Found problems: 85335
1958 Poland - Second Round, 5
Outside triangle $ ABC $ equilateral triangles $ BMC $, $ CNA $, and $ APB $ are constructed. Prove that the centers $ S $, $ T $, $ U $ of these triangles form an equilateral triangle.
1975 Putnam, B3
Let $n$ be a positive integer. Let $S=\{a_1,\ldots, a_{k}\}$ be a finite collection of at least $n$ not necessarily distinct positive real numbers. Let
$$f(S)=\left(\sum_{i=1}^{k} a_{i}\right)^{n}$$ and
$$g(S)=\sum_{1\leq i_{1}<\ldots<i_{n} \leq k} a_{i_{1}}\cdot\ldots\cdot a_{i_{n}}.$$ Determine $\sup_{S} \frac{g(S)}{f(S)}$.
2020 MOAA, TO4
Over all real numbers $x$, let $k$ be the minimum possible value of the expression $$\sqrt{x^2 + 9} +\sqrt{x^2 - 6x + 45}.$$
Determine $k^2$.
2017 EGMO, 3
Let $n\geq1$ be an integer and let $t_1<t_2<\dots<t_n$ be positive integers. In a group of $t_n+1$ people, some games of chess are played. Two people can play each other at most once. Prove that it is possible for the following two conditions to hold at the same time:
(i) The number of games played by each person is one of $t_1,t_2,\dots,t_n$.
(ii) For every $i$ with $1\leq i\leq n$, there is someone who has played exactly $t_i$ games of chess.
2011 National Olympiad First Round, 36
There are $14$ students with different heights. At each step, two adjacent students will be swapped. Whatever the first arrangement is, in at least how many steps the students can be lined up?
$\textbf{(A)}\ 42 \qquad\textbf{(B)}\ 43 \qquad\textbf{(C)}\ 45 \qquad\textbf{(D)}\ 52 \qquad\textbf{(E)}\ \text{None}$
2013 Purple Comet Problems, 5
A picture with an area of $160$ square inches is surrounded by a $2$ inch border. The picture with its border is a rectangle twice as long as it is wide. How many inches long is that rectangle?
2013 HMNT, 1
What is the smallest non-square positive integer that is the product of four prime numbers (not necessarily distinct)?
1991 AMC 8, 20
In the addition problem, each digit has been replaced by a letter. If different letters represent different digits then $C=$
[asy]
unitsize(18);
draw((-1,0)--(3,0));
draw((-3/4,1/2)--(-1/4,1/2)); draw((-1/2,1/4)--(-1/2,3/4));
label("$A$",(0.5,2.1),N); label("$B$",(1.5,2.1),N); label("$C$",(2.5,2.1),N);
label("$A$",(1.5,1.1),N); label("$B$",(2.5,1.1),N); label("$A$",(2.5,0.1),N);
label("$3$",(0.5,-.1),S); label("$0$",(1.5,-.1),S); label("$0$",(2.5,-.1),S);
[/asy]
$\text{(A)}\ 1 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 9$
2018 China Western Mathematical Olympiad, 1
Real numbers $x_1, x_2, \dots, x_{2018}$ satisfy $x_i + x_j \geq (-1)^{i+j}$ for all $1 \leq i < j \leq 2018$.
Find the minimum possible value of $\sum_{i=1}^{2018} ix_i$.
2011 Vietnam National Olympiad, 1
Prove that if $x>0$ and $n\in\mathbb N,$ then we have
\[\frac{x^n(x^{n+1}+1)}{x^n+1}\leq\left(\frac {x+1}{2}\right)^{2n+1}.\]
III Soros Olympiad 1996 - 97 (Russia), 10.5
Solve the system of equations
$$\begin{cases} \dfrac{x+y}{1+xy}=\dfrac{1-2y}{2-y} \\ \dfrac{x-y}{1-xy}=\dfrac{1-3x}{3-x} \end{cases}$$
2007 Croatia Team Selection Test, 2
Prove that the sequence $a_{n}=\lfloor n\sqrt 2 \rfloor+\lfloor n\sqrt 3 \rfloor$ contains infintely many even and infinitely many odd numbers.
2013 SDMO (Middle School), 3
Let $ABCD$ be a square, and let $\Gamma$ be the circle that is inscribed in square $ABCD$. Let $E$ and $F$ be points on line segments $AB$ and $AD$, respectively, so that $EF$ is tangent to $\Gamma$. Find the ratio of the area of triangle $CEF$ to the area of square $ABCD$.
