Found problems: 85335
1973 AMC 12/AHSME, 13
The fraction $ \frac{2(\sqrt2 \plus{} \sqrt6)}{3\sqrt{2\plus{}\sqrt3}}$ is equal to
$ \textbf{(A)}\ \frac{2\sqrt2}{3} \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ \frac{2\sqrt3}3 \qquad
\textbf{(D)}\ \frac43 \qquad
\textbf{(E)}\ \frac{16}{9}$
2012 Switzerland - Final Round, 10
Let $O$ be an inner point of an acute-angled triangle $ABC$. Let $A_1, B_1$ and $C_1$ be the projections of $O$ on the sides $BC, AC$ and $AB$ respectively . Let $P$ be the intersection of the perpendiculars on $B_1C_1$ and $A_1C_1$ from points$ A$ and $B$ respectilvey. Let $H$ be the projection of $P$ on $AB$. Show that points $A_1, B_1, C_1$ and $H$ lie on a circle.
2022 DIME, 2
Let $P(x) = x^2-1$ be a polynomial, and let $a$ be a positive real number satisfying$$P(P(P(a))) = 99.$$ The value of $a^2$ can be written as $m+\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.
[i]Proposed by [b]HrishiP[/b][/i]
2013 NIMO Problems, 5
Zang is at the point $(3,3)$ in the coordinate plane. Every second, he can move one unit up or one unit right, but he may never visit points where the $x$ and $y$ coordinates are both composite. In how many ways can he reach the point $(20, 13)$?
[i]Based on a proposal by Ahaan Rungta[/i]
2014 China National Olympiad, 1
Let $ABC$ be a triangle with $AB>AC$. Let $D$ be the foot of the internal angle bisector of $A$. Points $F$ and $E$ are on $AC,AB$ respectively such that $B,C,F,E$ are concyclic. Prove that the circumcentre of $DEF$ is the incentre of $ABC$ if and only if $BE+CF=BC$.
1994 Swedish Mathematical Competition, 4
Find all integers $m, n$ such that $2n^3 - m^3 = mn^2 + 11$.
2007 Purple Comet Problems, 3
A bowl contained $10\%$ blue candies and $25\%$ red candies. A bag containing three quarters red candies and one quarter blue candies was added to the bowl. Now the bowl is $16\%$ blue candies. What percentage of the candies in the bowl are now red?
2002 Mid-Michigan MO, 10-12
[b]p1.[/b] Find all integer solutions of the equation $a^2 - b^2 = 2002$.
[b]p2.[/b] Prove that the disks drawn on the sides of a convex quadrilateral as on diameters cover this quadrilateral.
[b]p3.[/b] $30$ students from one school came to Mathematical Olympiad. In how many different ways is it possible to place them in four rooms?
[b]p4.[/b] A $12$ liter container is filled with gasoline. How to split it in two equal parts using two empty $5$ and $8$ liter containers?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2001 AIME Problems, 13
In quadrilateral $ABCD$, $\angle{BAD}\cong\angle{ADC}$ and $\angle{ABD}\cong\angle{BCD}$, $AB=8$, $BD=10$, and $BC=6$. The length $CD$ may be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Kyiv City MO 1984-93 - geometry, 1993.9.3
The circle divides each side of an equilateral triangle into three equal parts. Prove that the sum of the squares of the distances from any point of this circle to the vertices of the triangle is constant.
2000 Slovenia National Olympiad, Problem 2
Find all functions $f:\mathbb R\to\mathbb R$ such that for all $x,y\in\mathbb R$,
$$f(x-f(y))=1-x-y.$$
2023 Sharygin Geometry Olympiad, 4
Points $D$ and $E$ lie on the lateral sides $AB$ and $BC$ respectively of an isosceles triangle $ABC$ in such a way that $\angle BED = 3\angle BDE$. Let $D'$ be the reflection of $D$ about $AC$. Prove that the line $D'E$ passes through the incenter of $ABC$.
1974 Putnam, A4
An unbiased coin is tossed $n$ times. What is the expected value of $|H-T|$, where $H$ is the number of heads and $T$ is the number of tails?
