Found problems: 85335
1989 IMO Shortlist, 28
Consider in a plane $ P$ the points $ O,A_1,A_2,A_3,A_4$ such that \[ \sigma(OA_iA_j) \geq 1 \quad \forall i, j \equal{} 1, 2, 3, 4, i \neq j.\] where $ \sigma(OA_iA_j)$ is the area of triangle $ OA_iA_j.$ Prove that there exists at least one pair $ i_0, j_0 \in \{1, 2, 3, 4\}$ such that \[ \sigma(OA_iA_j) \geq \sqrt{2}.\]
Today's calculation of integrals, 877
Let $f(x)=\lim_{n\to\infty} \frac{\sin^{n+2}x+\cos^{n+2}x}{\sin^n x+\cos^n x}$ for $0\leq x\leq \frac{\pi}2.$
Evaluate $\int_0^{\frac{\pi}2} f(x)\ dx.$
2021 SAFEST Olympiad, 2
Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$.
[i]South Africa [/i]
2023 Denmark MO - Mohr Contest, 5
Georg has a circular game board with 100 squares labelled $1, 2, . . . , 100$. Georg chooses three numbers $a, b, c$ among the numbers $1, 2, . . . , 99$. The numbers need not be distinct. Initially there is a piece on the square labelled $100$. First, Georg moves the piece $a$ squares forward $33$ times and puts a caramel on each of the squares the piece lands on. Then he moves the piece $b$ squares forward $33$ times and puts a caramel on each of the squares the piece lands on. Finally, he moves the piece $c$ squares forward $33$ times and puts a caramel on each of the squares the piece lands on. Thus he puts a total of $99$ caramels on the board. Georg wins all the caramels on square number $1$. How many caramels can Georg win, at most?
[img]https://cdn.artofproblemsolving.com/attachments/d/c/af438e5feadca5b1bfc98ae427f6fc24655e29.png[/img]
LMT Guts Rounds, 18
Congruent unit circles intersect in such a way that the center of each circle lies on the circumference of the other. Let $R$ be the region in which two circles overlap. Determine the perimeter of $R.$
2020 USEMO, 4
A function $f$ from the set of positive real numbers to itself satisfies
$$f(x + f(y) + xy) = xf(y) + f(x + y)$$
for all positive real numbers $x$ and $y$. Prove that $f(x) = x$ for all positive real numbers $x$.
2005 Baltic Way, 15
Let the lines $e$ and $f$ be perpendicular and intersect each other at $H$. Let $A$ and $B$ lie on $e$ and $C$ and $D$ lie on $f$, such that all five points $A,B,C,D$ and $H$ are distinct. Let the lines $b$ and $d$ pass through $B$ and $D$ respectively, perpendicularly to $AC$; let the lines $a$ and $c$ pass through $A$ and $C$ respectively, perpendicularly to $BD$. Let $a$ and $b$ intersect at $X$ and $c$ and $d$ intersect at $Y$. Prove that $XY$ passes through $H$.
2024 JBMO TST - Turkey, 1
In the acute-angled triangle $ABC$, $P$ is the midpoint of segment $BC$ and the point $K$ is the foot of the altitude from $A$. $D$ is a point on segment $AP$ such that $\angle BDC=90$. Let $(ADK) \cap BC=E$ and $(ABC) \cap AE=F$. Prove that $\angle AFD=90$.
1999 Mongolian Mathematical Olympiad, Problem 3
Does there exist a sequence $(a_n)_{n\in\mathbb N}$ of distinct positive integers such that:
(i) $a_n<1999n$ for all $n$;
(ii) none of the $a_n$ contains three decimal digits $1$?
2021 Latvia Baltic Way TST, P1
Prove that for positive real numbers $a,b,c$ satisfying $abc=1$ the following inequality holds:
$$ \frac{a}{b(1+c)} +\frac{b}{c(1+a)}+\frac{c}{a(1+b)} \ge \frac{3}{2} $$
1997 Federal Competition For Advanced Students, Part 2, 1
Let $a$ be a fixed integer. Find all integer solutions $x, y, z$ of the system
\[5x + (a + 2)y + (a + 2)z = a,\]\[(2a + 4)x + (a^2 + 3)y + (2a + 2)z = 3a - 1,\]\[(2a + 4)x + (2a + 2)y + (a^2 + 3)z = a + 1.\]
1993 All-Russian Olympiad, 2
A convex quadrilateral intersects a circle at points $A_1,A_2,B_1,B_2,C_1,C_2,D_1,$ and $D_2$. (Note that for some letter $N$, points $N_1$ and $N_2$ are on one side of the quadrilateral. Also, the points lie in that specific order on the circle.) Prove that if $A_1B_2=B_1C_2=C_1D_2= D_1A_2$, then quadrilateral formed by these four segments is cyclic.
2017 ASDAN Math Tournament, 3
What is the remainder when $2^{1023}$ is divided by $1023$?
2024 Putnam, A6
Let $c_0,\,c_1,\,c_2,\,\ldots$ be a sequence defined so that
\[
\frac{1-3x-\sqrt{1-14x+9x^2}}{4}=\sum_{k=0}^\infty c_kx^k
\]
for sufficiently small $x$. For a positive integer $n$, let $A$ be the $n$-by-$n$ matrix with $i,j$-entry $c_{i+j-1}$ for $i$ and $j$ in $\{1,\,\ldots,\,n\}$. Find the determinant of $A$.
