Found problems: 85335
1989 Iran MO (2nd round), 2
Let $n$ be a positive integer. Prove that the polynomial
\[P(x)= \frac{x^n}{n!}+\frac{x^{n-1}}{(n-1)!}+...+x+1 \]
Does not have any rational root.
2016 NIMO Problems, 7
Given two positive integers $m$ and $n$, we say that $m\mid\mid n$ if $m\mid n$ and $\gcd(m,\, n/m)=1$. Compute the smallest integer greater than \[\sum_{d\mid 2016}\sum_{m\mid\mid d}\frac{1}{m}.\]
[i]Proposed by Michael Ren[/i]
2015 ASDAN Math Tournament, 3
You have a circular necklace with $10$ beads on it, all of which are initially unpainted. You randomly select $5$ of these beads. For each selected bead, you paint that selected bead and the two beads immediately next to it (this means we may paint a bead multiple times). Once you have finished painting, what is the probability that every bead is painted?
2006 Hanoi Open Mathematics Competitions, 9
Let $x,y,z$ be real numbers such that $x^2+y^2+z^2=1$.Find the largest posible value of
$$|x^3+y^3+z^3-xyz|$$
1997 Bundeswettbewerb Mathematik, 2
Find a prime number $p$ such that $\frac{p+1}{2}$ and $\frac{p^2+1}{2}$ are perfect square
2016 Brazil Undergrad MO, 1
Let \((a_n)_{n \geq 1}\) s sequence of reals such that \(\sum_{n \geq 1}{\frac{a_n}{n}}\) converges. Show that
\(\lim_{n \rightarrow \infty}{\frac{1}{n} \cdot \sum_{k=1}^{n}{a_k}} = 0\)
1998 Cono Sur Olympiad, 2
Let $H$ be the orthocenter of the triangle $ABC$, $M$ is the midpoint of the segment $BC$. Let $X$ be the point of the intersection of the line $HM$ with arc $BC$(without $A$) of the circumcircle of $ABC$, let $Y$ be the point of intersection of the line $BH$ with the circle, show that $XY = BC$.
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 $P_S$.