There are $n$ students $ A_1 , A_2 , \cdots , A_n $ and some of them shaked hands with each other. ($ A_i $ and $ A_j$ can shake hands more than one time.) Let the student $ A_i $ shaked hands $ d_i $ times. Suppose $ d_1 + d_2 + \cdots + d_n > 0 $. Prove that there exist $ 1 \le i < j \le n $ satisfying the following conditions:
(a) Two students $ A_i $ and $ A_j $ shaked hands each other.
(b) $ \frac{(d_1 + d_2 + \cdots + d_n ) ^2 }{n^2 } \le d_i d_j $
1985 AMC 8, 2
$ 90\plus{}91\plus{}92\plus{}93\plus{}94\plus{}95\plus{}96\plus{}97\plus{}98\plus{}99\equal{}$
\[ \textbf{(A)}\ 845 \qquad
\textbf{(B)}\ 945 \qquad
\textbf{(C)}\ 1005 \qquad
\textbf{(D)}\ 1025 \qquad
\textbf{(E)}\ 1045
\]
2015 China Team Selection Test, 2
Let $X$ be a non-empty and finite set, $A_1,...,A_k$ $k$ subsets of $X$, satisying:
(1) $|A_i|\leq 3,i=1,2,...,k$
(2) Any element of $X$ is an element of at least $4$ sets among $A_1,....,A_k$.
Show that one can select $[\frac{3k}{7}] $ sets from $A_1,...,A_k$ such that their union is $X$.
2018 Denmark MO - Mohr Contest, 3
The positive integers $a, b$ and $c$ satisfy that the three fractions $\frac{b}{a}$, $\frac{c + 100}{b}$ and $\frac{a + b + 169}{2c + 200}$ are all integers. Determine all possible values of $a$.
1979 Canada National Olympiad, 2
It is known in Euclidean geometry that the sum of the angles of a triangle is constant. Prove, however, that the sum of the dihedral angles of a tetrahedron is not constant.
2007 National Olympiad First Round, 18
How many integers $n$ are there such that $n^3+8$ has at most $3$ positive divisors?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 1
\qquad\textbf{(E)}\ \text{None of the above}
$
2022 Princeton University Math Competition, A8
A permutation $\pi : \{1,2,\ldots,N\} \rightarrow \{1,2, \ldots,N\}$ is [i]very odd[/i] if the smallest positive integer $k$ such that $\pi^k(a) = a$ for all $1 \le a \le N$ is odd, where $\pi^k$ denotes $\pi$ composed with itself $k$ times. Let $X_0 = 1,$ and for $i \ge 1,$ let $X_i$ be the fraction of all permutations of $\{1,2,\ldots,i\}$ that are very odd. Let $S$ denote the set of all ordered $4$-tuples $(A,B,C,D)$ of nonnegative integers such that $A+B +C +D =2023.$ Find the last three digits of the integer $$2023\sum_{(A,B,C,D) \in S} X_AX_BX_CX_D.$$
2020 Costa Rica - Final Round, 1
Find all the $4$-digit natural numbers, written in base $10$, that are equal to the cube of the sum of its digits.
1980 IMO, 3
Prove that the equation \[ x^n + 1 = y^{n+1}, \] where $n$ is a positive integer not smaller then 2, has no positive integer solutions in $x$ and $y$ for which $x$ and $n+1$ are relatively prime.
2011 Kazakhstan National Olympiad, 6
We call a square table of a binary, if at each cell is written a single number 0 or 1. The binary table is called regular if each row and each column exactly two units. Determine the number of regular size tables $n\times n$ ($n> 1$ - given a fixed positive integer). (We can assume that the rows and columns of the tables are numbered: the cases of coincidence in turn, reflect, and so considered different).