2008 Harvard-MIT Mathematics Tournament, 3
There are $ 5$ dogs, $ 4$ cats, and $ 7$ bowls of milk at an animal gathering. Dogs and cats are distinguishable, but all bowls of milk are the same. In how many ways can every dog and cat be paired with either a member of the other species or a bowl of milk such that all the bowls of milk are taken?
1967 Putnam, B6
Let $f$ be a real-valued function having partial derivatives and which is defined for $x^2 +y^2 \leq1$ and is such that $|f(x,y)|\leq 1.$ Show that there exists a point $(x_0, y_0 )$ in the interior of the unit circle such that
$$\left( \frac{ \partial f}{\partial x}(x_0 ,y_0 ) \right)^{2}+ \left( \frac{ \partial f}{\partial y}(x_0 ,y_0 ) \right)^{2} \leq 16.$$
2018 Czech-Polish-Slovak Match, 2
Let $ABC$ be an acute scalene triangle. Let $D$ and $E$ be points on the sides $AB$ and $AC$, respectively, such that $BD=CE$. Denote by $O_1$ and $O_2$ the circumcentres of the triangles $ABE$ and $ACD$, respectively. Prove that the circumcircles of the triangles $ABC, ADE$, and $AO_1O_2$ have a common point different from $A$.
[i]Proposed by Patrik Bak, Slovakia[/i]
2024 Romania National Olympiad, 1
Solve over the real numbers the equation $$3^{\log_5(5x-10)}-2=5^{-1+\log_3x}.$$
2024 China Team Selection Test, 20
A positive integer is a good number, if its base $10$ representation can be split into at least $5$ sections, each section with a non-zero digit, and after interpreting each section as a positive integer (omitting leading zero digits), they can be split into two groups, such that each group can be reordered to form a geometric sequence (if a group has $1$ or $2$ numbers, it is also a geometric sequence), for example $20240327$ is a good number, since after splitting it as $2|02|403|2|7$, $2|02|2$ and $403|7$ form two groups of geometric sequences.
If $a>1$, $m>2$, $p=1+a+a^2+\dots+a^m$ is a prime, prove that $\frac{10^{p-1}-1}{p}$ is a good number.
1960 Czech and Slovak Olympiad III A, 4
Determine the (real) domain of a function $$y=\sqrt{1-\frac{x}{4}|x|+\sqrt{1-\frac{x}{2}|x|\,}\,}-\sqrt{1-\frac{x}{4}|x|-\sqrt{1-\frac{x}{2}|x|\,}\,}$$ and draw its graph.
2022 Indonesia TST, G
Given an acute triangle $ABC$. with $H$ as its orthocenter, lines $\ell_1$ and $\ell_2$ go through $H$ and are perpendicular to each other. Line $\ell_1$ cuts $BC$ and the extension of $AB$ on $D$ and $Z$ respectively. Whereas line $\ell_2$ cuts $BC$ and the extension of $AC$ on $E$ and $X$ respectively. If the line through $D$ and parallel to $AC$ and the line through $E$ parallel to $AB$ intersects at $Y$, prove that $X,Y,Z$ are collinear.
2009 AMC 12/AHSME, 7
The first three terms of an arithmetic sequence are $ 2x\minus{}3$, $ 5x\minus{}11$, and $ 3x\plus{}1$ respectively. The $ n$th term of the sequence is $ 2009$. What is $ n$?
$ \textbf{(A)}\ 255 \qquad
\textbf{(B)}\ 502 \qquad
\textbf{(C)}\ 1004 \qquad
\textbf{(D)}\ 1506 \qquad
\textbf{(E)}\ 8037$
2014 Vietnam National Olympiad, 3
Find all sets of not necessary distinct 2014 rationals such that:if we remove an arbitrary number in the set, we can divide remaining 2013 numbers into three sets such that each set has exactly 671 elements and the product of all elements in each set are the same.
2019 IMO Shortlist, C6
Let $n>1$ be an integer. Suppose we are given $2n$ points in the plane such that no three of them are collinear. The points are to be labelled $A_1, A_2, \dots , A_{2n}$ in some order. We then consider the $2n$ angles $\angle A_1A_2A_3, \angle A_2A_3A_4, \dots , \angle A_{2n-2}A_{2n-1}A_{2n}, \angle A_{2n-1}A_{2n}A_1, \angle A_{2n}A_1A_2$. We measure each angle in the way that gives the smallest positive value (i.e. between $0^{\circ}$ and $180^{\circ}$). Prove that there exists an ordering of the given points such that the resulting $2n$ angles can be separated into two groups with the sum of one group of angles equal to the sum of the other group.
1982 National High School Mathematics League, 1
For a convex polygon with $n$ edges $F$, if all its diagonals have the equal length, then
$\text{(A)}F\in \{\text{quadrilaterals}\}$
$\text{(B)}F\in \{\text{pentagons}\}$
$\text{(C)}F\in \{\text{pentagons}\} \cup\{\text{quadrilaterals}\}$
$\text{(D)}F\in \{\text{convex polygons that have all edges' length equal}\} \cup\{\text{convex polygons that have all inner angles equal}\}$