Found problems: 85335
2020 Brazil National Olympiad, 5
Let $ABC$ be a triangle and $M$ the midpoint of $AB$. Let circumcircles of triangles $CMO$ and $ABC$ intersect at $K$ where $O$ is the circumcenter of $ABC$. Let $P$ be the intersection of lines $OM$ and $CK$. Prove that $\angle{PAK} = \angle{MCB}$.
2022 Thailand Online MO, 2
Let $ABCD$ be a trapezoid such that $AB \parallel CD$ and $AB > CD$. Points $X$ and $Y$ are on the side $AB$ such that $XY = AB-CD$ and $X$ lies between $A$ and $Y$. Prove that one intersection of the circumcircles of triangles $AYD$ and $BXC$ is on line $CD$.
2012 ELMO Shortlist, 3
Let $s(k)$ be the number of ways to express $k$ as the sum of distinct $2012^{th}$ powers, where order does not matter. Show that for every real number $c$ there exists an integer $n$ such that $s(n)>cn$.
[i]Alex Zhu.[/i]
I Soros Olympiad 1994-95 (Rus + Ukr), 10.2
Find the smallest positive number $a$ for which $$\sin a^o = \sin a$$
(on the left ($a^o$) is an angle of $a$ degrees, on the right is an angle in $a$ radians).
2018 India PRMO, 19
Let $N=6+66+666+....+666..66$, where there are hundred $6's$ in the last term in the sum. How many times does the digit $7$ occur in the number $N$
2013 Online Math Open Problems, 1
Let $x$ be the answer to this problem. For what real number $a$ is the answer to this problem also $a-x$?
[i]Ray Li[/i]
2005 JHMT, 6
Line $DE$ cuts through triangle $ABC$, with $DF$ parallel to $BE$. Given that $BD =DF = 10$ and $AD = BE = 25$, find $BC$.
[img]https://cdn.artofproblemsolving.com/attachments/0/e/d6e3d7c1f9bd15f4573ccd5fc67c190b9cf7e9.png[/img]
2019 Caucasus Mathematical Olympiad, 5
Vasya has a numeric expression
$$ \Box \cdot \Box +\Box \cdot \Box $$
and 4 cards with numbers that can be put on 4 free places in the expression. Vasya tried to put cards in all possible ways and all the time obtained the same value as a result. Prove that equal numbers are written on three of his cards.
2006 IMO Shortlist, 10
Assign to each side $b$ of a convex polygon $P$ the maximum area of a triangle that has $b$ as a side and is contained in $P$. Show that the sum of the areas assigned to the sides of $P$ is at least twice the area of $P$.
2003 Moldova Team Selection Test, 1
Let $ n\in N^*$. A permutation $ (a_1,a_2,...,a_n)$ of the numbers $ (1,2,...,n)$ is called [i]quadratic [/i] iff at least one of the numbers $ a_1,a_1\plus{}a_2,...,a_1\plus{}a_2\plus{}a\plus{}...\plus{}a_n$ is a perfect square. Find the greatest natural number $ n\leq 2003$, such that every permutation of $ (1,2,...,n)$ is quadratic.
2006 Princeton University Math Competition, 8
Find all integers $n$ (not necessarily positive) such that $7n^3-3n^2-3n-1$ is a perfect cube.
1998 IMO Shortlist, 4
Let $ M$ and $ N$ be two points inside triangle $ ABC$ such that
\[ \angle MAB \equal{} \angle NAC\quad \mbox{and}\quad \angle MBA \equal{} \angle NBC.
\]
Prove that
\[ \frac {AM \cdot AN}{AB \cdot AC} \plus{} \frac {BM \cdot BN}{BA \cdot BC} \plus{} \frac {CM \cdot CN}{CA \cdot CB} \equal{} 1.
\]
2012 China Second Round Olympiad, 1
Let $P$ be a point on the graph of the function $y=x+\frac{2}{x}(x>0)$. $PA,PB$ are perpendicular to line $y=x$ and $x=0$, respectively, the feet of perpendicular being $A$ and $B$. Find the value of $\overrightarrow{PA}\cdot \overrightarrow{PB}$.
2011 Iran MO (3rd Round), 4
We say the point $i$ in the permutation $\sigma$ [b]ongoing[/b] if for every $j<i$ we have $\sigma (j)<\sigma (i)$.
