Found problems: 85335
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 $DEFG$.
1993 Tournament Of Towns, (372) 4
Three piles of stones are given. One may add to, or remove from one of the piles in one operation the number of stones in the other two piles. For example $[12,3,5]$ can become$ [12,20,5]$ by adding $17 = 12 + 5$ stones to pile 2 or $[4,3,5]$ by removing $8 = 3 + 5$ stones from pile $1$. Is it possible starting from the piles with $1993$, $199$ and $19$ stones to get one empty heap after several operations?
(MN Gusarov)
1995 Argentina National Olympiad, 3
Let ABCD be a parallelogram, and P a point such that
$2 PDA=ABP$ and
$2 PAD=PCD$
Show that $AB=BP=CP$
XMO (China) 2-15 - geometry, 14.3
In quadrilateral $ABCD$, $E$ and $F$ are midpoints of $AB$ and $CD$, and $G$ is the intersection of $AD$ with $BC$. $P$ is a point within the quadrilateral, such that $PA=PB$, $PC=PD$, and $\angle APB+\angle CPD=180^{\circ}$. Prove that $PG$ and $EF$ are parallel.
2021 Thailand Online MO, P10
Each cell of the board with $2021$ rows and $2022$ columns contains exactly one of the three letters $T$, $M$, and $O$ in a way that satisfies each of the following conditions:
[list]
[*] In total, each letter appears exactly $2021\times 674$ of times on the board.
[*] There are no two squares that share a common side and contain the same letter.
[*] Any $2\times 2$ square contains all three letters $T$, $M$, and $O$.
[/list]
Prove that each letter $T$, $M$, and $O$ appears exactly $674$ times on every row.
2013 USAMTS Problems, 1
Alex is trying to open a lock whose code is a sequence that is three letters long, with each of the letters being one of $\text A$, $\text B$ or $\text C$, possibly repeated. The lock has three buttons, labeled $\text A$, $\text B$ and $\text C$. When the most recent $3$ button-presses form the code, the lock opens. What is the minimum number of total button presses Alex needs to guarantee opening the lock?
2020 Yasinsky Geometry Olympiad, 6
A cube whose edge is $1$ is intersected by a plane that does not pass through any of its vertices, and its edges intersect only at points that are the midpoints of these edges. Find the area of the formed section. Consider all possible cases.
(Alexander Shkolny)
2003 ITAMO, 4
There are two sorts of people on an island: [i]knights[/i], who always talk truth, and [i]scoundrels[/i], who always lie. One day, the people establish a council consisting of $2003$ members. They sit around a round table, and during the council each member said: "Both my neighbors are scoundrels". In a later day, the council meets again, but one member could not come due to illness, so only $2002$ members were present. They sit around the round table, and everybody said: "Both my neighbors belong to the sort different from mine". Is the absent member a knight or a scoundrel?
2011 Germany Team Selection Test, 1
Two circles $\omega , \Omega$ intersect in distinct points $A,B$ a line through $B$ intersects $\omega , \Omega$ in $C,D$ respectively such that $B$ lies between $C,D$ another line through $B$ intersects $\omega , \Omega$ in $E,F$ respectively such that $E$ lies between $B,F$ and $FE=CD$. Furthermore $CF$ intersects $\omega , \Omega$ in $P,Q$ respectively and $M,N$ are midpoints of the arcs $PB,QB$. Prove that $CNMF$ is a cyclic quadrilateral.
2003 National High School Mathematics League, 7
The solution set for inequality $|x|^3-2x^2-4|x|+3<0$ is________.
2007 Pan African, 2
For which positive integers $n$ is $231^n-222^n-8 ^n -1$ divisible by $2007$?
2020-21 KVS IOQM India, 5
Find the number of positive integers $n$ such that the highest power of $7$ dividing $n!$ is $8$.
2025 Benelux, 4
Let $a_0, a_1, \ldots, a_{10}$ be integers such that, for each $i \in \{0,1,\ldots,2047\}$, there exists a subset $S \subseteq \{0,1,\ldots,10\}$ with
\[
\sum_{j \in S} a_j \equiv i \pmod{2048}.
\]
Show that for each $i \in \{0,1,\ldots,10\}$, there is exactly one $j \in \{0,1,\ldots,10\}$ such that $a_j$ is divisible by $2^i$ but not by $2^{i+1}$.
Note: $\sum_{j \in S} a_j$ is the summation notation, for instance, $\sum_{j \in \{2,5\}} a_j = a_2 + a_5$, while for the empty set $\varnothing$, one defines $\sum_{j \in \varnothing} a_j = 0$.
1998 IMO Shortlist, 6
Let $ABCDEF$ be a convex hexagon such that $\angle B+\angle D+\angle F=360^{\circ }$ and \[ \frac{AB}{BC} \cdot \frac{CD}{DE} \cdot \frac{EF}{FA} = 1. \] Prove that \[ \frac{BC}{CA} \cdot \frac{AE}{EF} \cdot \frac{FD}{DB} = 1. \]
2010 Sharygin Geometry Olympiad, 5
Let $AH$, $BL$ and $CM$ be an altitude, a bisectrix and a median in triangle $ABC$. It is known that lines $AH$ and $BL$ are an altitude and a bisectrix of triangle $HLM$. Prove that line $CM$ is a median of this triangle.
2023 Myanmar IMO Training, 6
Let points $M$ and $N$ lie on sides $AB$ and $BC$ of triangle $ABC$ in such a way that $MN||AC$. Points $M'$ and $N'$ are the reflections of $M$ and $N$ about $BC$ and $AB$ respectively. Let $M'A$ meet $BC$ at $X$, and let $N'C$ meet $AB$ at $Y$. Prove that $A,C,X,Y$ are concyclic.
V Soros Olympiad 1998 - 99 (Russia), 11.3
For what a from the interval $[0,\pi]$ do there exist $a$ and $b$ that are not simultaneously equal to zero, for which the inequality
$$a \cos x + b \cos 2x \le 0$$ is satisfied for all $x$ belonging to the segment $[a, \pi]$?
1994 India Regional Mathematical Olympiad, 4
Solve the system of equations for real $x$ and $y$: \begin{eqnarray*} 5x \left( 1 + \frac{1}{x^2 + y^2}\right) &=& 12 \\ 5y \left( 1 - \frac{1}{x^2+y^2} \right) &=& 4 . \end{eqnarray*}
2018 Germany Team Selection Test, 1
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.
[i]Proposed by Jeck Lim, Singapore[/i]