Found problems: 85335
PEN B Problems, 6
Suppose that $m$ does not have a primitive root. Show that \[a^{ \frac{\phi(m)}{2}}\equiv 1 \; \pmod{m}\] for every $a$ relatively prime $m$.
2018 Latvia Baltic Way TST, P3
Let $a_1,a_2,...$ be an infinite sequence of integers that satisfies $a_{n+2}=a_{n+1}+a_n$ for all $n \ge 1$. There exists a positive integer $k$ such that $a_k=a_{k+2018}$. Prove that there exists a term of the sequence which is equal to zero.
2015 ASDAN Math Tournament, 5
The eight corners of a cube are cut off, yielding a polyhedron with $6$ octagonal faces and $8$ triangular faces. Given that all polyhedron's edges have length $2$, compute the volume of the polyhedron.
2004 Federal Competition For Advanced Students, P2, 4
Show that there is an infinite sequence $a_1,a_2,...$ of natural numbers such that $a^2_1+a^2_2+ ...+a^2_N$ is a perfect square for all $N$. Give a recurrent formula for one such sequence.
1989 IMO Shortlist, 2
Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. If the two rooms have dimensions of 38 feet by 55 feet and 50 feet by 55 feet, what are the carpet dimensions?
1987 IMO Shortlist, 4
Let $ABCDEFGH$ be a parallelepiped with $AE \parallel BF \parallel CG \parallel DH$. Prove the inequality
\[AF + AH + AC \leq AB + AD + AE + AG.\]
In what cases does equality hold?
[i]Proposed by France.[/i]
2023 Flanders Math Olympiad, 4
There are $12$ mathematicians living in a village, each of whom belongs to the $\sqrt2$-clan or belong to the $\pi$-clan. Moreover every mathematician's birthday is in a different month and every mathematician has an odd number of friends among them the mathematicians. We agree that if mathematician $A$ is a friend of mathematician $B$, then so is $B$ is a friend of $A$. On his birthday, every mathematician looks at which clan the majority of his friends belong to, and decides to join that clan until his next birthday. Prove that the mathematicians no longer change clans after a certain point.
2024 Austrian MO National Competition, 1
Determine the smallest real constant $C$ such that the inequality
\[(X+Y)^2(X^2+Y^2+C)+(1-XY)^2 \ge 0\]
holds for all real numbers $X$ and $Y$. For which values of $X$ and $Y$ does equality hold for this smallest constant $C$?
[i](Walther Janous)[/i]
2011 Morocco National Olympiad, 1
Solve the following equation in $\mathbb{R}^+$ :
\[\left\{\begin{matrix}
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2010\\
x+y+z=\frac{3}{670}
\end{matrix}\right.\]
2015 Purple Comet Problems, 18
Define the determinant $D_1$ = $|1|$, the determinant $D_2$ =
$|1 1|$
$|1 3|$
, and the determinant $D_3=$
|1 1 1|
|1 3 3|
|1 3 5|
.
In general, for positive integer n, let the determinant $D_n$ have 1s in every position of its first row and first
column, 3s in the remaining positions of the second row and second column, 5s in the remaining positions of the third row and third column, and so forth. Find the least n so that $D_n$ $\geq$ 2015.
Today's calculation of integrals, 863
For $0<t\leq 1$, let $F(t)=\frac{1}{t}\int_0^{\frac{\pi}{2}t} |\cos 2x|\ dx.$
(1) Find $\lim_{t\rightarrow 0} F(t).$
(2) Find the range of $t$ such that $F(t)\geq 1.$
2005 Cuba MO, 1
Determine all the quadrilaterals that can be divided by a diagonal into two triangles of equal area and equal perimeter.
2004 USA Team Selection Test, 6
Define the function $f: \mathbb N \cup \{0\} \to \mathbb{Q}$ as follows: $f(0) = 0$ and \[ f(3n+k) = -\frac{3f(n)}{2} + k , \] for $k = 0, 1, 2$. Show that $f$ is one-to-one and determine the range of $f$.
PEN N Problems, 12
The sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{n}= 1+2^{2}+3^{3}+\cdots+n^{n}.\] Prove that there are infinitely many $n$ such that $a_{n}$ is composite.
