Found problems: 85335
2014 Saudi Arabia BMO TST, 5
Let $n > 3$ be an odd positive integer not divisible by $3$. Determine if it is possible to form an $n \times n$ array of numbers such that
[list]
[*] [b](a)[/b] the set of the numbers in each row is a permutation of $0, 1, \dots , n - 1$;
the set of the numbers in each column is a permutation of $0, 1, \dots , n-1$;
[*] [b](b)[/b] the board is [i]totally non-symmetric[/i]: for $1 \le i < j \le n$ and $1 \le i' < j' \le n$, if $(i, j) \neq (i', j')$ then $(a_{i,j} , a_{j,i}) \neq (a_{i',j'} , a_{j',i'})$ where $a_{i,j}$ denotes the entry in the $i^\text{th}$ row and $j^\text{th}$ column.[/list]
1957 AMC 12/AHSME, 42
If $ S \equal{} i^n \plus{} i^{\minus{}n}$, where $ i \equal{} \sqrt{\minus{}1}$ and $ n$ is an integer, then the total number of possible distinct values for $ S$ is:
$ \textbf{(A)}\ 1\qquad
\textbf{(B)}\ 2\qquad
\textbf{(C)}\ 3\qquad
\textbf{(D)}\ 4\qquad
\textbf{(E)}\ \text{more than 4}$
2018 AMC 8, 6
On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a coastal access road. He drove three times as fast on the highway as on the coastal road. If Anh spent 30 minutes driving on the coastal road, how many minutes did his entire trip take?
$\textbf{(A) }50\qquad\textbf{(B) }70\qquad\textbf{(C) }80\qquad\textbf{(D) }90\qquad \textbf{(E) }100$
2009 Vietnam Team Selection Test, 1
Let an acute triangle $ ABC$ with curcumcircle $ (O)$. Call $ A_1,B_1,C_1$ are foots of perpendicular line from $ A,B,C$ to opposite side. $ A_2,B_2,C_2$ are reflect points of $ A_1,B_1,C_1$ over midpoints of $ BC,CA,AB$ respectively. Circle $ (AB_2C_2),(BC_2A_2),(CA_2B_2)$ cut $ (O)$ at $ A_3,B_3,C_3$ respectively.
Prove that: $ A_1A_3,B_1B_3,C_1C_3$ are concurent.
2023 Romania National Olympiad, 3
Determine all positive integers $n$ for which the number
\[
N = \frac{1}{n \cdot (n + 1)}
\]
can be represented as a finite decimal fraction.
2014 Taiwan TST Round 3, 3
Let $M$ be any point on the circumcircle of triangle $ABC$. Suppose the tangents from $M$ to the incircle meet $BC$ at two points $X_1$ and $X_2$. Prove that the circumcircle of triangle $MX_1X_2$ intersects the circumcircle of $ABC$ again at the tangency point of the $A$-mixtilinear incircle.
2023 OlimphÃada, 4
We all know the Fibonacci sequence. However, a slightly less known sequence is the $k$-bonacci sequence. In it, we have $F_1^{(k)} = F_2^{(k)} = \cdots = F_{k-1}^{(k)} = 0, F_k^{(k)} = 1$ and $$F^{(k)}_{n+k} = F^{(k)}_{n+k-1} + F^{(k)}_{n+k-2} + \cdots + F^{(k)}_n,$$for all $n \geq 1$. Find all positive integers $k$ for which there exists a constant $N$ such that $$F^{(k)}_{n-1}F^{(k)}_{n+1} - (F ^{(k)}_n)^2 = (-1)^n$$ for every positive integer $n \geq N$.
2020 Kosovo National Mathematical Olympiad, 3
Let $a$ and $b$ be real numbers such that $a+b=\log_2( \log_2 3)$. What is the minimum value of $2^a + 3^b$ ?
Champions Tournament Seniors - geometry, 2011.2
Let $ABC$ be an isosceles triangle in which $AB = AC$. On its sides $BC$ and $AC$ respectively are marked points $P$ and $Q$ so that $PQ\parallel AB$. Let $F$ be the center of the circle circumscribed about the triangle $PQC$, and $E$ the midpoint of the segment $BQ$. Prove that $\angle AEF = 90^o $.
2023 Balkan MO Shortlist, C2
For an integer $n>2$, the tuple $(1, 2, \ldots, n)$ is written on a blackboard. On each turn, one can choose two numbers from the tuple such that their sum is a perfect square and swap them to obtain a new tuple. Find all integers $n > 2$ for which all permutations of $\{1, 2,\ldots, n\}$ can appear on the blackboard in this way.
2020 Durer Math Competition Finals, 5
On a piece of paper, we write down all positive integers $n$ such that all proper divisors of $n$ are less than $30$. We know that the sum of all numbers on the paper having exactly one proper divisor is $2397$. What is the sum of all numbers on the paper having exactly two proper divisors?
We say that $k$ is a proper divisor of the positive integer $n$ if $k | n$ and $1 < k < n$.
