Found problems: 85335
2024 Caucasus Mathematical Olympiad, 2
In an acute-angled triangle $ABC$ let $BL$ be the bisector, and let $BK$ be the altitude. Let the lines $BL$ and $BK$ meet the circumcircle of $ABC$ again at $W$ and $T$, respectively. Given that $BC = BW$, prove that $TL \perp BC$.
2021 Dutch IMO TST, 3
Let $ABC$ be an acute-angled and non-isosceles triangle with orthocenter $H$. Let $O$ be the center of the circumscribed circle of triangle $ABC$ and let $K$ be center of the circumscribed circle of triangle $AHO$. Prove that the reflection of $K$ wrt $OH$ lies on $BC$.
2016 IMO Shortlist, N8
Find all polynomials $P(x)$ of odd degree $d$ and with integer coefficients satisfying the following property: for each positive integer $n$, there exists $n$ positive integers $x_1, x_2, \ldots, x_n$ such that $\frac12 < \frac{P(x_i)}{P(x_j)} < 2$ and $\frac{P(x_i)}{P(x_j)}$ is the $d$-th power of a rational number for every pair of indices $i$ and $j$ with $1 \leq i, j \leq n$.
1953 AMC 12/AHSME, 19
In the expression $ xy^2$, the values of $ x$ and $ y$ are each decreased $ 25\%$; the value of the expression is:
$ \textbf{(A)}\ \text{decreased } 50\% \qquad\textbf{(B)}\ \text{decreased }75\%\\
\textbf{(C)}\ \text{decreased }\frac{37}{64}\text{ of its value} \qquad\textbf{(D)}\ \text{decreased }\frac{27}{64}\text{ of its value}\\
\textbf{(E)}\ \text{none of these}$
2015 IMO Shortlist, G6
Let $ABC$ be an acute triangle with $AB > AC$. Let $\Gamma $ be its circumcircle, $H$ its orthocenter, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\Gamma$ such that $\angle HQA = 90^{\circ}$ and let $K$ be the point on $\Gamma$ such that $\angle HKQ = 90^{\circ}$. Assume that the points $A$, $B$, $C$, $K$ and $Q$ are all different and lie on $\Gamma$ in this order.
Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other.
Proposed by Ukraine
2021 IMO, 1
Let $n \geqslant 100$ be an integer. Ivan writes the numbers $n, n+1, \ldots, 2 n$ each on different cards. He then shuffles these $n+1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.
2011 Sharygin Geometry Olympiad, 8
Using only the ruler, divide the side of a square table into $n$ equal parts.
All lines drawn must lie on the surface of the table.
1998 VJIMC, Problem 4-M
Prove the inequality
$$\frac{n\pi}4-\frac1{\sqrt{8n}}\le\frac12+\sum_{k=1}^{n-1}\sqrt{1-\frac{k^2}{n^2}}\le\frac{n\pi}4$$for every integer $n\ge2$.
2017 China Girls Math Olympiad, 6
Given a finite set $X$, two positive integers $n,k$, and a map $f:X\to X$. Define $f^{(1)}(x)=f(x),f^{(i+1)}(x)=f^{(i)}(x)$,$i=1,2,3,\ldots$. It is known that for any $x\in X$,$f^{(n)}(x)=x$.
Define $m_j$ the number of $x\in X$ satisfying $f^{(j)}(x)=x$.
Prove that:
(1)$\frac{1}n \sum_{j=1}^n m_j\sin {\frac{2kj\pi}{n}}=0$
(2)$\frac{1}n \sum_{j=1}^n m_j\cos {\frac{2kj\pi}{n}}$ is a non-negative integer.
2008 Serbia National Math Olympiad, 6
In a convex pentagon $ ABCDE$, let $ \angle EAB \equal{} \angle ABC \equal{} 120^{\circ}$, $ \angle ADB \equal{} 30^{\circ}$ and $ \angle CDE \equal{} 60^{\circ}$. Let $ AB \equal{} 1$. Prove that the area of the pentagon is less than $ \sqrt {3}$.
2015 ELMO Problems, 4
Let $a > 1$ be a positive integer. Prove that for some nonnegative integer $n$, the number $2^{2^n}+a$ is not prime.
[i]Proposed by Jack Gurev[/i]
2025 Kosovo National Mathematical Olympiad`, P4
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ for which these two conditions hold simultaneously
(i) For all $m,n \in \mathbb{N}$ we have:
$$ \frac{f(mn)}{\gcd(m,n)} = \frac{f(m)f(n)}{f(\gcd(m,n))};$$
(ii) For all prime numbers $p$, there exists a prime number $q$ such that $f(p^{2025})=q^{2025}$.
2004 Romania National Olympiad, 2
Let $P(n)$ be the number of functions $f: \mathbb{R} \to \mathbb{R}$, $f(x)=a x^2 + b x + c$, with $a,b,c \in \{1,2,\ldots,n\}$ and that have the property that $f(x)=0$ has only integer solutions. Prove that $n<P(n)<n^2$, for all $n \geq 4$.
