Found problems: 85335
2005 India IMO Training Camp, 2
Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying
\[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\]
for any two positive integers $ m$ and $ n$.
[i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers:
$ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$.
By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$).
[i]Proposed by Mohsen Jamali, Iran[/i]
2018 Belarusian National Olympiad, 10.3
For a fixed integer $n\geqslant2$ consider the sequence $a_k=\text{lcm}(k,k+1,\ldots,k+(n-1))$. Find all $n$ for which the sequence $a_k$ increases starting from some number.
2015 Taiwan TST Round 3, 1
For any positive integer $n$, let $a_n=\sum_{k=1}^{\infty}[\frac{n+2^{k-1}}{2^k}]$, where $[x]$ is the largest integer that is equal or less than $x$. Determine the value of $a_{2015}$.
2023 Auckland Mathematical Olympiad, 9
Quadrillateral $ABCD$ is inscribed in a circle with centre $O$. Diagonals $AC$ and $BD$ are perpendicular. Prove that the distance from the centre $O$ to $AD$ is half the length of $BC$.
MathLinks Contest 7th, 6.1
Let $ \{x_n\}_{n\geq 1}$ be a sequences, given by $ x_1 \equal{} 1$, $ x_2 \equal{} 2$ and
\[ x_{n \plus{} 2} \equal{} \frac { x_{n \plus{} 1}^2 \plus{} 3 }{x_n} .
\]
Prove that $ x_{2008}$ is the sum of two perfect squares.
VMEO II 2005, 11
Given $P$ a real polynomial with degree greater than $ 1$.
Find all pairs $(f,Q)$ with function $f : R \to R$ and the real polynomial $Q$ satisfying the following two conditions:
i) for all $x, y \in R$, we have $f(P(x) + f(y)) = y + Q(f(x))$.
ii) there exists $x_0 \in R$ such that $f(P(x_0)) = Q(f(x_0))$.
2014 South East Mathematical Olympiad, 6
Let $a,b$ and $c$ be integers and $r$ a real number such that $ar^2+br+c=0$ with $ac\not =0$.Prove that $\sqrt{r^2+c^2}$ is an irrational number
2019 Tuymaada Olympiad, 6
Prove that the expression
$$ (1^4+1^2+1)(2^4+2^2+1)\dots(n^4+n^2+1)$$
is not square for all $n \in \mathbb{N}$
2010 Princeton University Math Competition, 6
In the following diagram, a semicircle is folded along a chord $AN$ and intersects its diameter $MN$ at $B$. Given that $MB : BN = 2 : 3$ and $MN = 10$. If $AN = x$, find $x^2$.
[asy]
size(120); defaultpen(linewidth(0.7)+fontsize(10));
pair D2(pair P) {
dot(P,linewidth(3)); return P;
}
real r = sqrt(80)/5;
pair M=(-1,0), N=(1,0), A=intersectionpoints(arc((M+N)/2, 1, 0, 180),circle(N,r))[0], C=intersectionpoints(circle(A,1),circle(N,1))[0], B=intersectionpoints(circle(C,1),M--N)[0];
draw(arc((M+N)/2, 1, 0, 180)--cycle); draw(A--N); draw(arc(C,1,180,180+2*aSin(r/2)));
label("$A$",D2(A),NW);
label("$B$",D2(B),SW);
label("$M$",D2(M),S);
label("$N$",D2(N),SE);
[/asy]
2018 IFYM, Sozopol, 4
The real numbers $a$, $b$, $c$ are such that $a+b+c+ab+bc+ca+abc \geq 7$. Prove that
$\sqrt{a^2+b^2+2}+\sqrt{b^2+c^2+2}+\sqrt{c^2+a^2+2} \geq 6$
2023 Stanford Mathematics Tournament, 6
We say that an integer $x\in\{1,\dots,102\}$ is $\textit{square-ish}$ if there exists some integer $n$ such that $x\equiv n^2+n\pmod{103}$. Compute the product of all $\textit{square-ish}$ integers modulo $103$.
1975 Swedish Mathematical Competition, 6
$f(x)$ is defined for $0 \leq x \leq 1$ and has a continuous derivative satisfying $|f'(x)| \leq C|f(x)|$ for some positive constant $C$. Show that if $f(0) = 0$, then $f(x)=0$ for the entire interval.
2014 NIMO Problems, 8
For positive integers $a$, $b$, and $c$, define \[ f(a,b,c)=\frac{abc}{\text{gcd}(a,b,c)\cdot\text{lcm}(a,b,c)}. \] We say that a positive integer $n$ is $f@$ if there exist pairwise distinct positive integers $x,y,z\leq60$ that satisfy $f(x,y,z)=n$. How many $f@$ integers are there?
