Found problems: 85335
2008 Harvard-MIT Mathematics Tournament, 21
Let $ ABC$ be a triangle with $ AB \equal{} 5$, $ BC \equal{} 4$ and $ AC \equal{} 3$. Let $ \mathcal P$ and $ \mathcal Q$ be squares inside $ ABC$ with disjoint interiors such that they both have one side lying on $ AB$. Also, the two squares each have an edge lying on a common line perpendicular to $ AB$, and $ \mathcal P$ has one vertex on $ AC$ and $ \mathcal Q$ has one vertex on $ BC$. Determine the minimum value of the sum of the areas of the two squares.
[asy]import olympiad;
import math;
import graph;
unitsize(1.5cm);
pair A, B, C;
A = origin;
B = A + 5 * right;
C = (9/5, 12/5);
pair X = .7 * A + .3 * B;
pair Xa = X + dir(135);
pair Xb = X + dir(45);
pair Ya = extension(X, Xa, A, C);
pair Yb = extension(X, Xb, B, C);
pair Oa = (X + Ya)/2;
pair Ob = (X + Yb)/2;
pair Ya1 = (X.x, Ya.y);
pair Ya2 = (Ya.x, X.y);
pair Yb1 = (Yb.x, X.y);
pair Yb2 = (X.x, Yb.y);
draw(A--B--C--cycle);
draw(Ya--Ya1--X--Ya2--cycle);
draw(Yb--Yb1--X--Yb2--cycle);
label("$A$", A, W);
label("$B$", B, E);
label("$C$", C, N);
label("$\mathcal P$", Oa, origin);
label("$\mathcal Q$", Ob, origin);[/asy]
1999 Taiwan National Olympiad, 3
There are $1999$ people participating in an exhibition. Among any $50$ people there are two who don't know each other. Prove that there are $41$ people, each of whom knows at most $1958$ people.
2010 Kyrgyzstan National Olympiad, 3
At the meeting, each person is familiar with 22 people. If two persons $A$ and $B$ know each with one another, among the remaining people they do not have a common friend. For each pair individuals $A$ and $B$ are not familiar with each other, there are among the remaining six common acquaintances. How many people were at the meeting?
2013 Austria Beginners' Competition, 3
Let $a$ and $ b$ be real numbers with $0\le a, b\le 1$. Prove that $$\frac{a}{b + 1}+\frac{b}{a + 1}\le 1$$ When does equality holds?
(K. Czakler, GRG 21, Vienna)
2011 Romanian Masters In 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]
2013 Harvard-MIT Mathematics Tournament, 23
Let $ABCD$ be a parallelogram with $AB=8$, $AD=11$, and $\angle BAD=60^\circ$. Let $X$ be on segment $CD$ with $CX/XD=1/3$ and $Y$ be on segment $AD$ with $AY/YD=1/2$. Let $Z$ be on segment $AB$ such that $AX$, $BY$, and $DZ$ are concurrent. Determine the area of triangle $XYZ$.
1982 National High School Mathematics League, 5
For any$\varphi\in(0,\frac{\pi}{2})$, we have
$\text{(A)}\sin\sin\varphi<\cos\varphi<\cos\cos\varphi\qquad\text{(B)}\sin\sin\varphi>\cos\varphi>\cos\cos\varphi$
$\text{(C)}\sin\cos\varphi>\cos\varphi>\cos\sin\varphi\qquad\text{(D)}\sin\cos\varphi<\cos\varphi<\cos\sin\varphi$
2002 Federal Math Competition of S&M, Problem 3
Let $m$ and $n$ be positive integers. Prove that the number $2n-1$ is divisible by $(2^m-1)^2$ if and only if $n$ is divisible by $m(2^m-1)$.
1964 Spain Mathematical Olympiad, 5
Given a regular pentagon, its five diagonals are drawn. How many triangles do appear in the figure? Classify the set of triangles in classes of equal triangles.
2014 ASDAN Math Tournament, 17
Given that the line $y=mx+k$ intersects the parabola $y=ax^2+bx+c$ at two points, compute the product of the two $x$-coordinates of these points in terms of $a$, $b$, $c$, $k$, and $m$.
MBMT Team Rounds, 2020.33
Circle $\omega_1$ with center $K$ of radius $4$ and circle $\omega_2$ of radius $6$ intersect at points $W$ and $U$. If the incenter of $\triangle KWU$ lies on circle $\omega_2$, find the length of $\overline{WU}$. (Note: The incenter of a triangle is the intersection of the angle bisectors of the angles of the triangle)
[i]Proposed by Bradley Guo[/i]
1993 Taiwan National Olympiad, 3
Find all $ x,y,z\in\mathbb{N}_{0}$ such that $ 7^{x} \plus{} 1 \equal{} 3^{y} \plus{} 5^{z}$.
