Found problems: 85335
2022 Bulgarian Spring Math Competition, Problem 12.2
Let $ABCDV$ be a regular quadrangular pyramid with $V$ as the apex. The plane $\lambda$ intersects the $VA$, $VB$, $VC$ and $VD$ at $M$, $N$, $P$, $Q$ respectively. Find $VQ : QD$, if $VM : MA = 2 : 1$, $VN : NB = 1 : 1$ and $VP : PC = 1 : 2$.
2014 Contests, 3
In triangle $ABC$, $I$ is the centre of the incircle. There is a circle tangent
to $AI$ at $I$ which passes through $B$. This circle intersects $AB$ once more
in $P$ and intersects $BC$ once more in $Q$. The line $QI$ intersects $AC$ in $R$.
Prove that $|AR|\cdot |BQ|=|P I|^2$
2014 Contests, 3
We have $4n + 5$ points on the plane, no three of them are collinear. The points are colored with two colors. Prove that from the points we can form $n$ empty triangles (they have no colored points in their interiors) with pairwise disjoint interiors, such that all points occurring as vertices of the $n$ triangles have the same color.
1957 AMC 12/AHSME, 6
An open box is constructed by starting with a rectangular sheet of metal $ 10$ in. by $ 14$ in. and cutting a square of side $ x$ inches from each corner. The resulting projections are folded up and the seams welded. The volume of the resulting box is:
$ \textbf{(A)}\ 140x \minus{} 48x^2 \plus{} 4x^3 \qquad \textbf{(B)}\ 140x \plus{} 48x^2 \plus{} 4x^3\qquad \\\textbf{(C)}\ 140x \plus{} 24x^2 \plus{} x^3\qquad \textbf{(D)}\ 140x \minus{} 24x^2 \plus{} x^3\qquad \textbf{(E)}\ \text{none of these}$
2020 AIME Problems, 7
A club consisting of $11$ men and $12$ women needs to choose a committee from among its members so that the number of women on the committee is one more than the number of men on the committee. The committee could have as few as $1$ member or as many as $23$ members. Let $N$ be the number of such committees that can be formed. Find the sum of the prime numbers that divide $N$.
2017 India IMO Training Camp, 1
Let $ABC$ be an acute angled triangle with incenter $I$. Line perpendicular to $BI$ at $I$ meets $BA$ and $BC$ at points $P$ and $Q$ respectively. Let $D, E$ be the incenters of $\triangle BIA$ and $\triangle BIC$ respectively. Suppose $D,P,Q,E$ lie on a circle. Prove that $AB=BC$.
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.