Found problems: 85335
2014 China Northern MO, 1
As shown in the figure, given $\vartriangle ABC$ with $\angle B$, $\angle C$ acute angles, $AD \perp BC$, $DE \perp AC$, $M$ midpoint of $DE$, $AM \perp BE$. Prove that $\vartriangle ABC$ is isosceles.
[img]https://cdn.artofproblemsolving.com/attachments/a/8/f553c33557979f6f7b799935c3bde743edcc3c.png[/img]
2005 Today's Calculation Of Integral, 75
A function $f(\theta)$ satisfies the following conditions $(a),(b)$.
$(a)\ f(\theta)\geq 0$
$(b)\ \int_0^{\pi} f(\theta)\sin \theta d\theta =1$
Prove the following inequality.
\[\int_0^{\pi} f(\theta)\sin n\theta \ d\theta \leq n\ (n=1,2,\cdots)\]
2017 Indonesia MO, 1
$ABCD$ is a parallelogram. $g$ is a line passing $A$. Prove that the distance from $C$ to $g$ is either the sum or the difference of the distance from $B$ to $g$, and the distance from $D$ to $g$.
2024/2025 TOURNAMENT OF TOWNS, P5
Given a circle ${\omega }_{1}$ , and a circle ${\omega }_{2}$ inside it. An arbitrary circle ${\omega }_{3}$ is chosen which is tangent to the two latter circles and both tangencies are internal. The tangency points are linked by a segment. A tangent line to ${\omega }_{2}$ is drawn through the meet point of this segment and the circle ${\omega }_{2}$ . Thus a chord of the circle ${\omega }_{3}$ is obtained. Prove that the ends of all such chords (obtained by all possible choices of ${\omega }_{3}$ ) belong to a fixed circle.
Pavel Kozhevnikov
2021 Purple Comet Problems, 16
Paula rolls three standard fair dice. The probability that the three numbers rolled on the dice are the side lengths of a triangle with positive area is $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2000 Romania Team Selection Test, 2
Let $P,Q$ be two monic polynomials with complex coefficients such that $P(P(x))=Q(Q(x))$ for all $x$. Prove that $P=Q$.
[i]Marius Cavachi[/i]
2013 IPhOO, 6
A fancy bathroom scale is calibrated in Newtons. This scale is put on a ramp, which is at a $40^\circ$ angle to the horizontal. A box is then put on the scale and the box-scale system is then pushed up the ramp by a horizontal force $F$. The system slides up the ramp at a constant speed. If the bathroom scale reads $R$ and the coefficient of static friction between the system and the ramp is $0.40$, what is $\frac{F}{R}$? Round to the nearest thousandth.
[i](Proposed by Ahaan Rungta)[/i]
1997 Baltic Way, 9
The worlds in the Worlds’ Sphere are numbered $1,2,3,\ldots $ and connected so that for any integer $n\ge 1$, Gandalf the Wizard can move in both directions between any worlds with numbers $n,2n$ and $3n+1$. Starting his travel from an arbitrary world, can Gandalf reach every other world?
1996 Cono Sur Olympiad, 5
We want to cover totally a square(side is equal to $k$ integer and $k>1$) with this rectangles:
$1$ rectangle ($1\times 1$), $2$ rectangles ($2\times 1$), $4$ rectangles ($3\times 1$),...., $2^n$ rectangles ($n + 1 \times 1$), such that the rectangles can't overlap and don't exceed the limits of square.
Find all $k$, such that this is possible and for each $k$ found you have to draw a solution
2003 Bulgaria Team Selection Test, 5
Let $ABCD$ be a circumscribed quadrilateral and let $P$ be the orthogonal projection of its in center on $AC$.
Prove that $\angle {APB}=\angle {APD}$
1976 Miklós Schweitzer, 9
Let $ D$ be a convex subset of the $ n$-dimensional space, and suppose that $ D'$ is obtained from $ D$ by applying a positive central dilatation and then a translation. Suppose also that the sum of the volumes of $ D$ and $ D'$ is $ 1$, and $ D \cap D'\not\equal{} \emptyset .$ Determine the supremum of the volume of the convex hull of $ D \cup D'$ taken for all such pairs of sets $ D,D'$.
[i]L. Fejes-Toth, E. Makai[/i]
1998 Czech and Slovak Match, 2
A polynomial $P(x)$ of degree $n \ge 5$ with integer coefficients has $n$ distinct integer roots, one of which is $0$. Find all integer roots of the polynomial $P(P(x))$.
