Found problems: 85335
2022 Saudi Arabia IMO TST, 1
Let $(a_n)$ be the integer sequence which is defined by $a_1= 1$ and
$$ a_{n+1}=a_n^2 + n \cdot a_n \,\, , \,\, \forall n \ge 1.$$
Let $S$ be the set of all primes $p$ such that there exists an index $i$ such that $p|a_i$.
Prove that the set $S$ is an infinite set and it is not equal to the set of all primes.
2015 IFYM, Sozopol, 6
Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that for $\forall$ $x,y\in \mathbb{R}$ :
$f(x+f(x+y))+xy=yf(x)+f(x)+f(y)+x$.
2006 JHMT, 8
Circles $P$, $Q$, and $R$ are externally tangent to one another. The external tangent of $P$ and $Q$ that does not intersect $R$ intersects $P$ and $Q$ at $P_Q$ and $Q_P$ , respectively. $Q_R$,$R_Q$,$R_P$ , and $P_R$ are defined similarly. If the radius of $Q$ is $4$ and $\overline{Q_PP_Q} \parallel \overline{R_QQ_R}$, compute $R_PP_R$.
1992 IMO Longlists, 29
Show that in the plane there exists a convex polygon of 1992 sides satisfying the following conditions:
[i](i)[/i] its side lengths are $ 1, 2, 3, \ldots, 1992$ in some order;
[i](ii)[/i] the polygon is circumscribable about a circle.
[i]Alternative formulation:[/i] Does there exist a 1992-gon with side lengths $ 1, 2, 3, \ldots, 1992$ circumscribed about a circle? Answer the same question for a 1990-gon.
2016 Hong Kong TST, 3
2016 circles with radius 1 are lying on the plane. Among these 2016 circles, show that one can select a collection $C$ of 27 circles satisfying the following: either every pair of two circles in $C$ intersect or every pair of two circles in $C$ does not intersect.
2019 Taiwan TST Round 1, 5
Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.
2011 Cuba MO, 2
A cube of dimensions $20 \times 20 \times 20$ is constructed with blocks of $1 \times 2 \times 2$. Prove that there is a line that passes through the cube but not any block.
2017 India Regional Mathematical Olympiad, 2
Show that the equation \(a^3+(a+1)^3+\ldots+(a+6)^3=b^4+(b+1)^4\) has no solutions in integers \(a,b\).
1999 Flanders Math Olympiad, 4
Let $a,b,m,n$ integers greater than 1. If $a^n-1$ and $b^m+1$ are both primes, give as much info as possible on $a,b,m,n$.
2012 China Team Selection Test, 2
Given an integer $k\ge 2$. Prove that there exist $k$ pairwise distinct positive integers $a_1,a_2,\ldots,a_k$ such that for any non-negative integers $b_1,b_2,\ldots,b_k,c_1,c_2,\ldots,c_k$ satisfying $a_1\le b_i\le 2a_i, i=1,2,\ldots,k$ and $\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i$, we have
\[k\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i.\]
1969 IMO Longlists, 57
Given triangle $ ABC $ with points $ M $ and $ N $ are in the sides $ AB $ and $ AC $ respectively.
If $ \dfrac{BM}{MA} +\dfrac{CN}{NA} = 1 $ , then prove that the centroid of $ ABC $ lies on $ MN $ .
2022/2023 Tournament of Towns, P5
On the sides of a regular nonagon $ABCDEFGHI$, triangles $XAB, YBC, ZCD$ and $TDE$ are constructed outside the nonagon. The angles at $X, Y, Z, T$ in these triangles are each $20^\circ$. The angles $XAB, YBC, ZCD$ and $TDE$ are such that (except for the first angle) each angle is $20^\circ$ greater than the one listed before it. Prove that the points $X, Y , Z, T$ lie on the same circle.
2022 Yasinsky Geometry Olympiad, 2
On the sides $AB$, $BC$, $CD$, $DA$ of the square $ABCD$ points $P, Q, R, T$ are chosen such that $$\frac{AP}{PB}=\frac{BQ}{QC}=\frac{CR}{RD}=\frac{DT}{TA}=\frac12.$$
The segments $AR$, $BT$, $CP$, $DQ$ in the intersection form the quadrilateral $KLMN$ (see figure). [img]https://cdn.artofproblemsolving.com/attachments/f/c/587a2358734c300fe7082c520f90c91f872b49.png[/img]
a) Prove that $KLMN$ is a square.
b) Find the ratio of the areas of the squares $KLMN$ and $ABCD$.