[i]Alternative formulation:[/i] Solve the equation $ 1\plus{}7^{x}\equal{}3^{y}\plus{}5^{z}$ in nonnegative integers $ x$, $ y$, $ z$.
2011 National Olympiad First Round, 35
Which of these has the smallest maxima on positive real numbers?
$\textbf{(A)}\ \frac{x^2}{1+x^{12}} \qquad\textbf{(B)}\ \frac{x^3}{1+x^{11}} \qquad\textbf{(C)}\ \frac{x^4}{1+x^{10}} \qquad\textbf{(D)}\ \frac{x^5}{1+x^{9}} \qquad\textbf{(E)}\ \frac{x^6}{1+x^{8}}$
2012 IFYM, Sozopol, 2
There are 20 towns on the bay of a circular island. Each town has 20 teams for a mathematical duel. No two of these teams are of equal strength. When two teams meet in a duel, the stronger one wins. For a given number $n\in \mathbb{N}$ one town $A$ can be called [i]“n-stronger”[/i] than $B$, if there exist $n$ different duels between a team from $A$ and team from $B$, for which the team from $A$ wins. Find the maximum value of $n$, for which it is possible for each town to be [i]n-stronger[/i] by its neighboring one clockwise.
1996 Greece Junior Math Olympiad, 4b
Determine whether exist a prime number $p$ and natural number $n$ such that $n^2 + n + p = 1996$.
2010 Ukraine Team Selection Test, 5
Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram.
Prove that $GR=GS$.
[i]Proposed by Hossein Karke Abadi, Iran[/i]
1965 IMO, 5
Consider $\triangle OAB$ with acute angle $AOB$. Thorugh a point $M \neq O$ perpendiculars are drawn to $OA$ and $OB$, the feet of which are $P$ and $Q$ respectively. The point of intersection of the altitudes of $\triangle OPQ$ is $H$. What is the locus of $H$ if $M$ is permitted to range over
a) the side $AB$;
b) the interior of $\triangle OAB$.
2003 Moldova National Olympiad, 12.1
For every natural number $n$ let:
$a_n=ln(1+2e+4e^4+\dots+2ne^{n^2})$. Find:
\[ \displaystyle{\lim_{n \to \infty}\frac{a_n}{n^2}} \].
2015 AMC 8, 8
What is the smallest whole number larger than the perimeter of any triangle with a side of length $ 5$ and a side of length $19$?
$\textbf{(A) }24\qquad\textbf{(B) }29\qquad\textbf{(C) }43\qquad\textbf{(D) }48\qquad \textbf{(E) }57$
1990 Chile National Olympiad, 1
Show that any triangle can be subdivided into isosceles triangles.
1995 VJIMC, Problem 1
Prove that the systems of hyperbolas
\begin{align*}x^2-y^2&=a\\xy&=b\end{align*}are orthogonal.
2012 Tournament of Towns, 6
We attempt to cover the plane with an infi nite sequence of rectangles, overlapping allowed.
(a) Is the task always possible if the area of the $n$th rectangle is $n^2$ for each $n$?
(b) Is the task always possible if each rectangle is a square, and for any number $N$, there exist squares with total area greater than $N$?
2010 India IMO Training Camp, 10
Let $ABC$ be a triangle. Let $\Omega$ be the brocard point. Prove that $\left(\frac{A\Omega}{BC}\right)^2+\left(\frac{B\Omega}{AC}\right)^2+\left(\frac{C\Omega}{AB}\right)^2\ge 1$
2017 Harvard-MIT Mathematics Tournament, 10
[b]D[/b]enote $\phi=\frac{\sqrt{5}+1}{2}$ and consider the set of all finite binary strings without leading zeroes. Each string $S$ has a “base-$\phi$” value $p(S)$. For example, $p(1101)=\phi^3+\phi^2+1$. For any positive integer n, let $f(n)$ be the number of such strings S that satisfy $p(S) =\frac{\phi^{48n}-1}{\phi^{48}-1}$. The sequence of fractions $\frac{f(n+1)}{f(n)}$ approaches a real number $c$ as $n$ goes to infinity. Determine the value of $c$.
1997 Vietnam Team Selection Test, 2
Find all pairs of positive real numbers $ (a, b)$ such that for every $ n \in\mathbb{N}^*$ and every real root $ x_n$ of the equation $ 4n^2x \equal{} \log_2(2n^2x \plus{} 1)$ we always have $ a^{x_n} \plus{} b^{x_n} \ge 2 \plus{} 3x_n$.