Found problems: 85335
2004 Federal Math Competition of S&M, 3
Let $A = \{1,2,3, . . . ,11\}$. How many subsets $B$ of $A$ are there, such that for each $n\in \{1,2, . . . ,8\}$, if $n$ and $n+2$ are in $B$ then at least one of the numbers $ n+1$ and $n+3$ is also in $B$?
2017 ELMO Shortlist, 1
Let $ABC$ be a triangle with orthocenter $H,$ and let $M$ be the midpoint of $\overline{BC}.$ Suppose that $P$ and $Q$ are distinct points on the circle with diameter $\overline{AH},$ different from $A,$ such that $M$ lies on line $PQ.$ Prove that the orthocenter of $\triangle APQ$ lies on the circumcircle of $\triangle ABC.$
[i]Proposed by Michael Ren[/i]
2009 Albania Team Selection Test, 3
Two people play a game as follows: At the beginning both of them have one point and in every move, one of them can double it's points, or when the other have more point than him, subtract to him his points. Can the two competitors have 2009 and 2002 points respectively? What about 2009 and 2003? Generally which couples of points can they have?
LMT Guts Rounds, 7
A team of four students goes to LMT, and each student brings a lunch. However, on the bus, the students’ lunches get mixed up, and during lunch time, each student chooses a random lunch to eat (no two students may eat the same lunch). What is the probability that each student chooses his or her own lunch correctly?
2020 Stanford Mathematics Tournament, 5
Find the smallest possible number of edges in a convex polyhedron that has an odd number of edges in total has an even number of edges on each face.
1952 AMC 12/AHSME, 22
On hypotenuse $ AB$ of a right triangle $ ABC$ a second right triangle $ ABD$ is constructed with hypotenuse $ AB$. If $ \overline{BC} \equal{} 1, \overline{AC} \equal{} b$, and $ \overline{AD} \equal{} 2$, then $ \overline{BD}$ equals:
$ \textbf{(A)}\ \sqrt {b^2 \plus{} 1} \qquad\textbf{(B)}\ \sqrt {b^2 \minus{} 3} \qquad\textbf{(C)}\ \sqrt {b^2 \plus{} 1} \plus{} 2$
$ \textbf{(D)}\ b^2 \plus{} 5 \qquad\textbf{(E)}\ \sqrt {b^2 \plus{} 3}$
2011 F = Ma, 1
A cyclist travels at a constant speed of $\text{22.0 km/hr}$ except for a $20$ minute stop. The cyclist’s average speed was $\text{17.5 km/hr}$. How far did the cyclist travel?
(A) $\text{28.5 km}$
(B) $\text{30.3 km}$
(C) $\text{31.2 km}$
(D) $\text{36.5 km}$
(E) $\text{38.9 km}$
2020 GQMO, 3
Let $A$ and $B$ be two distinct points in the plane. Let $M$ be the midpoint of the segment $AB$, and let $\omega$ be a circle that goes through $A$ and $M$. Let $T$ be a point on $\omega$ such that the line $BT$ is tangent to $\omega$. Let $X$ be a point (other than $B$) on the line $AB$ such that $TB = TX$, and let $Y$ be the foot of the perpendicular from $A$ onto the line $BT$.
Prove that the lines $AT$ and $XY$ are parallel.
[i]Navneel Singhal, India[/i]
2021 Romanian Master of Mathematics Shortlist, N2
We call a set of positive integers [i]suitable [/i] if none of its elements is coprime to the sum of all
elements of that set. Given a real number $\varepsilon \in (0,1)$, prove that, for all large enough positive
integers $N$, there exists a suitable set of size at least $\varepsilon N$, each element of which is at most $N$.
2009 AMC 12/AHSME, 6
Suppose that $ P\equal{}2^m$ and $ Q\equal{}3^n$. Which of the following is equal to $ 12^{mn}$ for every pair of integers $ (m,n)$?
$ \textbf{(A)}\ P^2Q \qquad
\textbf{(B)}\ P^nQ^m \qquad
\textbf{(C)}\ P^nQ^{2m} \qquad
\textbf{(D)}\ P^{2m}Q^n \qquad
\textbf{(E)}\ P^{2n}Q^m$
2013 Danube Mathematical Competition, 4
Show that there exists a proper non-empty subset $S$ of the set of real numbers such that, for every real number $x$, the set $\{nx + S : n \in N\}$ is finite, where $nx + S =\{nx + s : s \in S\}$
2007 QEDMO 4th, 11
Let $S_{1},$ $S_{2},$ $...,$ $S_{n}$ be finitely many subsets of $\mathbb{N}$ such that $S_{1}\cup S_{2}\cup...\cup S_{n}=\mathbb{N}.$ Prove that there exists some $k\in\left\{ 1,2,...,n\right\} $ such that for each positive integer $m,$ the set $S_{k}$ contains infinitely many multiples of $m.$
2012 IFYM, Sozopol, 5
Let $c_0,c_1>0$. And suppose the sequence $\{c_n\}_{n\ge 0}$ satisfies
\[ c_{n+1}=\sqrt{c_n}+\sqrt{c_{n-1}}\quad \text{for} \;n\ge 1 \]
Prove that $\lim_{n\to \infty}c_n$ exists and find its value.