(Alexander Shkolny)
1989 Greece National Olympiad, 2
A collection of short stories written by Papadiamantis contains $70$ short stories, one of $1$ page, one of $2$ pages, ... one of $70$ pages . and not nessecarily in that order. Every short story starts on a new page and numbering of pages of the book starts from the first page . What is the maximum number of short stories that start on page with odd number?
2010 Contests, 2
Every non-negative integer is coloured white or red, so that:
• there are at least a white number and a red number;
• the sum of a white number and a red number is white;
• the product of a white number and a red number is red.
Prove that the product of two red numbers is always a red number, and the sum of two red numbers is always a red number.
2008 Saint Petersburg Mathematical Olympiad, 5
Given are distinct natural numbers $a$, $b$, and $c$. Prove that
\[ \gcd(ab+1, ac+1, bc+1)\le \frac{a+b+c}{3}\]
OIFMAT III 2013, 9
Let $ a, b \in Z $, prove that if the expression $ a \cdot 2013^n + b $ is a perfect square for all natural $n$, then $ a $ is zero.
2006 QEDMO 3rd, 12
Per and Kari each have $n$ pieces of paper. They both write down the numbers from $1$ to $2n$ in an arbitrary order, one number on each side. Afterwards, they place the pieces of paper on a table showing one side. Prove that they can always place them so that all the numbers from $1$ to $2n$ are visible at once.
2008 Macedonia National Olympiad, 2
Positive numbers $ a$, $ b$, $ c$ are such that $ \left(a \plus{} b\right)\left(b \plus{} c\right)\left(c \plus{} a\right) \equal{} 8$. Prove the inequality
\[ \frac {a \plus{} b \plus{} c}{3}\ge\sqrt [27]{\frac {a^3 \plus{} b^3 \plus{} c^3}{3}}
\]
2014 AMC 10, 24
The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is [i]bad[/i] if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?
$ \textbf {(A) } 1 \qquad \textbf {(B) } 2 \qquad \textbf {(C) } 3 \qquad \textbf {(D) } 4 \qquad \textbf {(E) } 5 $
2015 India IMO Training Camp, 2
Find all functions from $\mathbb{N}\cup\{0\}\to\mathbb{N}\cup\{0\}$ such that $f(m^2+mf(n))=mf(m+n)$, for all $m,n\in \mathbb{N}\cup\{0\}$.
1998 Slovenia National Olympiad, Problem 2
find all functions $f(x)$ satisfying:
$(\forall x\in R) f(x)+xf(1-x)=x^2+1$
2020 Kazakhstan National Olympiad, 1
There are $n$ lamps and $k$ switches in a room. Initially, each lamp is either turned on or turned off. Each lamp is connected by a wire with $2020$ switches. Switching a switch changes the state of a lamp, that is connected to it, to the opposite state. It is known that one can switch the switches so that all lamps will be turned on. Prove, that it is possible to achieve the same result by switching the switches no more than $ \left \lfloor \dfrac{k}{2} \right \rfloor$ times.
[i]Proposed by T. Zimanov[/i]
2009 Cuba MO, 5
Prove that there are infinitely many positive integers $n$ such that $\frac{5^n-1}{n+2}$ is an integer.
2014 BMT Spring, 6
Pick a $3$-digit number $abc$, which contains no $0$'s. The probability that this is a winning number is $\frac1a\cdot\frac1b\cdot\frac1c$. However, the BMT problem writer tries to balance out the chances for the numbers in the following ways:
[list]
[*] For the lowest digit $n$ in the number, he rolls an $n$-sided die for each time that the digit appears, and gives the number $0$ probability of winning if an $n$ is rolled.
[*] For the largest digit $m$ in the number, he rolls an $m$-sided die once and scales the probability of winning by that die roll.
[/list]
If you choose optimally, what is the probability that your number is a winning number?