Found problems: 85335
2004 Harvard-MIT Mathematics Tournament, 3
A swimming pool is in the shape of a circle with diameter $60$ ft. The depth varies linearly along the east-west direction from $3$ ft at the shallow end in the east to $15$ ft at the diving end in the west (this is so that divers look impressive against the sunset) but does not vary at all along the north-south direction. What is the volume of the pool, in ft$^3$?
2011 Romanian Master of Mathematics, 1
Prove that there exist two functions $f,g \colon \mathbb{R} \to \mathbb{R}$, such that $f\circ g$ is strictly decreasing and $g\circ f$ is strictly increasing.
[i](Poland) Andrzej Komisarski and Marcin Kuczma[/i]
2022 China Girls Math Olympiad, 2
Let $n$ be a positive integer. There are $3n$ women's volleyball teams in the tournament, with no more than one match between every two teams (there are no ties in volleyball). We know that there are $3n^2$ games played in this tournament.
Proof: There exists a team with at least $\frac{n}{4}$ win and $\frac{n}{4}$ loss
LMT Speed Rounds, 11
Let $LEX INGT_1ONMAT_2H$ be a regular $13$-gon. Find $\angle LMT_1$, in degrees.
[i]Proposed by Edwin Zhao[/i]
2005 IMAR Test, 1
The incircle of triangle $ABC$ touches the sides $BC,CA,AB$ at the points $D,E,F$, respectively. Let $K$ be a point on the side $BC$ and let $M$ be the point on the line segment $AK$ such that $AM=AE=AF$. Denote by $L,N$ the incenters of triangles $ABK,ACK$, respectively.
Prove that $K$ is the foot of the altitude from $A$ if and only if $DLMN$ is a square.
[hide="Remark"]This problem is slightly connected to [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=344774#p344774]GMB-IMAR 2005, Juniors, Problem 2[/url]
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[i]Bogdan Enescu[/i]
MathLinks Contest 4th, 4.2
We say that two triangles $T_1$ and $T_2$ are contained one in each other, and we write $T_1 \subset T_2$, if and only if all the points of the triangle $T_1$ lie on the sides or in the interior of the triangle $T_2$.
Let $\Delta$ be a triangle of area $S$, and let $\Delta_1 \subset \Delta$ be the largest equilateral triangle with this property, and let $\Delta \subset \Delta_2$ be the smallest equilateral triangle with this property (in terms of areas). Let $S_1, S_2$ be the areas of $\Delta_1, \Delta_2$ respectively.
Prove that $S_1S_2 = S^2$.
Bonus question: : Does this statement hold for quadrilaterals (and squares)?
2010 CHMMC Fall, 14
A $4$-dimensional hypercube of edge length $1$ is constructed in $4$-space with its edges parallel to
the coordinate axes and one vertex at the origin. The coordinates of its sixteen vertices are given
by $(a, b, c, d)$, where each of $a, b, c,$ and $d$ is either $0$ or $1$. The $3$-dimensional hyperplane given
by $x + y + z + w = 2$ intersects the hypercube at $6$ of its vertices. Compute the $3$-dimensional
volume of the solid formed by the intersection.
2018 AIME Problems, 4
In equiangular octagon $CAROLINE$, $CA = RO = LI = NE = \sqrt{2}$ and $AR = OL = IN = EC = 1$. The self-intersecting octagon $CORNELIA$ encloses six non-overlapping triangular regions. Let $K$ be the area enclosed by $CORNELIA$, that is, that total area of the six triangular regions. Then $K=\tfrac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a + b$.
2007 iTest Tournament of Champions, 3
For each positive integer $n$, let $g(n)$ be the sum of the digits when $n$ is written in binary. For how many positive integers $n$, where $1\leq n\leq 2007$, is $g(n)\geq 3$?
2021 Indonesia TST, N
Let $n$ be a positive integer. Prove that $$\gcd(\underbrace{11\dots 1}_{n \text{times}},n)\mid 1+10^k+10^{2k}+\dots+10^{(n-1)k}$$ for all positive integer $k$.
2010 USAJMO, 5
Two permutations $a_1,a_2,\dots,a_{2010}$ and $b_1,b_2,\dots,b_{2010}$ of the numbers $1,2,\dots,2010$ are said to [i]intersect[/i] if $a_k=b_k$ for some value of $k$ in the range $1\le k\le 2010$. Show that there exist $1006$ permutations of the numbers $1,2,\dots,2010$ such that any other such permutation is guaranteed to intersect at least one of these $1006$ permutations.
1966 IMO Longlists, 38
Two concentric circles have radii $R$ and $r$ respectively. Determine the greatest possible number of circles that are tangent to both these circles and mutually nonintersecting. Prove that this number lies between $\frac 32 \cdot \frac{\sqrt R +\sqrt r }{\sqrt R -\sqrt r } -1$ and $\frac{63}{20} \cdot \frac{R+r}{R-r}.$
1993 Italy TST, 2
Suppose that $p,q$ are prime numbers such that $\sqrt{p^2 +7pq+q^2}+\sqrt{p^2 +14pq+q^2}$ is an integer.
