Found problems: 85335
1975 All Soviet Union Mathematical Olympiad, 206
Given a triangle $ABC$ with the unit area. The first player chooses a point $X$ on the side $[AB]$, than the second -- $Y$ on $[BC]$ side, and, finally, the first chooses a point $Z$ on $[AC]$ side. The first tries to obtain the greatest possible area of the $XYZ$ triangle, the second -- the smallest. What area can obtain the first for sure and how?
MBMT Guts Rounds, 2015.22
In rhombus ABCD, $\angle A = 60^\circ$. Rhombus $BEFG$ is constructed, where $E$ and $G$ are the midpoints of $BC$ and $AB$, respectively. Rhombus $BHIJ$ is constructed, where $H$ and $J$ are the midpoints of $BE$ and $BG$, respectively. This process is repeated forever. If the area of $ABCD$ and the sum of the areas of all of the rhombi are both integers, compute the smallest possible value of $AB$.
2013 Benelux, 2
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
\[f(x + y) + y \le f(f(f(x)))\]
holds for all $x, y \in \mathbb{R}$.
2012 ELMO Shortlist, 1
In acute triangle $ABC$, let $D,E,F$ denote the feet of the altitudes from $A,B,C$, respectively, and let $\omega$ be the circumcircle of $\triangle AEF$. Let $\omega_1$ and $\omega_2$ be the circles through $D$ tangent to $\omega$ at $E$ and $F$, respectively. Show that $\omega_1$ and $\omega_2$ meet at a point $P$ on $BC$ other than $D$.
[i]Ray Li.[/i]
2009 USAMTS Problems, 2
Find, with proof, a positive integer $n$ such that
\[\frac{(n + 1)(n + 2) \cdots (n + 500)}{500!}\]
is an integer with no prime factors less than $500$.
VMEO III 2006 Shortlist, N13
Prove the following two inequalities:
1) If $n > 49$, then exist positive integers $a, b > 1$ such that $a+b=n$ and $$\frac{\phi (a)}{a}+\frac{\phi (b)}{b}<1$$
2) If $n > 4$, then exist integer integers $a, b > 1$ such that $a+b=n$ and $$\frac{\phi (a)}{a}+\frac{\phi (b)}{b}>1$$
2023 JBMO TST - Turkey, 2
Let $ABC$ is acute angled triangle and $K,L$ is points on $AC,BC$ respectively such that $\angle{AKB}=\angle{ALB}$. $P$ is intersection of $AL$ and $BK$ and $Q$ is the midpoint of segment $KL$. Let $T,S$ are the intersection $AL,BK$ with $(ABC)$ respectively. Prove that $TK,SL,PQ$ are concurrent.
1975 Miklós Schweitzer, 3
Let $ S$ be a semigroup without proper two-sided ideals and suppose that for every $ a,b \in S$ at least one of the products $ ab$ and $ ba$ is equal to one of the elements $ a,b$. Prove that either $ ab\equal{}a$ for all $ a,b \in S$ or $ ab\equal{}b$ for all $ a,b \in S$.
[i]L. Megyesi[/i]
MOAA Team Rounds, TO5
For a real number $x$, the minimum value of the expression $$\frac{2x^2 + x - 3}{x^2 - 2x + 3}$$ can be written in the form $\frac{a-\sqrt{b}}{c}$, where $a, b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime. Find $a + b + c$
1998 IberoAmerican, 2
Find the maximal possible value of $n$ such that there exist points $P_1,P_2,P_3,\ldots,P_n$ in the plane and real numbers $r_1,r_2,\ldots,r_n$ such that the distance between any two different points $P_i$ and $P_j$ is $r_i+r_j$.
2012 NIMO Problems, 5
In convex hexagon $ABCDEF$, $\angle A \cong \angle B$, $\angle C \cong \angle D$, and $\angle E \cong \angle F$. Prove that the perpendicular bisectors of $\overline{AB}$, $\overline{CD}$, and $\overline{EF}$ pass through a common point.
[i]Proposed by Lewis Chen[/i]
1989 IberoAmerican, 1
Determine all triples of real numbers that satisfy the following system of equations:
\[x+y-z=-1\\ x^2-y^2+z^2=1\\ -x^3+y^3+z^3=-1\]
2024 Miklos Schweitzer, 10
Let $A > 0$ and $B = (3 + 2\sqrt{2})A$. Prove that in the finite sequence $a_k = \lfloor k / \sqrt{2} \rfloor$ for $k \in (A, B) \cap \mathbb{Z}$, the number of even and odd terms differs by at most $2$.