[i]Proposed by Sadovnichy-Grigorian-Konyagin[/i]
2003 AMC 10, 15
What is the probability that an integer in the set $ \{1,2,3,\ldots,100\}$ is divisible by $ 2$ and not divisible by $ 3$?
$ \textbf{(A)}\ \frac{1}{6} \qquad
\textbf{(B)}\ \frac{33}{100} \qquad
\textbf{(C)}\ \frac{17}{50} \qquad
\textbf{(D)}\ \frac{1}{2} \qquad
\textbf{(E)}\ \frac{18}{25}$
2006 Purple Comet Problems, 5
Find the sum of all positive integers less than $2006$ which are both multiples of six and one more than a multiple of seven.
1996 Vietnam National Olympiad, 2
The triangle ABC has BC=1 and $ \angle BAC \equal{} a$. Find the shortest distance between its incenter and its centroid. Denote this shortest distance by $ f(a)$. When a varies in the interval $ (\frac {\pi}{3},\pi)$, find the maximum value of $ f(a)$.
2008 ISI B.Math Entrance Exam, 8
Let $a^2+b^2=1$ , $c^2+d^2=1$ , $ac+bd=0$
Prove that
$a^2+c^2=1$ , $b^2+d^2=1$ , $ab+cd=0$ .
2001 Moldova National Olympiad, Problem 2
Prove that the sum of two consecutive prime numbers is never a product of two prime numbers.
2013 Tuymaada Olympiad, 7
Points $A_1$, $A_2$, $A_3$, $A_4$ are the vertices of a regular tetrahedron of edge length $1$. The points $B_1$ and $B_2$ lie inside the figure bounded by the plane $A_1A_2A_3$ and the spheres of radius $1$ and centres $A_1$, $A_2$, $A_3$.
Prove that $B_1B_2 < \max\{B_1A_1, B_1A_2, B_1A_3, B_1A_4\}$.
[i] A. Kupavsky [/i]
1992 Putnam, B3
For any pair $(x,y)$ of real numbers, a sequence $(a_{n}(x,y))$ is defined as follows:
$$a_{0}(x,y)=x, \;\;\;\; a_{n+1}(x,y) =\frac{a_{n}(x,y)^{2} +y^2 }{2} \;\, \text{for}\, n\geq 0$$
Find the area of the region $\{(x,y)\in \mathbb{R}^{2} \, |\, (a_{n}(x,y)) \,\, \text{converges} \}$.
1941 Moscow Mathematical Olympiad, 086
Given three points $H_1, H_2, H_3$ on a plane. The points are the reflections of the intersection point of the heights of the triangle $\vartriangle ABC$ through its sides. Construct $\vartriangle ABC$.
2014 AMC 10, 1
What is $10 \cdot \left(\tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{10}\right)^{-1}?$
${ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}}\ \frac{170}{3}\qquad\textbf{(E)}\ 170$
1960 IMO, 2
For what values of the variable $x$ does the following inequality hold: \[ \dfrac{4x^2}{(1-\sqrt{2x+1})^2}<2x+9 \ ? \]
2000 AIME Problems, 7
Given that \[ \frac 1{2!17!}+\frac 1{3!16!}+\frac 1{4!15!}+\frac 1{5!14!}+\frac 1{6!13!}+\frac 1{7!12!}+\frac 1{8!11!}+\frac 1{9!10!}=\frac N{1!18!} \] find the greatest integer that is less than $\frac N{100}.$
2008 CentroAmerican, 6
Let $ ABC$ be an acute triangle. Take points $ P$ and $ Q$ inside $ AB$ and $ AC$, respectively, such that $ BPQC$ is cyclic. The circumcircle of $ ABQ$ intersects $ BC$ again in $ S$ and the circumcircle of $ APC$ intersects $ BC$ again in $ R$, $ PR$ and $ QS$ intersect again in $ L$. Prove that the intersection of $ AL$ and $ BC$ does not depend on the selection of $ P$ and $ Q$.