Show that $p = q$.
1969 IMO Shortlist, 38
$(HUN 5)$ Let $r$ and $m (r \le m)$ be natural numbers and $Ak =\frac{2k-1}{2m}\pi$. Evaluate $\frac{1}{m^2}\displaystyle\sum_{k=1}^{m}\displaystyle\sum_{l=1}^{m}\sin(rA_k)\sin(rA_l)\cos(rA_k-rA_l)$
2014 Miklós Schweitzer, 11
Let $U$ be a random variable that is uniformly distributed on the interval $[0,1]$, and let
\[S_n= 2\sum_{k=1}^n \sin(2kU\pi).\]
Show that, as $n\to \infty$, the limit distribution of $S_n$ is the Cauchy distribution with density function $f(x)=\frac1{\pi(1+x^2)}$.
2004 Hong kong National Olympiad, 4
Let $S=\{1,2,...,100\}$ . Find number of functions $f: S\to S$ satisfying the following conditions
a)$f(1)=1$
b)$f$ is bijective
c)$f(n)=f(g(n))f(h(n))\forall n\in S$, where $g(n),h(n)$ are positive integer numbers such that $g(n)\leq h(n),n=g(n)h(n)$ that minimize $h(n)-g(n)$.
1992 Balkan MO, 1
For all positive integers $m,n$ define $f(m,n) = m^{3^{4n}+6} - m^{3^{4n}+4} - m^5 + m^3$. Find all numbers $n$ with the property that $f(m, n)$ is divisible by 1992 for every $m$.
[i]Bulgaria[/i]
1996 AMC 12/AHSME, 22
Four distinct points, $A$, $B$, $C$, and $D$, are to be selected from $1996$ points evenly spaced around a circle. All quadruples are equally likely to be chosen. What is the probability that the chord $AB$ intersects the chord $CD$?
$\text{(A)}\ \frac 14 \qquad \text{(B)}\ \frac 13 \qquad \text{(C)}\ \frac 12 \qquad \text{(D)}\ \frac 23\qquad \text{(E)}\ \frac 34$
2020 LMT Fall, 30
$\triangle ABC$ has the property that $\angle ACB = 90^{\circ}$. Let $D$ and $E$ be points on $AB$ such that $D$ is on ray $BA$, $E$ is on segment $AB$, and $\angle DCA = \angle ACE$. Let the circumcircle of $\triangle CDE$ hit $BC$ at $F \ne C$, and $EF$ hit $AC$ and $DC$ at $P$ and $Q$, respectively. If $EP = FQ$, then the ratio $\frac{EF}{PQ}$ can be written as $a+\sqrt{b}$ where $a$ and $b$ are positive integers. Find $a+b$.
[i]Proposed by Kevin Zhao[/i]
2017 AIME Problems, 9
Let $a_{10} = 10$, and for each integer $n >10$ let $a_n = 100a_{n - 1} + n$. Find the least $n > 10$ such that $a_n$ is a multiple of $99$.
2019 Oral Moscow Geometry Olympiad, 6
In the acute triangle $ABC$, the point $I_c$ is the center of excircle on the side $AB$, $A_1$ and $B_1$ are the tangency points of the other two excircles with sides $BC$ and $CA$, respectively, $C'$ is the point on the circumcircle diametrically opposite to point $C$. Prove that the lines $I_cC'$ and $A_1B_1$ are perpendicular.
2018 Iran MO (1st Round), 20
In the convex and cyclic quadrilateral $ABCD$, we have $\angle B = 110^{\circ}$. The intersection of $AD$ and $BC$ is $E$ and the intersection of $AB$ and $CD$ is $F$. If the perpendicular from $E$ to $AB$ intersects the perpendicular from $F$ to $BC$ on the circumcircle of the quadrilateral at point $P$, what is $\angle PDF$ in degrees?
1991 Greece National Olympiad, 4
Find all positive intger solutions of $3^x+29=2^y$.
2016 May Olympiad, 3
In a triangle $ABC$, let $D$ and $E$ point in the sides $BC$ and $AC$ respectively. The segments $AD$ and $BE$ intersects in $O$, let $r$ be line (parallel to $AB$) such that $r$ intersects $DE$ in your midpoint, show that the triangle $ABO$ and the quadrilateral $ODCE$ have the same area.
2023 Mexico National Olympiad, 3
Let $ABCD$ be a convex quadrilateral. If $M, N, K$ are the midpoints of the segments $AB, BC$, and $CD$, respectively, and there is also a point $P$ inside the quadrilateral $ABCD$ such that, $\angle BPN= \angle PAD$ and $\angle CPN=\angle PDA$. Show that $AB \cdot CD=4PM\cdot PK$.