2000 Federal Competition For Advanced Students, Part 2, 1
In a non-equilateral acute-angled triangle $ABC$ with $\angle C = 60^\circ$, $U$ is the circumcenter, $H$ the orthocenter and $D$ the intersection of $AH$ and $BC$. Prove that the Euler line $HU$ bisects the angle $BHD$.
2015 ASDAN Math Tournament, 16
Find the maximum value of $c$ such that
\begin{align*}
1&=-cx+y\\
-7&=x^2+y^2+8y
\end{align*}
has a unique real solution $(x,y)$.
2021 Thailand TST, 1
Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$.
[i]South Africa [/i]
2007 Bulgaria Team Selection Test, 2
Find all $a\in\mathbb{R}$ for which there exists a non-constant function $f: (0,1]\rightarrow\mathbb{R}$ such that \[a+f(x+y-xy)+f(x)f(y)\leq f(x)+f(y)\] for all $x,y\in(0,1].$
1995 Putnam, 1
For a partition $\pi$ of $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$, let $\pi(x)$ be the number of elements in the part containing $x$. Prove that for any two partitions $\pi$ and $\pi^{\prime}$, there are two distinct numbers $x$ and $y$ in $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ such that $\pi(x) = \pi(y)$ and $\pi^{\prime}(x) = \pi^{\prime}(y)$.
2000 AMC 12/AHSME, 12
Let $ A$, $ M$, and $ C$ be nonnegative integers such that $ A \plus{} M \plus{} C \equal{} 12$. What is the maximum value of $ A \cdot M \cdot C \plus{} A\cdot M \plus{} M \cdot C \plus{} C\cdot A$?
$ \textbf{(A)}\ 62 \qquad \textbf{(B)}\ 72 \qquad \textbf{(C)}\ 92 \qquad \textbf{(D)}\ 102 \qquad \textbf{(E)}\ 112$
2014 Chile TST Ibero, 1
Consider a function $f: \mathbb{R} \to \mathbb{R}$ satisfying for all $x \in \mathbb{R}$:
\[
f(x+1) = \frac{1}{2} + \sqrt{f(x) - f(x)^2}.
\]
Prove that there exists a $b > 0$ such that $f(x + b) = f(x)$ for all $x \in \mathbb{R}$.
2022 Princeton University Math Competition, B1
Let $q$ be the sum of the expressions $a_1^{-a_2^{a_3^{a_4}}}$ over all permutations $(a_1, a_2, a_3, a_4)$ of $(1,2,3,4).$ Determine $\lfloor q \rfloor.$
2009 Purple Comet Problems, 2
Find the least positive integer $n$ such that for every prime number $p, p^2 + n$ is never prime.
2000 Moldova National Olympiad, Problem 3
For any $n\in\mathbb N$, denote by $a_n$ the sum $2+22+222+\cdots+22\ldots2$, where the last summand consists of $n$ digits of $2$. Determine the greatest $n$ for which $a_n$ contains exactly $222$ digits of $2$.
2016 PUMaC Individual Finals A, 2
Let $m, k$, and $c$ be positive integers with $k > c$, and let $\lambda$ be a positive, non-integer real root of the equation $\lambda^{m+1} - k \lambda^m - c = 0$. Let $f : Z^+ \to Z$ be defined by $f(n) = \lfloor \lambda n \rfloor$ for all $n \in Z^+$. Show that $f^{m+1}(n) \equiv cn - 1$ (mod $k$) for all $n \in Z^+$. (Here, $Z^+$ denotes the set of positive integers, $ \lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $f^{m+1}(n) = f(f(... f(n)...))$ where $f$ appears $m + 1$ times.)
1963 AMC 12/AHSME, 29
A particle projected vertically upward reaches, at the end of $t$ seconds, an elevation of $s$ feet where $s = 160 t - 16t^2$. The highest elevation is:
$\textbf{(A)}\ 800 \qquad
\textbf{(B)}\ 640\qquad
\textbf{(C)}\ 400 \qquad
\textbf{(D)}\ 320 \qquad
\textbf{(E)}\ 160$