Found problems: 85335
2000 AIME Problems, 9
Given that $z$ is a complex number such that $z+\frac 1z=2\cos 3^\circ,$ find the least integer that is greater than $z^{2000}+\frac 1{z^{2000}}.$
1960 IMO Shortlist, 1
Determine all three-digit numbers $N$ having the property that $N$ is divisible by 11, and $\dfrac{N}{11}$ is equal to the sum of the squares of the digits of $N$.
2014 ASDAN Math Tournament, 7
Let $ABCD$ be a square piece of paper with side length $4$. Let $E$ be a point on $AB$ such that $AE=3$ and let $F$ be a point on $CD$ such that $DF=1$. Now, fold $AEFD$ over the line $EF$. Compute the area of the resulting shape.
2007 Iran MO (3rd Round), 2
Let $ m,n$ be two integers such that $ \varphi(m) \equal{}\varphi(n) \equal{} c$. Prove that there exist natural numbers $ b_{1},b_{2},\dots,b_{c}$ such that $ \{b_{1},b_{2},\dots,b_{c}\}$ is a reduced residue system with both $ m$ and $ n$.
2025 CMIMC Algebra/NT, 6
Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $[-1,1].$ Find the probability that $$|x|+|y|+1 \le 3\min\{|x+y+1|, |x+y-1|\}.$$
2005 Romania National Olympiad, 3
Let $X_1,X_2,\ldots,X_m$ a numbering of the $m=2^n-1$ non-empty subsets of the set $\{1,2,\ldots,n\}$, $n\geq 2$. We consider the matrix $(a_{ij})_{1\leq i,j\leq m}$, where $a_{ij}=0$, if $X_i \cap X_j = \emptyset$, and $a_{ij}=1$ otherwise. Prove that the determinant $d$ of this matrix does not depend on the way the numbering was done and compute $d$.
1987 IMO Longlists, 55
Two moving bodies $M_1,M_2$ are displaced uniformly on two coplanar straight lines. Describe the union of all straight lines $M_1M_2.$
VI Soros Olympiad 1999 - 2000 (Russia), 10.7
The numbers $1, 2, 3, ..., 99, 100$ are randomly arranged in the cells of a square table measuring $10\times 10$ (each number is used only once). Prove that there are three cells in the table whose sum of numbers does not exceed 1$82$. The centers of these cells form an isosceles right triangle, the legs of which are parallel to the edges of the table.
2023 Dutch BxMO TST, 3
We play a game of musical chairs with $n$ chairs numbered $1$ to $n$. You attach $n$ leaves, numbered $1$ to $n$, to the chairs in such a way that the number on a leaf does not match the number on the chair it is attached to. One player sits on each chair. Every time you clap, each player looks at the number on the leaf attached to his current seat and moves to sit on the seat with that number. Prove that, for any $m$ that is not a prime power with$ 1 < m \leq n$, it is possible to attach the leaves to the seats in such a way that after $m$ claps everyone has returned to the chair they started on for the first time.
2017 Hanoi Open Mathematics Competitions, 9
Prove that the equilateral triangle of area $1$ can be covered by five arbitrary equilateral triangles having the total area $2$.
MOAA Team Rounds, 2018.3
Let $BE$ and $CF$ be altitudes in triangle $ABC$. If $AE = 24$, $EC = 60$, and $BF = 31$, determine $AF$.
2017 Dutch IMO TST, 1
Let $a, b,c$ be distinct positive integers, and suppose that $p = ab+bc+ca$ is a prime number.
$(a)$ Show that $a^2,b^,c^2$ give distinct remainders after division by $p$.
(b) Show that $a^3,b^3,c^3$ give distinct remainders after division by $p$.
2018 Taiwan APMO Preliminary, 7
$240$ students are participating a big performance show. They stand in a row and face to their coach. The coach askes them to count numbers from left to right, starting from $1$. (Of course their counts be like $1,2,3,...$)The coach askes them to remember their number and do the following action:
First, if your number is divisible by $3$ then turn around.
Then, if your number is divisible by $5$ then turn around.
Finally, if your number is divisible by $7$ then turn around.
(a) How many students are face to coach now?
(b) What is the number of the $66^{\text{th}}$ student counting from left who is face to coach?
2021 Francophone Mathematical Olympiad, 4
Let $\mathbb{N}_{\ge 1}$ be the set of positive integers.
Find all functions $f \colon \mathbb{N}_{\ge 1} \to \mathbb{N}_{\ge 1}$ such that, for all positive integers $m$ and $n$:
(a) $n = \left(f(2n)-f(n)\right)\left(2 f(n) - f(2n)\right)$,
(b)$f(m)f(n) - f(mn) = \left(f(2m)-f(m)\right)\left(2 f(n) - f(2n)\right) + \left(f(2n)-f(n)\right)\left(2 f(m) - f(2m)\right)$,
(c) $m-n$ divides $f(2m)-f(2n)$ if $m$ and $n$ are distinct odd prime numbers.