2007 Spain Mathematical Olympiad, Problem 2
Determine all the possible non-negative integer values that are able to satisfy the expression:
$\frac{(m^2+mn+n^2)}{(mn-1)}$
if $m$ and $n$ are non-negative integers such that $mn \neq 1$.
2002 Olympic Revenge, 1
Show that there is no function \(f:\mathbb{N}^* \rightarrow \mathbb{N}^*\) such that \(f^n(n)=n+1\) for all \(n\) (when \(f^n\) is the \(n\)th iteration of \(f\))
2004 Singapore MO Open, 1
Let $m,n$ be integers so that $m \ge n > 1$. Let $F_1,...,F_k$ be a collection of $n$-element subsets of $\{1,...,m\}$ so that $F_i\cap F_j$ contains at most $1$ element, $1 \le i < j \le k$. Show that $k\le \frac{m(m-1)}{n(n-1)} $
1988 Canada National Olympiad, 4
Let $x_{n + 1} = 4x_n - x_{n - 1}$, $x_0 = 0$, $x_1 = 1$, and $y_{n + 1} = 4y_n - y_{n - 1}$, $y_0 = 1$, $y_1 = 2$. Show that for all $n \ge 0$ that $y_n^2 = 3x_n^2 + 1$.
2015 China Team Selection Test, 1
The circle $\Gamma$ through $A$ of triangle $ABC$ meets sides $AB,AC$ at $E$,$F$ respectively, and circumcircle of $ABC$ at $P$. Prove: Reflection of $P$ across $EF$ is on $BC$ if and only if $\Gamma$ passes through $O$ (the circumcentre of $ABC$).
1969 All Soviet Union Mathematical Olympiad, 127
Let $h_k$ be an apothem of the regular $k$-gon inscribed into a circle with radius $R$. Prove that $$(n + 1)h_{n+1} - nh_n > R$$
1971 IMO Longlists, 45
A broken line $A_1A_2 \ldots A_n$ is drawn in a $50 \times 50$ square, so that the distance from any point of the square to the broken line is less than $1$. Prove that its total length is greater than $1248.$
2022 Purple Comet Problems, 5
Below is a diagram showing a $6 \times 8$ rectangle divided into four $6 \times 2$ rectangles and one diagonal line. Find the total perimeter of the four shaded trapezoids.
2020 CHMMC Winter (2020-21), 6
Suppose that
\[
\prod_{n=1}^{\infty}\left(\frac{1+i\cot\left(\frac{n\pi}{2n+1}\right)}{1-i\cot\left(\frac{n\pi}{2n+1}\right)}\right)^{\frac{1}{n}} = \left(\frac{p}{q}\right)^{i \pi},
\]
where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
[i]Note: for a complex number $z = re^{i \theta}$ for reals $r > 0, 0 \le \theta < 2\pi$, we define $z^{n} = r^{n} e^{i \theta n}$ for all positive reals $n$.[/i]
1971 Spain Mathematical Olympiad, 6
The velocities of a submerged and surfaced submarine are, respectively, $v$ and $kv$. It is situated at a point $P$ at $30$ miles from the center $O$ of a circle of $60$ mile radius. The surveillance of an enemy squadron forces him to navigate submerged while inside the circle. Discuss, according to the values of $k$, the fastest path to move to the opposite end of the diameter that passes through $P$ . (Consider the case particular $k =\sqrt5$.)
2008 Sharygin Geometry Olympiad, 4
(A.Zaslavsky) Given three points $ C_0$, $ C_1$, $ C_2$ on the line $ l$. Find the locus of incenters of triangles $ ABC$ such that points $ A$, $ B$ lie on $ l$ and the feet of the median, the bisector and the altitude from $ C$ coincide with $ C_0$, $ C_1$, $ C_2$.
2023 Portugal MO, 3
A crate with a base of $4 \times 2$ and a height of $2$ is open at the top. Tomas wants to completely fill the crate with some of his cubes. It has $16$ equal cubes of volume $1$ and two equal cubes of volume $8$. A cube of volume $1$ can only be placed on the top layer if the cube on the bottom layer has already been placed. In how many ways can Tom'as fill the box with cubes, placing them one by one?
2014 China Team Selection Test, 6
Let $k$ be a fixed even positive integer, $N$ is the product of $k$ distinct primes $p_1,...,p_k$, $a,b$ are two positive integers, $a,b\leq N$. Denote
$S_1=\{d|$ $d|N, a\leq d\leq b, d$ has even number of prime factors$\}$,
$S_2=\{d|$ $d|N, a\leq d\leq b, d$ has odd number of prime factors$\}$,
Prove: $|S_1|-|S_2|\leq C^{\frac{k}{2}}_k$