Found problems: 85335
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.
2020 LMT Fall, A30
A large gathering of people stand in a triangular array with $2020$ rows, such that the first row has $1$ person, the second row has $2$ people, and so on. Every day, the people in each row infect all of the people adjacent to them in their own row. Additionally, the people at the ends of each row infect the people at the ends of the rows in front of and behind them that are at the same side of the row as they are. Given that two people are chosen at random to be infected with COVID at the beginning of day 1, what is the earliest possible day that the last uninfected person will be infected with COVID?
[i]Proposed by Richard Chen[/i]
1993 AMC 12/AHSME, 17
Amy painted a dart board over a square clock face using the "hour positions" as boundaries. [See figure.] If $t$ is the area of one of the eight triangular regions such as that between $12$ o'clock and $1$ o'clock, and $q$ is the area of one of the four corner quadrilaterals such as that between $1$ o'clock and $2$ o'clock, then $\frac{q}{t}=$
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draw((0,0)--(4,0)--(4,4)--(0,4)--(0,0)--(.9,0)--(3.1,4)--(.9,4)--(3.1,0)--(2,0)--(2,4));
draw((0,3.1)--(4,.9)--(4,3.1)--(0,.9)--(0,2)--(4,2));
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$ \textbf{(A)}\ 2\sqrt{3}-2 \qquad\textbf{(B)}\ \frac{3}{2} \qquad\textbf{(C)}\ \frac{\sqrt{5}+1}{2} \qquad\textbf{(D)}\ \sqrt{3} \qquad\textbf{(E)}\ 2 $