Found problems: 85335
2018 PUMaC Live Round, 2.3
Sophie has $20$ indistinguishable pairs of socks in a laundry bag. She pulls them out one at a time. After pulling out $30$ socks, the expected number of unmatched socks among the socks that she has pulled out can be expressed in simplest form as $\tfrac{m}{n}$. Find $m+n$.
2021 Purple Comet Problems, 13
Two infinite geometric series have the same sum. The first term of the first series is $1$, and the first term of the second series is $4$. The fifth terms of the two series are equal. The sum of each series can be written as $m + \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.
1983 Austrian-Polish Competition, 7
Let $P_1,P_2,P_3,P_4$ be four distinct points in the plane. Suppose $\ell_1,\ell_2, … , \ell_6$ are closed segments in that plane with the following property: Every straight line passing through at least one of the points $P_i$ meets the union $\ell_1 \cup \ell_2\cup … \cup\ell_6$ in exactly two points. Prove or disprove that the segments $\ell_i$ necessarily form a hexagon.
2000 Harvard-MIT Mathematics Tournament, 7
Suppose you are given a fair coin and a sheet of paper with the polynomial $x^m$ written on it. Now for each toss of the coin, if heads show up, you must erase the polynomial $x^r$ (where $r$ is going to change with time - initially it is $m$) written on the paper and replace it with $x^{r-1}$. If tails show up, replace it with $x^{r+1}$. What is the expected value of the polynomial I get after $m$ such tosses? (Note: this is a different concept from the most probable value)
2016 Macedonia JBMO TST, 5
Solve the following equation in the set of positive integers
$x + y^2 + (GCD(x, y))^2 = xy \cdot GCD(x, y)$.
2020 Baltic Way, 7
A mason has bricks with dimensions $2\times5\times8$ and other bricks with dimensions $2\times3\times7$. She also has a box with dimensions $10\times11\times14$. The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box filling its entire volume and with no bricks sticking out.
Find all possible values of the total number of bricks that she can pack.
2009 239 Open Mathematical Olympiad, 3
$200$ sticks are given whose lengths are $1, 2, 4, \ldots , 2^{199}$. What is the smallest number of sticks needed to be broken so that out of all the resulting sticks, several triangles could be created, if each stick could be broken only once, and each triangle can be created out of only three sticks?
2017 May Olympiad, 3
Let $ABCD$ be a rhombus of sides $AB = BC = CD= DA = 13$. On the side $AB$ construct the rhombus $BAFE$ outside $ABCD$ and such that the side $AF$ is parallel to the diagonal $BD$ of $ABCD$. If the area of $BAFE$ is equal to $65$, calculate the area of $ABCD$.
2023 Quang Nam Province Math Contest (Grade 11), Problem 1
Solve the system of equations:$$\left\{ \begin{array}{l}
({x^2} + y)\sqrt {y - 2x} - 4 = 2{x^2} + 2x + y\\
{x^3} - {x^2} - y + 6 = 4\sqrt {x + 1} + 2\sqrt {y - 1}
\end{array} \right.(x,y \in \mathbb{R}).$$
1990 Mexico National Olympiad, 4
Find $0/1 + 1/1 + 0/2 + 1/2 + 2/2 + 0/3 + 1/3 + 2/3 + 3/3 + 0/4 + 1/4 + 2/4 + 3/4 + 4/4 + 0/5 + 1/5 + 2/5 + 3/5 + 4/5 + 5/5 + 0/6 + 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6$
2002 JBMO ShortLists, 6
Let $ a_1,a_2,...,a_6$ be real numbers such that:
$ a_1 \not \equal{} 0, a_1a_6 \plus{} a_3 \plus{} a_4 \equal{} 2a_2a_5 \ \mathrm{and}\ a_1a_3 \ge a_2^2$
Prove that $ a_4a_6\le a_5^2$. When does equality holds?
1998 Greece National Olympiad, 4
Let a function $g:\mathbb{N}_0\to\mathbb{N}_0$ satisfy $g(0)=0$ and $g(n)=n-g(g(n-1))$ for all $n\ge 1$. Prove that:
a) $g(k)\ge g(k-1)$ for any positive integer $k$.
b) There is no $k$ such that $g(k-1)=g(k)=g(k+1)$.
2020 Ukrainian Geometry Olympiad - April, 4
Inside triangle $ABC$, the point $P$ is chosen such that $\angle PAB = \angle PCB =\frac14 (\angle A+ \angle C)$. Let $BL$ be the bisector of $\vartriangle ABC$. Line $PL$ intersects the circumcircle of $\vartriangle APC$ at point $Q$. Prove that the line $QB$ is the bisector of $\angle AQC$.
