Found problems: 85335
2019 Spain Mathematical Olympiad, 2
Determine if there exists a finite set $S$ formed by positive prime numbers so that for each integer $n\geq2$, the number $2^2 + 3^2 +...+ n^2$ is a multiple of some element of $S$.
2006 France Team Selection Test, 3
Let $M=\{1,2,\ldots,3 \cdot n\}$. Partition $M$ into three sets $A,B,C$ which $card$ $A$ $=$ $card$ $B$ $=$ $card$ $C$ $=$ $n .$
Prove that there exists $a$ in $A,b$ in $B, c$ in $C$ such that or $a=b+c,$ or $b=c+a,$ or $c=a+b$
[i]Edited by orl.[/i]
1966 IMO Shortlist, 51
Consider $n$ students with numbers $1, 2, \ldots, n$ standing in the order $1, 2, \ldots, n.$ Upon a command, any of the students either remains on his place or switches his place with another student. (Actually, if student $A$ switches his place with student $B,$ then $B$ cannot switch his place with any other student $C$ any more until the next command comes.)
Is it possible to arrange the students in the order $n,1, 2, \ldots, n-1$ after two commands ?
1989 IMO Longlists, 52
Let $ f$ be a function from the real numbers to the real numbers such that $ f(1) \equal{} 1, f(a\plus{}b) \equal{} f(a)\plus{}f(b)$ for all $ a, b,$ and $ f(x)f \left( \frac{1}{x} \right) \equal{} 1$ for all $ x \neq 0.$ Prove that $ f(x) \equal{} x$ for all real numbers $ x.$
2007 China Team Selection Test, 2
Let $ x_1, \ldots, x_n$ be $ n>1$ real numbers satisfying $ A\equal{}\left |\sum^n_{i\equal{}1}x_i\right |\not \equal{}0$ and $ B\equal{}\max_{1\leq i<j\leq n}|x_j\minus{}x_i|\not \equal{}0$. Prove that for any $ n$ vectors $ \vec{\alpha_i}$ in the plane, there exists a permutation $ (k_1, \ldots, k_n)$ of the numbers $ (1, \ldots, n)$ such that \[ \left |\sum_{i\equal{}1}^nx_{k_i}\vec{\alpha_i}\right | \geq \dfrac{AB}{2A\plus{}B}\max_{1\leq i\leq n}|\alpha_i|.\]
2017 Germany Team Selection Test, 3
Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
2010 Slovenia National Olympiad, 4
Find the smallest three-digit number such that the following holds:
If the order of digits of this number is reversed and the number obtained by this is added to the original number, the resulting number consists of only odd digits.
2006 AMC 8, 8
The table shows some of the results of a survey by radiostation KAMC. What percentage of the males surveyed listen to the station?
$ \begin{tabular}{|c|c|c|c|}
\hline & Listen & Don't Listen & Total\\
\hline Males & ? & 26 & ?\\
\hline Females & 58 & ? & 96\\
\hline Total & 136 & 64 & 200\\
\hline
\end{tabular}$
$ \textbf{(A)}\ 39 \qquad
\textbf{(B)}\ 48 \qquad
\textbf{(C)}\ 52 \qquad
\textbf{(D)}\ 55 \qquad
\textbf{(E)}\ 75$
2022 Bulgarian Autumn Math Competition, Problem 8.2
It's given a right-angled triangle $ABC (\angle{C}=90^{\circ})$ and area $S$. Let $S_1$ be the area of the circle with diameter $AB$ and $k=\frac{S_1}{S}$\\
a) Compute the angles of $ABC$, if $k=2\pi$
b) Prove it is not possible for k to be $3$
2011 District Olympiad, 1
a) Prove that $\{x+y\}-\{y\}$ can only be equal to $\{x\}$ or $\{x\}-1$ for any $x,y\in \mathbb{R}$.
b) Let $\alpha\in \mathbb{R}\backslash \mathbb{Q}$. Denote $a_n=\{n\alpha\}$ for all $n\in \mathbb{N}^*$ and define the sequence $(x_n)_{n\ge 1}$ by
\[x_n=(a_2-a_1)(a_3-a_2)\cdot \ldots \cdot (a_{n+1}-a_n)\]
Prove that the sequence $(x_n)_{n\ge 1}$ is convergent and find it's limit.
2008 All-Russian Olympiad, 7
In convex quadrilateral $ ABCD$, the rays $ BA,CD$ meet at $ P$, and the rays $ BC,AD$ meet at $ Q$. $ H$ is the projection of $ D$ on $ PQ$. Prove that there is a circle inscribed in $ ABCD$ if and only if the incircles of triangles $ ADP,CDQ$ are visible from $ H$ under the same angle.
2024 Princeton University Math Competition, A2 / B4
Let $f$ and $g$ be two polynomials such that $f(g(x))=g(f(x))$. If $g(x)$ is linear but not identically equal to $x$, and $f(x)=x^3+60x^2+1000x+c$ for some $c$, find the value of $c$.
