Found problems: 85335
2025 Czech-Polish-Slovak Junior Match., 4
Three non-negative integers are written on the board. In every step, the three numbers $(a, b, c)$ are being replaced with $a+b, b+c, c+a$. Find the biggest number of steps, after which the number $111$ will appear on the board.
2004 India IMO Training Camp, 1
Let $x_1, x_2 , x_3, .... x_n$ be $n$ real numbers such that $0 < x_j < \frac{1}{2}$. Prove that \[ \frac{ \prod\limits_{j=1}^{n} x_j } { \left( \sum\limits_{j=1}^{n} x_j \right)^n} \leq \frac{ \prod\limits_{j=1}^{n} (1-x_j) } { \left( \sum\limits_{j=1}^{n} (1 - x_j) \right)^n} \]
2004 Baltic Way, 16
Through a point $P$ exterior to a given circle pass a secant and a tangent to the circle. The secant intersects the circle at $A$ and $B$, and the tangent touches the circle at $C$ on the same side of the diameter through $P$ as the points $A$ and $B$. The projection of the point $C$ on the diameter is called $Q$. Prove that the line $QC$ bisects the angle $\angle AQB$.
2022 China Team Selection Test, 2
Let $ABCD$ be a convex quadrilateral, the incenters of $\triangle ABC$ and $\triangle ADC$ are $I,J$, respectively. It is known that $AC,BD,IJ$ concurrent at a point $P$. The line perpendicular to $BD$ through $P$ intersects with the outer angle bisector of $\angle BAD$ and the outer angle bisector $\angle BCD$ at $E,F$, respectively. Show that $PE=PF$.
2003 Switzerland Team Selection Test, 6
Let $ a,b,c $ be positive real numbers satisfying $ a+b+c=2 $. Prove the inequality \[ \frac{1}{1+ab}+\frac{1}{1+bc}+\frac{1}{1+ca} \ge \frac{27}{13} \]
2022 Austrian MO Regional Competition, 2
Determine the number of ten-digit positive integers with the following properties:
$\bullet$ Each of the digits $0, 1, 2, . . . , 8$ and $9$ is contained exactly once.
$\bullet$ Each digit, except $9$, has a neighbouring digit that is larger than it.
(Note. For example, in the number $1230$, the digits $1$ and $3$ are the neighbouring digits of $2$ while $2$ and $0$ are the neighbouring digits of $3$. The digits $1$ and $0$ have only one neighbouring digit.)
[i](Karl Czakler)[/i]
2008 ISI B.Stat Entrance Exam, 6
Evaluate: $\lim_{n\to\infty} \frac{1}{2n} \ln\binom{2n}{n}$
1968 All Soviet Union Mathematical Olympiad, 100
The sequence $a_1,a_2,a_3,...$, is constructed according to the rule $$a_1=1, a_2=a_1+1/a_1, ... , a_{n+1}=a_n+1/a_n, ...$$
Prove that $a_{100} > 14$.
2007 Putnam, 4
A [i]repunit[/i] is a positive integer whose digits in base $ 10$ are all ones. Find all polynomials $ f$ with real coefficients such that if $ n$ is a repunit, then so is $ f(n).$
2017 Tuymaada Olympiad, 6
Let $\sigma(n) $ denote the sum of positive divisors of a number $n $. A positive integer $N=2^rb $ is given,where $r $ and $b $ are positive integers and $b $ is odd. It is known that $\sigma(N)=2N-1$. Prove that $b$ and $\sigma (b) $ are coprime.
Tuymaada Q6 Juniors
2019 District Olympiad, 3
Let $a,b,c$ be distinct complex numbers with $|a|=|b|=|c|=1.$ If $|a+b-c|^2+|b+c-a|^2+|c+a-b|^2=12,$ prove that the points of affixes $a,b,c$ are the vertices of an equilateral triangle.
2007 Cuba MO, 8
For each positive integer $n$, let $S(n)$ be the sum of the digits of $n^2 +1$. A sequence $\{a_n\}$ is defined, with $a_0$ an arbitrary positive integer and $a_{n+1} = S(a_n)$. Prove that the sequence $\{a_n\}$ is eventually periodic with period three.
2002 Croatia National Olympiad, Problem 3
Points $E$ and $F$ are taken on the diagonals $AB_1$ and $CA_1$ of the lateral faces $ABB_1A_1$ and $CAA_1C_1$ of a triangular prism $ABCA_1B_1C_1$ so that $EF\parallel BC_1$. Find the ratio of the lengths of $EF$ and $BC_1$.