1997 Tournament Of Towns, (563) 4
(a) Several identical napkins, each in the shape of a regular hexagon, are put on a table (the napkins may overlap). Each napkin has one side which is parallel to a fixed line. Is it always possible to hammer a few nails into the table so that each napkin is nailed with exactly one nail?
(b) The same question for regular pentagons.
(A Kanel)
2022 Bulgarian Autumn Math Competition, Problem 10.4
The European zoos with exactly $100$ types of species each are separated into two groups $\hat{A}$ and $\hat{B}$ in such a way that every pair of zoos $(A, B)$ $(A\in\hat{A}, B\in\hat{B})$ have some animal in common. Prove that we can colour the cages in $3$ colours (all animals of the same type live in the same cage) such that no zoo has cages of only one colour
MathLinks Contest 2nd, 2.2
Let $\{a_n\}_{n\ge 0}$ be a sequence of rational numbers given by $a_0 = a_1 = a_2 = a_3 = 1$ and for all $n \ge 4$ we have $a_{n-4}a_n = a_{n-3}a_{n-1} + a^2_{n-2}$. Prove that all the terms of the sequence are integers.
2001 Tuymaada Olympiad, 3
Let ABC be an acute isosceles triangle ($AB=BC$) inscribed in a circle with center $O$ . The line through the midpoint of the chord $AB$ and point $O$ intersects the line $AC$ at $L$ and the circle at the point $P$. Let the bisector of angle $BAC$ intersects the circle at point $K$. Lines $AB$ and $PK$ intersect at point $D$. Prove that the points $L,B,D$ and $P$ lie on the same circle.
2019 May Olympiad, 4
You have to divide a square paper into three parts, by two straight cuts, so that by locating these parts properly, without gaps or overlaps, an obtuse triangle is formed. Indicate how to cut the square and how to assemble the triangle with the three parts.
2020 USAMTS Problems, 2:
Infinitely many math beasts stand in a line, all six feet apart, wearing masks, and with clean hands. Grogg starts at the front of the line, holding $n$ pieces of candy, $ n \ge 1,$ and everyone else has none. He passes his candy to the beasts behind him, one piece each to the next $n$ beasts in line. Then, Grogg leaves the line. The other beasts repeat this process: the beast in front, who has $k$ pieces of candy, passes one piece each to the next $k$ beasts in line, and then leaves the line. For some values of $n,$ another beast, besides Grogg, temporarily holds all the candy. For which values of $n$ does this occur?
2016 HMIC, 3
Denote by $\mathbb{N}$ the positive integers. Let $f:\mathbb{N} \rightarrow \mathbb{N}$ be a function such that, for any $w,x,y,z \in \mathbb{N}$, \[ f(f(f(z)))f(wxf(yf(z)))=z^{2}f(xf(y))f(w). \] Show that $f(n!) \ge n!$ for every positive integer $n$.
[i]Pakawut Jiradilok[/i]
Today's calculation of integrals, 881
Evaluate $\int_{-\pi}^{\pi} \left(\sum_{k=1}^{2013} \sin kx\right)^2dx$.
2012 Math Prize For Girls Problems, 3
What is the least positive integer $n$ such that $n!$ is a multiple of $2012^{2012}$?
2022 Bulgarian Autumn Math Competition, Problem 11.1
Find all real numbers $q$, such that for all real $p \geq 0$, the equation $x^2-2px+q^2+q-2=0$ has at least one real root in $(-1;0)$.
2021 CMIMC, 1.5
Let $\gamma_1, \gamma_2, \gamma_3$ be three circles with radii $3, 4, 9,$ respectively, such that $\gamma_1$ and $\gamma_2$ are externally tangent at $C,$ and $\gamma_3$ is internally tangent to $\gamma_1$ and $\gamma_2$ at $A$ and $B,$ respectively. Suppose the tangents to $\gamma_3$ at $A$ and $B$ intersect at $X.$ The line through $X$ and $C$ intersect $\gamma_3$ at two points, $P$ and $Q.$ Compute the length of $PQ.$
[i]Proposed by Kyle Lee[/i]
2006 South africa National Olympiad, 1
Reduce the fraction
\[\frac{2121212121210}{1121212121211}\]
to its simplest form.