[b]a)[/b] prove that the number of permutations of the set $\{1,....,n\}$ with exactly $r$ ongoing points is $s(n,r)$.
[b]b)[/b] prove that the number of $n$-letter words with letters $\{a_1,....,a_k\},a_1<.....<a_k$. with exactly $r$ ongoing points is $\sum_{m}\dbinom{k}{m} S(n,m) s(m,r)$.
2019 China Second Round Olympiad, 3
Point $A,B,C,D,E$ lie on a line in this order, such that $BC=CD=\sqrt{AB\cdot DE},$ $P$ doesn't lie on the line, and satisfys that $PB=PD.$ Point $K,L$ lie on the segment $PB,PD,$ respectively, such that $KC$ bisects $\angle BKE,$ and $LC$ bisects $\angle ALD.$
Prove that $A,K,L,E$ are concyclic.
2003 India IMO Training Camp, 1
Let $A',B',C'$ be the midpoints of the sides $BC, CA, AB$, respectively, of an acute non-isosceles triangle $ABC$, and let $D,E,F$ be the feet of the altitudes through the vertices $A,B,C$ on these sides respectively. Consider the arc $DA'$ of the nine point circle of triangle $ABC$ lying outside the triangle. Let the point of trisection of this arc closer to $A'$ be $A''$. Define analogously the points $B''$ (on arc $EB'$) and $C''$(on arc $FC'$). Show that triangle $A''B''C''$ is equilateral.
The Golden Digits 2024, P1
On a table, there are $2025$ empty boxes numbered $1,2,\dots ,2025$, and $2025$ balls with weights $1,2,\dots ,2025$. Starting with Vadim, Vadim and Marian take turns selecting a ball from the table and placing it into an empty box. After all $2025$ turns, there is exactly one ball in each box. Denote the weight of the ball in box $i$ by $w_i$. Marian wins if $$\sum_{i=1}^{2025}i\cdot w_i\equiv 0 \pmod{23}.$$ If both players play optimally, can Marian guarantee a win?
[i]Proposed by Pavel Ciurea[/i]
PEN A Problems, 57
Prove that for every $n \in \mathbb{N}$ the following proposition holds: $7|3^n +n^3$ if and only if $7|3^{n} n^3 +1$.
1979 Spain Mathematical Olympiad, 2
A certain Oxford professor, assigned to espionage cryptography services British, role played by Dirk Bogarde in a film, recruits his proposing small attention exercises, such as mentally reading a word the other way around. Frequently he does it with his own name: $SEBASTIAN$, what will there be to read $NAITSABES$.
He wonders if there is any movement of the plane or of space that transforms one of these words in the other, just as they appear written. And if it had been called $AVITO$, like a certain Unamuno character? Give a reasoned explanation for each answer.
2022 Saudi Arabia JBMO TST, 1
Find all pairs of positive prime numbers $(p, q)$ such that
$$p^5 + p^3 + 2 = q^2 - q.$$
Kvant 2019, M2582
An integer $1$ is written on the blackboard. We are allowed to perform the following operations:to change the number $x$ to $3x+1$ of to $[\frac{x}{2}]$. Prove that we can get all positive integers using this operations.
1984 USAMO, 1
The product of two of the four roots of the quartic equation $x^4 - 18x^3 + kx^2+200x-1984=0$ is $-32$. Determine the value of $k$.
2009 Greece JBMO TST, 4
Find positive real numbers $x,y,z$ that are solutions of the system
$x+y+z=xy+yz+zx$ and $xyz=1$ , and have the smallest possible sum.
1974 Chisinau City MO, 73
For the real numbers $a_1,...,a_n, b_1,...,b_m$ , the following relations hold:
1) $|a_i|= |b_j|=1$, $i=1,...,n$ ,$j=1,...,m$
2) $a_1\sqrt{2+a_2\sqrt{2+...+a_n\sqrt2}}=b_1\sqrt{2+b_2\sqrt{2+...+b_m\sqrt2}}$
Prove that $n = m$ and $a_i=b_i$ , $i=1,...,n$
2025 District Olympiad, P1
Solve in real numbers the equation $$\log_7 (6^x+1)=\log_6(7^x-1).$$
[i]Mathematical Gazette[/i]