2020 Olympic Revenge, 4
Let $n$ be a positive integer and $A$ a set of integers such that the set $\{x = a + b\ |\ a, b \in A\}$ contains $\{1^2, 2^2, \dots, n^2\}$. Prove that there is a positive integer $N$ such that if $n \ge N$, then $|A| > n^{0.666}$.
1986 Swedish Mathematical Competition, 2
The diagonals $AC$ and $BD$ of a quadrilateral $ABCD$ intersect at $O$. If $S_1$ and $S_2$ are the areas of triangles $AOB$ and $COD$ and S that of $ABCD$, show that $\sqrt{S_1} + \sqrt{S_2} \le \sqrt{S}$. Prove that equality holds if and only if $AB$ and $CD$ are parallel.
PEN K Problems, 15
Find all functions $f: \mathbb{Z}\to \mathbb{Z}$ such that for all $m,n\in \mathbb{Z}$: \[f(m+f(n))=f(m)-n.\]
2003 All-Russian Olympiad Regional Round, 10.5
Find all $x$ for which the equation $ x^2 + y^2 + z^2 + 2xyz = 1$ (relative to $z$) has a valid solution for any $y$.
2015 Korea Junior Math Olympiad, 3
For all nonnegative integer $i$, there are seven cards with $2^i$ written on it.
How many ways are there to select the cards so that the numbers add up to $n$?
2023 Romania National Olympiad, 2
Determine all triples $(a,b,c)$ of integers that simultaneously satisfy the following relations:
\begin{align*}
a^2 + a = b + c, \\
b^2 + b = a + c, \\
c^2 + c = a + b.
\end{align*}
2010 District Olympiad, 4
We consider the quadrilateral $ABCD$, with $AD = CD = CB$ and $AB \parallel CD$. Points $E$ and $F$ belong to the segments $CD$ and $CB$ so that angles $\angle ADE = \angle AEF$. Prove that:
a) $4CF \le CB$ ,
b) if $4CF = CB$, then $AE$ is the bisector of the angle $\angle DAF$.
2023 Regional Olympiad of Mexico Southeast, 2
Let $ABC$ be an acute-angled triangle, $D$ be the foot of the altitude from $A$, the circle with diameter $AD$ intersect $AB$ at $F$ and $AC$ at $E$. Let $P$ be the orthocenter of triangle $AEF$ and $O$ be the circumcenter of $ABC$. Prove that $A, P,$ and $O$ are collinear.
PEN E Problems, 33
Prove that there are no positive integers $a$ and $b$ such that for all different primes $p$ and $q$ greater than $1000$, the number $ap+bq$ is also prime.
2014 District Olympiad, 1
[list=a]
[*]Prove that for any real numbers $a$ and $b$ the following inequality
holds:
\[ \left( a^{2}+1\right) \left( b^{2}+1\right) +50\geq2\left( 2a+1\right)\left( 3b+1\right)\]
[*]Find all positive integers $n$ and $p$ such that:
\[ \left( n^{2}+1\right) \left( p^{2}+1\right) +45=2\left( 2n+1\right)\left( 3p+1\right) \][/list]
2019 India IMO Training Camp, P1
Let $a_1,a_2,\ldots, a_m$ be a set of $m$ distinct positive even numbers and $b_1,b_2,\ldots,b_n$ be a set of $n$ distinct positive odd numbers such that
\[a_1+a_2+\cdots+a_m+b_1+b_2+\cdots+b_n=2019\]
Prove that
\[5m+12n\le 581.\]
2021 Sharygin Geometry Olympiad, 9
Points $E$ and $F$ lying on sides $BC$ and $AD$ respectively of a parallelogram $ABCD$ are such that $EF=ED=DC$. Let $M$ be the midpoint of $BE$ and $MD$ meet $EF$ at $G$. Prove that $\angle EAC=\angle GBD$.
2011 AMC 10, 18
Circles $A, B,$ and $C$ each have radius 1. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}$. What is the area inside Circle $C$ but outside circle $A$ and circle $B$ ?
[asy]
pathpen = linewidth(.7); pointpen = black;
pair A=(-1,0), B=-A, C=(0,1); fill(arc(C,1,0,180)--arc(A,1,90,0)--arc(B,1,180,90)--cycle, gray(0.5)); D(CR(D("A",A,SW),1)); D(CR(D("B",B,SE),1)); D(CR(D("C",C,N),1));[/asy]
${
\textbf{(A)}\ 3 - \frac{\pi}{2} \qquad
\textbf{(B)}\ \frac{\pi}{2} \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ \frac{3\pi}{4} \qquad
\textbf{(E)}\ 1+\frac{\pi}{2}} $
PEN O Problems, 31
Prove that, for any integer $a_{1}>1$, there exist an increasing sequence of positive integers $a_{1}, a_{2}, a_{3}, \cdots$ such that \[a_{1}+a_{2}+\cdots+a_{n}\; \vert \; a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}\] for all $n \in \mathbb{N}$.