Found problems: 85335
2001 China Team Selection Test, 2
Let $\theta_i \in \left ( 0,\frac{\pi}{4} \right ]$ for $i=1,2,3,4$. Prove that:
$\tan \theta _1 \tan \theta _2 \tan \theta _3 \tan \theta _4 \le (\frac{\sin^8 \theta _1+\sin^8 \theta _2+\sin^8 \theta _3+\sin^8 \theta _4}{\cos^8 \theta _1+\cos^8 \theta _2+\cos^8 \theta _3+\cos^8 \theta _4})^\frac{1}{2}$
[hide=edit]@below, fixed now. There were some problems (weird characters) so aops couldn't send it.[/hide]
1987 Vietnam National Olympiad, 3
Let be given $ n \ge 2$ lines on a plane, not all concurrent and no two parallel. Prove that there is a point which belongs to exactly two of the given lines.
2020 Malaysia IMONST 2, 4
Given are four circles $\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4$. Circles $\Gamma_1$ and $\Gamma_2$ are externally tangent at point $A$. Circles $\Gamma_2$ and
$\Gamma_3$ are externally tangent at point $B$. Circles $\Gamma_3$ and $\Gamma_4$ are externally tangent at point $C$. Circles $\Gamma_4$ and
$\Gamma_1$ are externally tangent at point $D$. Prove that $ABCD$ is cyclic.
2015 BMT Spring, 11
Write down $1, 2, 3, ... , 2015$ in a row on a whiteboard. Every minute, select a pair of adjacent numbers at random, erase them, and insert their sum where you selected the numbers. (For instance, selecting $3$ and $4$ from $1, 2, 3, 4, 5$ would result in $1, 2, 7, 5$.) Repeat this process until you have two numbers remaining. What is the probability that the smaller number is less than or equal to $2015$?
1989 AIME Problems, 1
Compute $\sqrt{(31)(30)(29)(28)+1}$.
2012 ELMO Shortlist, 2
Determine whether it's possible to cover a $K_{2012}$ with
a) 1000 $K_{1006}$'s;
b) 1000 $K_{1006,1006}$'s.
[i]David Yang.[/i]
1994 Poland - First Round, 7
(a) Find out, whether there exists a differentiable function $f: R \longrightarrow R$, not equaling $0$ for all $x \in R$, satisfying the conditions $2f(f(x)) = f(x) \geq 0$ for all $x \in R$.
(b) Find out, whether there exists a differentiable function $f: R \longrightarrow R$, not equaling $0$ for all $x \in R$, satisfying the conditions $-1 \leq 2f(f(x)) = f(x) \leq 1$ for all $x \in R$.
2023 Regional Olympiad of Mexico Southeast, 1
Victor writes down all $7-$digit numbers using the digits $1, 2, 3, 4, 5, 6,$ and $7$ exactly once. Prove that there are no two numbers among them where one is a multiple of the other.
2013 BMT Spring, 1
Find the value of $a$ satisfying
\begin{align*}
a+b&=3\\
b+c&=11\\
c+a&=61
\end{align*}
2006 AMC 10, 12
The lines $ x \equal{} \frac 14y \plus{} a$ and $ y \equal{} \frac 14x \plus{} b$ intersect at the point $ (1,2)$. What is $ a \plus{} b$?
$ \textbf{(A) } 0 \qquad \textbf{(B) } \frac 34 \qquad \textbf{(C) } 1 \qquad \textbf{(D) } 2 \qquad \textbf{(E) } \frac 94$
2024-25 IOQM India, 17
Consider an isosceles triangle $ABC$ with sides $BC = 30, CA = AB = 20$. Let $D$ be the foot of the perpendicular from $A$ to $BC$, and let $M$ be the midpoint of $AD$. Let $PQ$ be a chord of the circumcircle of triangle $ABC$, such that $M$ lies on $PQ$ and $PQ$ is parallel to $BC$. The length of $PQ$ is:
2007 Thailand Mathematical Olympiad, 6
A triangle has perimeter $2s$, inradius $r$, and incenter $I$. If $s_a, s_b$ and $s_c$ are the distances from $I$ to the three vertices, then show that $$\frac34 +\frac{r}{s_a}+\frac{r}{s_b}+\frac{r}{s_c} \le \frac{s^2}{12r^2}$$
2017 Harvard-MIT Mathematics Tournament, 10
Let $ABC$ be a triangle in the plane with $AB = 13$, $BC = 14$, $AC = 15$. Let $M_n$ denote the smallest possible value of $(AP^n + BP^n + CP^n)^{\frac{1}{n}}$ over all points $P$ in the plane. Find $\lim_{n \to \infty} M_n$.