2007 ITest, 20
Find the largest integer $n$ such that $2007^{1024}-1$ is divisible by $2^n$.
$\textbf{(A) }1\hspace{14em}\textbf{(B) }2\hspace{14em}\textbf{(C) }3$
$\textbf{(D) }4\hspace{14em}\textbf{(E) }5\hspace{14em}\textbf{(F) }6$
$\textbf{(G) }7\hspace{14em}\textbf{(H) }8\hspace{14em}\textbf{(I) }9$
$\textbf{(J) }10\hspace{13.7em}\textbf{(K) }11\hspace{13.5em}\textbf{(L) }12$
$\textbf{(M) }13\hspace{13.3em}\textbf{(N) }14\hspace{13.4em}\textbf{(O) }15$
$\textbf{(P) }16\hspace{13.6em}\textbf{(Q) }55\hspace{13.4em}\textbf{(R) }63$
$\textbf{(S) }64\hspace{13.7em}\textbf{(T) }2007$
2018 India PRMO, 14
If $x = cos 1^o cos 2^o cos 3^o...cos 89^o$ and $y = cos 2^o cos 6^o cos 10^o...cos 86^o$, then what is the integer nearest to $\frac27 \log_2 \frac{y}{x}$ ?
2016 APMC, 3
Let $a_1,a_2,\cdots$ be a strictly increasing sequence on positive integers.
Is it always possible to partition the set of natural numbers $\mathbb{N}$ into infinitely many subsets with infinite cardinality $A_1,A_2,\cdots$, so that for every subset $A_i$, if we denote $b_1<b_2<\cdots$ be the elements of $A_i$, then for every $k\in \mathbb{N}$ and for every $1\le i\le a_k$, it satisfies $b_{i+1}-b_{i}\le k$?
1953 Miklós Schweitzer, 10
[b]10.[/b] Consider a point performing a random walk on a planar triangular lattice and suppose that it moves away with equal probability from any lattice point along any one of the six lattice lines issuing from it. Prove that if the walk is continued indefinitely, then the point will return to its starting point with probability 1. [b](P. 5)[/b]
1984 AMC 12/AHSME, 29
Find the largest value for $\frac{y}{x}$ for pairs of real numbers $(x,y)$ which satisfy \[(x-3)^2 + (y-3)^2 = 6.\]
$\textbf{(A) }3 + 2 \sqrt 2\qquad
\textbf{(B) } 2 + \sqrt 3\qquad
\textbf{(C ) }3 \sqrt 3\qquad
\textbf{(D) }6\qquad
\textbf{(E) }6 + 2 \sqrt 3$
1988 Tournament Of Towns, (165) 2
We are given convex quadrilateral $ABCD$. The midpoints of $BC$ and $DA$ are $M$ and $N$ respectively. The diagonal $AC$ divides $MN$ in half. Prove that the areas of triangles $ABC$ and $ACD$ are equal .
2019 South East Mathematical Olympiad, 2
$ABCD$ is a parallelogram with $\angle BAD \neq 90$. Circle centered at $A$ radius $BA$ denoted as $\omega _1$ intersects the extended side of $AB,CB$ at points $E,F$ respectively. Suppose the circle centered at $D$ with radius $DA$, denoted as $\omega _2$, intersects $AD,CD$ at points $M,N$ respectively. Suppose $EN,FM$ intersects at $G$, $AG$ intersects $ME$ at point $T$. $MF$ intersects $\omega _1$ at $Q \neq F$, and $EN$ intersects $\omega _2$ at $P \neq N$. Prove that $G,P,T,Q$ concyclic.
PEN O Problems, 7
Show that for each $n \ge 2$, there is a set $S$ of $n$ integers such that $(a-b)^2$ divides $ab$ for every distinct $a, b\in S$.
2008 Paraguay Mathematical Olympiad, 2
Find for which values of $n$, an integer larger than $1$ but smaller than $100$, the following expression has its minimum value:
$S = |n-1| + |n-2| + \ldots + |n-100|$
2006 Vietnam Team Selection Test, 1
Given an acute angles triangle $ABC$, and $H$ is its orthocentre. The external bisector of the angle $\angle BHC$ meets the sides $AB$ and $AC$ at the points $D$ and $E$ respectively. The internal bisector of the angle $\angle BAC$ meets the circumcircle of the triangle $ADE$ again at the point $K$. Prove that $HK$ is through the midpoint of the side $BC$.
1974 IMO Longlists, 52
A fox stands in the centre of the field which has the form of an equilateral triangle, and a rabbit stands at one of its vertices. The fox can move through the whole field, while the rabbit can move only along the border of the field. The maximal speeds of the fox and rabbit are equal to $u$ and $v$, respectively. Prove that:
(a) If $2u>v$, the fox can catch the rabbit, no matter how the rabbit moves.
(b) If $2u\le v$, the rabbit can always run away from the fox.