1999 Harvard-MIT Mathematics Tournament, 5
You are trapped in a room with only one exit, a long hallway with a series of doors and land mines. To get out you must open all the doors and disarm all the mines. In the room is a panel with $3$ buttons, which conveniently contains an instruction manual. The red button arms a mine, the yellow button disarms two mines and closes a door, and the green button opens two doors. Initially $3$ doors are closed and $3$ mines are armed. The manual warns that attempting to disarm two mines or open two doors when only one is armed/closed will reset the system to its initial state. What is the minimum number of buttons you must push to get out?
1997 Brazil Team Selection Test, Problem 5
Let $ABC$ be an acute-angled triangle with incenter $I$. Consider the point $A_1$ on $AI$ different from $A$, such that the midpoint of $AA_1$ lies on the circumscribed circle of $ABC$. Points $B_1$ and $C_1$ are defined similarly.
(a) Prove that $S_{A_1B_1C_1}=(4R+r)p$, where $p$ is the semi-perimeter, $R$ is the circumradius and $r$ is the inradius of $ABC$.
(b) Prove that $S_{A_1B_1C_1}\ge9S_{ABC}$.
2016 India National Olympiad, P2
For positive real numbers $a,b,c$ which of the following statements necessarily implies $a=b=c$: (I) $a(b^3+c^3)=b(c^3+a^3)=c(a^3+b^3)$, (II) $a(a^3+b^3)=b(b^3+c^3)=c(c^3+a^3)$ ? Justify your answer.
2018 China Team Selection Test, 4
Let $p$ be a prime and $k$ be a positive integer. Set $S$ contains all positive integers $a$ satisfying $1\le a \le p-1$, and there exists positive integer $x$ such that $x^k\equiv a \pmod p$.
Suppose that $3\le |S| \le p-2$. Prove that the elements of $S$, when arranged in increasing order, does not form an arithmetic progression.
2010 LMT, 19
Two integers are called [i]relatively prime[/i] if they share no common factors other than $1.$ Determine the sum of all positive integers less than $162$ that are relatively prime to $162.$
1945 Moscow Mathematical Olympiad, 105
A circle rolls along a side of an equilateral triangle. The radius of the circle is equal to the height of the triangle. Prove that the measure of the arc intercepted by the sides of the triangle on this circle is equal to $60^o$ at all times.
2022 USA TSTST, 6
Let $O$ and $H$ be the circumcenter and orthocenter, respectively, of an acute scalene triangle $ABC$. The perpendicular bisector of $\overline{AH}$ intersects $\overline{AB}$ and $\overline{AC}$ at $X_A$ and $Y_A$ respectively. Let $K_A$ denote the intersection of the circumcircles of triangles $OX_AY_A$ and $BOC$ other than $O$.
Define $K_B$ and $K_C$ analogously by repeating this construction two more times. Prove that $K_A$, $K_B$, $K_C$, and $O$ are concyclic.
[i]Hongzhou Lin[/i]
2014 Dutch IMO TST, 3
Let $a$, $b$ and $c$ be rational numbers for which $a+bc$, $b+ac$ and $a+b$ are all non-zero and for which we have
\[\frac{1}{a+bc}+\frac{1}{b+ac}=\frac{1}{a+b}.\]
Prove that $\sqrt{(c-3)(c+1)}$ is rational.
2009 Korea National Olympiad, 1
Let $ A = \{ 1, 2, 3, \cdots , 12 \} $. Find the number of one-to-one function $ f :A \to A $ satisfying following condition: for all $ i \in A $, $ f(i)-i $ is not a multiple of $ 3 $.
2019 PUMaC Combinatorics A, 3
Marko lives on the origin of the Cartesian plane. Every second, Marko moves $1$ unit up with probability $\tfrac{2}{9}$, $1$ unit right with probability $\tfrac{2}{9}$, $1$ unit up and $1$ unit right with probability $\tfrac{4}{9}$, and he doesn’t move with probability $\tfrac{1}{9}$. After $2019$ seconds, Marko ends up on the point $(A, B)$. What is the expected value of $A\cdot B$?
2009 Postal Coaching, 4
All the integers from $1$ to $100$ are arranged in a $10 \times 10$ table as shown below. Prove that if some ten numbers are removed from the table, the remaining $90$ numbers contain 10 numbers in Arithmetic Progression.
$1 \,\,\,\,2\,\, \,\,3 \,\,\,\,... \,\,10$
$11 \,\,12 \,\,13 \,\,... \,\,20$
$\,\,.\,\,\,\,.\,\,\,.$
$\,\,.\,\,\,\,.\,\,\,\,.$
$91 \,\,92 \,\,93\,\, ... \,\,100$