1977 Polish MO Finals, 3
Consider the set $A = \{0, 1, 2, . . . , 2^{2n} - 1\}$. The function $f : A \rightarrow A$ is given by: $f(x_0 + 2x_1 + 2^2x_2 + ... + 2^{2n-1}x_{2n-1})=$$(1 - x_0) + 2x_1 + 2^2(1 - x_2) + 2^3x_3 + ... + 2^{2n-1}x_{2n-1}$
for every $0-1$ sequence $(x_0, x_1, . . . , x_{2n-1})$. Show that if $a_1, a_2, . . . , a_9$ are consecutive terms of an arithmetic progression, then the sequence $f(a_1), f(a_2), . . . , f(a_9)$ is not increasing.
2019 China Team Selection Test, 4
Prove that there exist a subset $A$ of $\{1,2,\cdots,2^n\}$ with $n$ elements, such that for any two different non-empty subset of $A$, the sum of elements of one subset doesn't divide another's.
2020 Durer Math Competition Finals, 5
The hexagon $ABCDEF$ has all angles equal . We know that four consecutive sides of the hexagon have lengths $7, 6, 3$ and $5$ in this order. What is the sum of the lengths of the two remaining sides?
2023 Assara - South Russian Girl's MO, 8
The girl continues the sequence of letters $ASSARA... $, adding one of the three letters $A$, $R$ or $S$. When adding the next letter, the girl makes sure that no two written sevens of consecutive letters coincide. At some point it turned out that it was impossible to add a new letter according to these rules. What letter could be written last?
1998 Belarus Team Selection Test, 3
Let $s,t$ be given nonzero integers, $(x,y)$ be any (ordered) pair of integers. A sequence of moves is performed as follows: per move $(x,y)$ changes to $(x+t, y-s)$. The pair (x,y) is said to be [i]good [/i] if after some (may be, zero) number of moves described a pair of integers arises that are not relatively prime.
a) Determine whether $(s,t)$ is itself a good pair;
bj Prove that for any nonzero $s$ and $t$ there is a pair $(x,y)$ which is not good.
2015 ELMO Problems, 1
Define the sequence $a_1 = 2$ and $a_n = 2^{a_{n-1}} + 2$ for all integers $n \ge 2$. Prove that $a_{n-1}$ divides $a_n$ for all integers $n \ge 2$.
[i]Proposed by Sam Korsky[/i]
2014 Singapore Senior Math Olympiad, 34
Let $x_1,x_2,\dots,x_{100}$ be real numbers such that $|x_1|=63$ and $|x_{n+1}|=|x_n+1|$ for $n=1,2\dots,99$.
Find the largest possible value of $(-x_1-x_2-\cdots-x_{100})$.
1966 IMO Longlists, 10
How many real solutions are there to the equation $x = 1964 \sin x - 189$ ?
2012 BMT Spring, 5
Let $p > 1$ be relatively prime to $10$. Let $n$ be any positive number and$ d$ be the last digit of $n$. Define $f(n) = \lfloor \frac{n}{10} \rfloor + d \cdot m$. Then, we can call $m$ a [i]divisibility multiplier[/i] for $p$, if $f(n)$ is divisible by $p$ if and only if $n$ is divisible by $p$. Find a divisibility multiplier for $2013$.
2010 SEEMOUS, Problem 4
Suppose that $A$ and $B$ are $n\times n$ matrices with integer entries, and $\det B\ne0$. Prove that there exists $m\in\mathbb N$ such that the product $AB^{-1}$ can be represented as
$$AB^{-1}=\sum_{k=1}^mN_k^{-1},$$where $N_k$ are $n\times n$ matrices with integer entries for all $k=1,\ldots,m$, and $N_i\ne N_j$ for $i\ne j$.
2021 Science ON all problems, 4
Consider positive real numbers $x,y,z$. Prove the inequality
$$\frac 1x+\frac 1y+\frac 1z+\frac{9}{x+y+z}\ge 3\left (\left (\frac{1}{2x+y}+\frac{1}{x+2y}\right )+\left (\frac{1}{2y+z}+\frac{1}{y+2z}\right )+\left (\frac{1}{2z+x}+\frac{1}{x+2z}\right )\right ).$$
[i] (Vlad Robu \& Sergiu Novac)[/i]
2005 Romania Team Selection Test, 3
Let $\mathbb{N}=\{1,2,\ldots\}$. Find all functions $f: \mathbb{N}\to\mathbb{N}$ such that for all $m,n\in \mathbb{N}$ the number $f^2(m)+f(n)$ is a divisor of $(m^2+n)^2$.