Mexican Quarantine Mathematical Olympiad, #3
Let $\Gamma_1$ and $\Gamma_2$ be circles intersecting at points $A$ and $B$. A line through $A$ intersects $\Gamma_1$ and $\Gamma_2$ at $C$ and $D$ respectively. Let $P$ be the intersection of the lines tangent to $\Gamma_1$ at $A$ and $C$, and let $Q$ be the intersection of the lines tangent to $\Gamma_2$ at $A$ and $D$. Let $X$ be the second intersection point of the circumcircles of $BCP$ and $BDQ$, and let $Y$ be the intersection of lines $AB$ and $PQ$. Prove that $C$, $D$, $X$ and $Y$ are concyclic.
[i]Proposed by Ariel García[/i]
2021 China Second Round A2, 1
As shown in the figure, in the acute angle $\vartriangle ABC$, $AB > AC$, $M$ is the midpoint of the minor arc $BC$ of the circumcircle $\Omega$ of $\vartriangle ABC$. $K$ is the intersection point of the bisector of the exterior angle $\angle BAC$ and the extension line of $BC$. From point $A$ draw a line perpendicular on $BC$ and take a point $D$ (different from $A$) on that line , such that $DM = AM$. Let the circumscribed circle of $\vartriangle ADK$ intersect the circle $\Omega$ at point $A$ and at another point $T$. Prove that $AT$ bisects line segment $BC$.
[img]https://cdn.artofproblemsolving.com/attachments/1/3/6fde30405101620828d63ae31b8c0ffcec972f.png[/img]
2011 Morocco National Olympiad, 1
Let $a$ and $b$ be two positive real numbers such that $a+b=ab$.
Prove that $\frac{a}{b^{2}+4}+\frac{b}{a^{2}+4}\geq \frac{1}{2}$.
2014 Hanoi Open Mathematics Competitions, 14
Let be given $a < b < c$ and $f(x) =\frac{c(x - a)(x - b)}{(c - a)(c - b)}+\frac{a(x - b)(x - c)}{(a - b)(a -c)}+\frac{b(x -c)(x - a)}{(b - c)(b - a)}$.
Determine $f(2014)$.
2020 China National Olympiad, 3
Let $S$ be a set, $|S|=35$. A set $F$ of mappings from $S$ to itself is called to be satisfying property $P(k)$, if for any $x,y\in S$, there exist $f_1, \cdots, f_k \in F$ (not necessarily different), such that $f_k(f_{k-1}(\cdots (f_1(x))))=f_k(f_{k-1}(\cdots (f_1(y))))$.
Find the least positive integer $m$, such that if $F$ satisfies property $P(2019)$, then it also satisfies property $P(m)$.
1974 Swedish Mathematical Competition, 3
Let $a_1=1$, $a_2=2^{a_1}$, $a_3=3^{a_2}$, $a_4=4^{a_3}$, $\dots$, $a_9 = 9^{a_8}$. Find the last two digits of $a_9$.
1996 Nordic, 3
The circle whose diameter is the altitude dropped from the vertex $A$ of the triangle $ABC$ intersects the sides
$AB$ and $AC$ at $D$ and $E$, respectively $(A\ne D, A \ne E)$. Show that the circumcenter of $ABC$ lies on the altitude drawn from the vertex $A$ of the triangle $ADE$, or on its extension.
2012 India IMO Training Camp, 3
Let $\mathbb{R}^{+}$ denote the set of all positive real numbers. Find all functions $f:\mathbb{R}^{+}\longrightarrow \mathbb{R}$ satisfying
\[f(x)+f(y)\le \frac{f(x+y)}{2}, \frac{f(x)}{x}+\frac{f(y)}{y}\ge \frac{f(x+y)}{x+y},\]
for all $x, y\in \mathbb{R}^{+}$.
2006 National Olympiad First Round, 27
If $x,y,z$ are positive real numbers such that $xy+yz+zx=5$, $x^2+y^2+z^2-xyz$ cannot be $\underline{\hspace{1cm}}$.
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 5
\qquad\textbf{(D)}\ 3\sqrt 3
\qquad\textbf{(E)}\ \text{None of above}
$
2001 India National Olympiad, 4
Show that given any nine integers, we can find four, $a, b, c, d$ such that $a + b - c - d$is divisible by $20$. Show that this is not always true for eight integers.
1990 National High School Mathematics League, 3
There are $n$ schools in a city. $i$th school dispatches $C_i(1\leq C_i\leq39,1\leq i\leq n)$ students to watch a football match. The number of all students $\sum_{i=1}^{n}C_{i}=1990$. In each line, there are $199$ seats, but students from the same school must sit in the same line. So, how many lines of seats we need to have to make sure all students have a seat.
2006 Canada National Olympiad, 2
Let $ABC$ be acute triangle. Inscribe a rectangle $DEFG$ in this triangle such that $D\in AB,E\in AC,F\in BC,G\in BC$. Describe the locus of (i.e., the curve occupied by) the intersections of the diagonals of all possible rectangles $DEFG$.