[i]Proposed by Michael Ren[/i]
2008 South East Mathematical Olympiad, 3
In $\triangle ABC$, side $BC>AB$. Point $D$ lies on side $AC$ such that $\angle ABD=\angle CBD$. Points $Q,P$ lie on line $BD$ such that $AQ\bot BD$ and $CP\bot BD$. $M,E$ are the midpoints of side $AC$ and $BC$ respectively. Circle $O$ is the circumcircle of $\triangle PQM$ intersecting side $AC$ at $H$. Prove that $O,H,E,M$ lie on a circle.
2008 AIME Problems, 15
Find the largest integer $ n$ satisfying the following conditions:
(i) $ n^2$ can be expressed as the difference of two consecutive cubes;
(ii) $ 2n\plus{}79$ is a perfect square.
1990 IMO Longlists, 56
For positive integers $n, p$ with $n \geq p$, define real number $K_{n, p}$ as follows:
$K_{n, 0} = \frac{1}{n+1}$ and $K_{n, p} = K_{n-1, p-1} -K_{n, p-1}$ for $1 \leq p \leq n.$
(i) Define $S_n = \sum_{p=0}^n K_{n,p} , \ n = 0, 1, 2, \ldots$ . Find $\lim_{n \to \infty} S_n.$
(ii) Find $T_n = \sum_{p=0}^n (-1)^p K_{n,p} , \ n = 0, 1, 2, \ldots$.
2004 Greece Junior Math Olympiad, 4
Determine the rational number $\frac{a}{b}$, where $a,b$ are positive integers, with minimal denominator, which is such that
$ \frac{52}{303} < \frac{a}{b}< \frac{16}{91}$
2024 Lusophon Mathematical Olympiad, 2
For each set of five integers $S= \{a_1, a_2, a_3, a_4, a_5\} $, let $P_S$ be the product of all differences between two of the elements, namely
$$P_S=(a_5-a_1)(a_4-a_1)(a_3-a_1)(a_2-a_1)(a_5-a_2)(a_4-a_2)(a_3-a_2)(a_5-a_3)(a_4-a_3)(a_5-a_4)$$
Determine the greatest integer $n$ such that given any set $S$ of five integers, $n$ divides $P_S$.
2008 Abels Math Contest (Norwegian MO) Final, 4b
A point $D$ lies on the side $BC$ , and a point $E$ on the side $AC$ , of the triangle $ABC$ , and $BD$ and $AE$ have the same length. The line through the centres of the circumscribed circles of the triangles $ADC$ and $BEC$ crosses $AC$ in $K$ and $BC$ in $L$. Show that $KC$ and $LC$ have the same length.
2001 Tournament Of Towns, 5
In a chess tournament, every participant played with each other exactly once, receiving $1$ point for a win, $1/2$ for a draw and $0$ for a loss.
[list][b](a)[/b] Is it possible that for every player $P$, the sum of points of the players who were beaten by P is greater than the sum of points of the players who beat $P$?
[b](b)[/b] Is it possible that for every player $P$, the first sum is less than the second one?[/list]
1992 Polish MO Finals, 3
Show that $(k^3)!$ is divisible by $(k!)^{k^2+k+1}$.
2002 AMC 10, 1
The ratio $ \dfrac{10^{2000}\plus{}10^{2002}}{10^{2001}\plus{}10^{2001}}$ is closest to which of the following numbers?
$ \text{(A)}\ 0.1\qquad
\text{(B)}\ 0.2\qquad
\text{(C)}\ 1\qquad
\text{(D)}\ 5\qquad
\text{(E)}\ 10$
2008 Princeton University Math Competition, B2
What is $3(2 \log_4 (2(2 \log_3 9)))$ ?
2015 Taiwan TST Round 3, 2
Let $\Omega$ and $O$ be the circumcircle and the circumcentre of an acute-angled triangle $ABC$ with $AB > BC$. The angle bisector of $\angle ABC$ intersects $\Omega$ at $M \ne B$. Let $\Gamma$ be the circle with diameter $BM$. The angle bisectors of $\angle AOB$ and $\angle BOC$ intersect $\Gamma$ at points $P$ and $Q,$ respectively. The point $R$ is chosen on the line $P Q$ so that $BR = MR$. Prove that $BR\parallel AC$.
(Here we always assume that an angle bisector is a ray.)
[i]Proposed by Sergey Berlov, Russia[/i]
1988 Tournament Of Towns, (169) 2
We are given triangle $ABC$. Two lines, symmetric with $AC$, relative to lines $AB$ and $BC$ are drawn, and meet at $K$ . Prove that the line $BK$ passes through the centre of the circumscribed circle of triangle $ABC$.
(V.Y. Protasov)