[i]Clarification[/i]: $g$ is not constant.
Estonia Open Junior - geometry, 1996.1.4
In a trapezoid, the two non parallel sides and a base have length $1$, while the other base and both the diagonals have length $a$. Find the value of $a$.
2016 NIMO Problems, 6
Consider a sequence $a_0$, $a_1$, $\ldots$, $a_9$ of distinct positive integers such that $a_0=1$, $a_i < 512$ for all $i$, and for every $1 \le k \le 9$ there exists $0 \le m \le k-1$ such that \[(a_k-2a_m)(a_k-2a_m-1) = 0.\] Let $N$ be the number of these sequences. Find the remainder when $N$ is divided by $1000$.
[i]Based on a proposal by Gyumin Roh[/i]
2019 Stars of Mathematics, 2
If $n\geqslant 3$ is an integer and $a_1,a_2,\dotsc ,a_n$ are non-zero integers such that
$$a_1a_2\cdots a_n\left( \frac{1}{a_1^2}+\frac{1}{a_2^2} +\cdots +\frac{1}{a_n^2}\right)$$is an integer, does it follow that the product $a_1a_2\cdots a_n$ is divisible by each $a_i^2$?
2014 Belarus Team Selection Test, 4
Thirty rays with the origin at the same point are constructed on a plane. Consider all angles between any two of these rays. Let $N$ be the number of acute angles among these angles. Find the smallest possible value of $N$.
(E. Barabanov)
2009 VJIMC, Problem 3
Let $k$ and $n$ be positive integers such that $k\le n-1$. Let $S:=\{1,2,\ldots,n\}$ and let $A_1,A_2,\ldots,A_k$ be nonempty subsets of $S$. Prove that it is possible to color some elements of $S$ using two colors, red and blue, such that the following conditions are satisfied:
(i) Each element of $S$ is either left uncolored or is colored red or blue.
(ii) At least one element of $S$ is colored.
(iii) Each set $A_i~(i=1,2,\ldots,k)$ is either completely uncolored or it contains at least one red and at least one blue element.
2019 Brazil Team Selection Test, 5
Determine all the functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[ f(x^2 + f(y)) = f(f(x)) + f(y^2) + 2f(xy) \]
for all real numbers $x$ and $y$.
1992 All Soviet Union Mathematical Olympiad, 570
Define the sequence $a_1 = 1, a_2, a_3, ...$ by $$a_{n+1} = a_1^2 + a_2 ^2 + a_3^2 + ... + a_n^2 + n$$ Show that $1$ is the only square in the sequence.
Russian TST 2017, P2
A regular hexagon is divided by straight lines parallel to its sides into $6n^2$ equilateral triangles. On them, there are $2n$ rooks, no two of which attack each other (a rook attacks in directions parallel to the sides of the hexagon). Prove that if we color the triangles black and white such that no two adjacent triangles have the same color, there will be as many rooks on the black triangles as on the white ones.
1960 AMC 12/AHSME, 33
You are given a sequence of $58$ terms; each term has the form $P+n$ where $P$ stands for the product $2 \times 3 \times 5 \times... \times 61$ of all prime numbers less than or equal to $61$, and $n$ takes, successively, the values $2, 3, 4, ...., 59$. let $N$ be the number of primes appearing in this sequence. Then $N$ is:
$ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 17\qquad\textbf{(D)}\ 57\qquad\textbf{(E)}\ 58 $
2004 May Olympiad, 1
Javier multiplies four digits, not necessarily different, and obtains a number ending in $7$. Determine how much the sum of the four digits that Javier multiplies can be worth. Give all the possibilities.
2017 IFYM, Sozopol, 7
Find all pairs $(x,y)$, $x,y\in \mathbb{N}$ for which
$gcd(n(x!-xy-x-y+2)+2,n(x!-xy-x-y+3)+3)>1$
for $\forall$ $n\in \mathbb{N}$.
2006 MOP Homework, 3
Let $ABC$ be a triangle with $AB\neq AC$, and let $A_{1}B_{1}C_{1}$ be the image of triangle $ABC$ through a rotation $R$ centered at $C$.
Let $M,E , F$ be the midpoints of the segments $BA_{1}, AC, BC_{1}$ respectively
Given that $EM = FM$, compute $\angle EMF$.
2019 SAFEST Olympiad, 1
Let $ABC$ be an isosceles triangle with $AB = AC$. Let $AD$ be the diameter of the circumcircle of $ABC$ and let $P$ be a point on the smaller arc $BD$. The line $DP$ intersects the rays $AB$ and $AC$ at points $M$ and $N$, respectively. The line $AD$ intersects the lines $BP$ and $CP$ at points $Q$ and $R$, respectively. Prove that the midpoint of $MN$ lies on the circumcircle of $PQR$