[i]Laurentiu Panaitopol[/i]
2002 Iran Team Selection Test, 4
$O$ is a point in triangle $ABC$. We draw perpendicular from $O$ to $BC,AC,AB$ which intersect $BC,AC,AB$ at $A_{1},B_{1},C_{1}$. Prove that $O$ is circumcenter of triangle $ABC$ iff perimeter of $ABC$ is not less than perimeter of triangles $AB_{1}C_{1},BC_{1}A_{1},CB_{1}A_{1}$.
2021 SEEMOUS, Problem 3
Let $A \in \mathcal{M}_n(\mathbb{C})$ be a matrix such that $(AA^*)^2=A^*A$, where $A^*=(\bar{A})^t$ denotes the Hermitian transpose (i.e., the conjugate transpose) of $A$.
(a) Prove that $AA^*=A^*A$.
(b) Show that the non-zero eigenvalues of $A$ have modulus one.
2001 239 Open Mathematical Olympiad, 3
The circles $ S_1 $ and $ S_2 $ intersect at points $ A $ and $ B $. Circle $ S_3 $ externally touches $ S_1 $ and $ S_2 $ at points $ C $ and $ D $ respectively. Let $ PQ $ be a chord cut by the line $ AB $ on circle $ S_3 $, and $ K $ be the midpoint of $ CD $. Prove that $ \angle PKC = \angle QKC $.
2013 USAJMO, 5
Quadrilateral $XABY$ is inscribed in the semicircle $\omega$ with diameter $XY$. Segments $AY$ and $BX$ meet at $P$. Point $Z$ is the foot of the perpendicular from $P$ to line $XY$. Point $C$ lies on $\omega$ such that line $XC$ is perpendicular to line $AZ$. Let $Q$ be the intersection of segments $AY$ and $XC$. Prove that \[\dfrac{BY}{XP}+\dfrac{CY}{XQ}=\dfrac{AY}{AX}.\]
2019 SIMO, Q3
In a scalene triangle $ABC$, the incircle touches $BC, AC$ and $AB$ at $D, E, F$ respectively. Let $K$ be the foot of the perpendicular from $A$ onto $BC$, and $M$ the midpoint of $BC$. Let $AD$ intersect the incircle again at $X$, and $BE$ at $Y$. Given that $E,F,K,M$ are concyclic, prove that $AX=XY=YD$.
2023 Harvard-MIT Mathematics Tournament, 3
Suppose $ABCD$ is a rectangle whose diagonals meet at $E$. The perimeter of triangle $ABE$ is $10\pi$ and the perimeter of triangle $ADE$ is $n$. Compute the number of possible integer values of $n$.
2007 Today's Calculation Of Integral, 221
Evaluate $ \int_{2}^{6}\ln\frac{\minus{}1\plus{}\sqrt{1\plus{}4x}}{2}\ dx$.
2025 Malaysian IMO Team Selection Test, 6
A sequence $2^{a_1}, 2^{a_2}, \cdots,2^{a_m}$ is called [i]good[/i], if $a_i$ are non-negative integers, and $a_{i+1}-a_{i}$ is either $0$ or $1$ for all $1\le i\le m-1$.
Fix a positive integer $n$, and Ivan has a whiteboard with some ones written on it. In each step, he may erase any good sequence $2^{a_1}, 2^{a_2}, \cdots,2^{a_m}$ that appears on the whiteboard, and then he writes the number $2^k$ such that $$2^{k-1}<2^{a_1}+2^{a_2}+\cdots+2^{a_m}\le 2^{k}$$ Suppose Ivan starts with the least possible number of ones to obtain $2^n$ after some steps, determine the minimum number of steps he will need in order to do so.
[i]Proposed by Ivan Chan Kai Chin[/i]
1996 IMO Shortlist, 3
Let $ a > 2$ be given, and starting $ a_0 \equal{} 1, a_1 \equal{} a$ define recursively:
\[ a_{n\plus{}1} \equal{} \left(\frac{a^2_n}{a^2_{n\minus{}1}} \minus{} 2 \right) \cdot a_n.\]
Show that for all integers $ k > 0,$ we have: $ \sum^k_{i \equal{} 0} \frac{1}{a_i} < \frac12 \cdot (2 \plus{} a \minus{} \sqrt{a^2\minus{}4}).$
2011 Serbia JBMO TST, 4
If a, b, c are positive real numbers with $ a+b+c=1 $. Find the minimum value of $ \sqrt{a}+\sqrt{b}+\sqrt{c}+\frac{1}{\sqrt{abc}} $
2020 Malaysia IMONST 1, 10
Given positive integers $a, b,$ and $c$ with $a + b + c = 20$.
Determine the number of possible integer values for $\frac{a + b}{c}.$
2016 District Olympiad, 4
Let $ f:[0,1]\longrightarrow [0,1] $ be a nondecreasing function. Prove that the sequence
$$ \left( \int_0^1 \frac{1+f^n(x)}{1+f^{1+n} (x)} \right)_{n\ge 1} $$
is convergent and calculate its limit.