2022 HMNT, 19
Define the [i]annoyingness[/i] of a permutation of the first $n$ integers to be the minimum number of copies of the permutation that are needed to be placed next to each other so that the subsequence $1,2 \ldots ,n$ appears. For instance, the annoyingness of $3,2,1$ is $3,$ and the annoyingness of $1,3,4,2$ is $2.$
A random permutation of $1,2, \ldots, 2022$ is selected. Compute the expected value of the annoyingness of this permutation.
2021 Science ON all problems, 3
Circles $\omega_1$ and $\omega_2$ are externally tangent to each other at $P$. A random line $\ell$ cuts $\omega_1$ at $A$ and $C$ and $\omega_2$ at $B$ and $D$ (points $A,C,B,D$ are in this order on $\ell$). Line $AP$ meets $\omega_2$ again at $E$ and line $BP$ meets $\omega_1$ again at $F$. Prove that the radical axis of circles $(PCD)$ and $(PEF)$ is parallel to $\ell$.
\\ \\
[i](Vlad Robu)[/i]
2000 Moldova Team Selection Test, 1
Find all integers that can be written in the form $\frac{(x+y+z)^2}{xyz}$, where $x,y,z$ are positive integers.
2016 USA Team Selection Test, 3
Let $ABC$ be an acute scalene triangle and let $P$ be a point in its interior. Let $A_1$, $B_1$, $C_1$ be projections of $P$ onto triangle sides $BC$, $CA$, $AB$, respectively. Find the locus of points $P$ such that $AA_1$, $BB_1$, $CC_1$ are concurrent and $\angle PAB + \angle PBC + \angle PCA = 90^{\circ}$.
1986 AMC 8, 4
The product $ (1.8)(40.3\plus{}.07)$ is closest to
\[ \textbf{(A)}\ 7 \qquad
\textbf{(B)}\ 42 \qquad
\textbf{(C)}\ 74 \qquad
\textbf{(D)}\ 84 \qquad
\textbf{(E)}\ 737
\]
2015 FYROM JBMO Team Selection Test, 3
Let $a, b$ and $c$ be positive real numbers. Prove that $\prod_{cyc}(16a^2+8b+17)\geq2^{12}\prod_{cyc}(a+1)$.
2016 PUMaC Team, 1
Quadrilateral $ABCD$ has integer side lengths, and angles $ABC, ACD$, and $BAD$ are right angles. Compute the smallest possible value of $AD$.
1964 Poland - Second Round, 1
Prove that if $ n $ is a natural number and the angle $ \alpha $ is not a multiple of $ \frac{180^{\circ}}{2^n} $, then
$$\frac{1}{\sin 2\alpha} + \frac{1}{\sin 4\alpha} + \frac{1}{\sin 8\alpha} + ... + = ctg \alpha - ctg 2^n \alpha.$$
1996 Tournament Of Towns, (486) 4
All vertices of a hexagon, whose sides may intersect at points other than the vertices, lie on a circle.
(a) Draw a hexagon such that it has the largest possible number of points of self-intersection.
(b) Prove that this number is indeed maximum.
(NB Vassiliev)
2011 Princeton University Math Competition, A5
Let $\sigma$ be a random permutation of $\{0, 1, \ldots, 6\}$. Let $L(\sigma)$ be the length of the longest initial monotonic consecutive subsequence of $\sigma$ not containing $0$; for example, \[L(\underline{2,3,4},6,5,1,0) = 3,\ L(\underline{3,2},4,5,6,1,0) = 2,\ L(0,1,2,3,4,5,6) = 0.\] If the expected value of $L(\sigma)$ can be written as $\frac mn$, where $m$ and $n$ are relatively prime positive integers, then find $m + n$.
2019 Jozsef Wildt International Math Competition, W. 50
Let $x$, $y$, $z > 0$, $\lambda \in (-\infty, 0) \cup (1,+\infty)$ such that $x + y + z = 1$. Then$$\sum \limits_{cyc} x^{\lambda}y^{\lambda}\sum \limits_{cyc}\frac{1}{(x+y)^{2\lambda}}\geq 9\left(\frac{1}{4}-\frac{1}{9}\sum \limits_{cyc}\frac{1}{(x+1)^2} \right)^{\lambda}$$
2012 Stanford Mathematics Tournament, 3
Let $ABC$ be an equilateral triangle of side 1. Draw three circles $O_a$, $O_b$, $O_c$ with diameters $BC$, $CA$, and $AB$, respectively. Let $S_a$ denote the area of the region inside $O_a$ and outside of $O_b$ and $O_c$. Define $S_b$ and $S_c$ similarly, and let $S$ be the area of intersection between the three circles. Find $S_a+S_b+S_c-S$.