Found problems: 85335
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$.
1999 Kazakhstan National Olympiad, 2
Prove that for any odd $ n $ there exists a unique polynomial $ P (x) $ $ n $ -th degree satisfying the equation $ P \left (x- \frac {1} {x} \right) = x ^ n- \frac {1} {x ^ n}. $ Is this true for any natural number $ n $?
2003 China Team Selection Test, 1
There are $n$($n\geq 3$) circles in the plane, all with radius $1$. In among any three circles, at least two have common point(s), then the total area covered by these $n$ circles is less than $35$.
1984 IMO Longlists, 63
Inside triangle $ABC$ there are three circles $k_1, k_2, k_3$ each of which is tangent to two sides of the triangle and to its incircle $k$. The radii of $k_1, k_2, k_3$ are $1, 4$, and $9$. Determine the radius of $k.$
1965 IMO, 2
Consider the sytem of equations
\[ a_{11}x_1+a_{12}x_2+a_{13}x_3 = 0 \]\[a_{21}x_1+a_{22}x_2+a_{23}x_3 =0\]\[a_{31}x_1+a_{32}x_2+a_{33}x_3 = 0 \] with unknowns $x_1, x_2, x_3$. The coefficients satisfy the conditions:
a) $a_{11}, a_{22}, a_{33}$ are positive numbers;
b) the remaining coefficients are negative numbers;
c) in each equation, the sum ofthe coefficients is positive.
Prove that the given system has only the solution $x_1=x_2=x_3=0$.