2022 Junior Macedonian Mathematical Olympiad, P3
Let $\triangle ABC$ be an acute triangle with orthocenter $H$. The circle $\Gamma$ with center $H$ and radius $AH$ meets the lines $AB$ and $AC$ at the points $E$ and $F$ respectively. Let $E'$, $F'$ and $H'$ be the reflections of the points $E$, $F$ and $H$ with respect to the line $BC$, respectively. Prove that the points $A$, $E'$, $F'$ and $H'$ lie on a circle.
[i]Proposed by Jasna Ilieva[/i]
2000 239 Open Mathematical Olympiad, 2
100 volleyball teams played a one-round tournament. No two matches held at the same time. It turned out that before each match the teams playing against each other had the same number of wins. Find all possible number of wins for the winner of this tournament.
1965 Poland - Second Round, 6
Prove that there is no polyhedron whose every plane section is a triangle.
2005 Today's Calculation Of Integral, 81
Prove the following inequality.
\[\frac{1}{12}(\pi -6+2\sqrt{3})\leq \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \ln (1+\cos 2x) dx\leq \frac{1}{4}(2-\sqrt{3})\]
1987 Czech and Slovak Olympiad III A, 1
Given a trapezoid, divide it by a line into two quadrilaterals in such a way that both of them are cyclic with the same circumradius. Discuss conditions of solvability.
1995 AMC 8, 5
Find the smallest whole number that is larger than the sum
\[2\dfrac{1}{2}+3\dfrac{1}{3}+4\dfrac{1}{4}+5\dfrac{1}{5}.\]
$\text{(A)}\ 14 \qquad \text{(B)}\ 15 \qquad \text{(C)}\ 16 \qquad \text{(D)}\ 17 \qquad \text{(E)}\ 18$
1978 IMO Shortlist, 12
In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$
2009 Sharygin Geometry Olympiad, 7
Given two intersecting circles with centers $O_1, O_2$. Construct the circle touching one of them externally and the second one internally such that the distance from its center to $O_1O_2$ is maximal.
(M.Volchkevich)
2017 VJIMC, 2
Prove or disprove the following statement. If $g:(0,1) \to (0,1)$ is an increasing function and satisfies
$g(x) > x$ for all $x \in (0,1)$, then there exists a continuous function $f:(0,1) \to \mathbb{R}$ satisfying $f(x) < f(g(x)) $ for all $x \in (0,1)$, but $f$ is not an increasing function.
2016 Japan MO Preliminary, 7
Let $a, b, c, d$ be real numbers satisfying the system of equation
$\[(a+b)(c+d)=2 \\
(a+c)(b+d)=3 \\
(a+d)(b+c)=4\]$
Find the minimum value of $a^2+b^2+c^2+d^2$.
2017 Canadian Mathematical Olympiad Qualification, 6
Let $N$ be a positive integer. There are $N$ tasks, numbered $1, 2, 3, \ldots, N$, to be completed. Each task takes one minute to complete and the tasks must be completed subjected to the following conditions:
[list]
[*] Any number of tasks can be performed at the same time.
[*] For any positive integer $k$, task $k$ begins immediately after all tasks whose numbers are divisors of $k$, not including $k$ itself, are completed.
[*] Task 1 is the first task to begin, and it begins by itself.
[/list]
Suppose $N = 2017$. How many minutes does it take for all of the tasks to complete? Which tasks are the last ones to complete?
2025 Al-Khwarizmi IJMO, 6
Let $a,b,c$ be real numbers such that \[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\] Find the smallest possible value of $a^2 + b^2 + c^2$.
[i]Binh Luan and Nhan Xet, Vietnam[/i]