Found problems: 85335
2002 China Team Selection Test, 1
In acute triangle $ ABC$, show that:
$ \sin^3{A}\cos^2{(B \minus{} C)} \plus{} \sin^3{B}\cos^2{(C \minus{} A)} \plus{} \sin^3{C}\cos^2{(A \minus{} B)} \leq 3\sin{A} \sin{B} \sin{C}$
and find out when the equality holds.
1998 Romania National Olympiad, 3
A ring $A$ is called Boolean if $x^2 = x$ for every $x \in A$. Prove that:
a) A finite set $A$ with $n \geq 2$ elements can be equipped with the structure of a Boolean ring if and only if $n = 2^k$ for some positive integer $k$.
b) The set of nonnegative integers can be equipped with the structure of a Boolean ring.
2014 Peru IMO TST, 5
$n$ vertices from a regular polygon with $2n$ sides are chosen and coloured red. The other $n$ vertices are coloured blue. Afterwards, the $\binom{n}{2}$ lengths of the segments formed with all pairs of red vertices are ordered in a non-decreasing sequence, and the same procedure is done with the $\binom{n}{2}$ lengths of the segments formed with all pairs of blue vertices. Prove that both sequences are identical.
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:
$(\forall x\in R) f(x)+xf(1-x)=x^2+1$