Found problems: 85335
1998 Turkey MO (2nd round), 2
If $0\le a\le b\le c$ real numbers, prove that $(a+3b)(b+4c)(c+2a)\ge 60abc$.
2017 Sharygin Geometry Olympiad, P5
A segment $AB$ is fixed on the plane. Consider all acute-angled triangles with side $AB$. Find the locus of
а) the vertices of their greatest angles,
b) their incenters.
2000 Harvard-MIT Mathematics Tournament, 29
What is the value of ${ \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+\cdots }}}}}} $?
2017 Bulgaria JBMO TST, 2
Let $k$ be the incircle of triangle $ABC$. It touches $AB=c, BC=a, AC=b$ at $C_1, A_1, B_1$, respectively. Suppose that $KC_1$ is a diameter of the incircle. Let $C_1A_1$ intersect $KB_1$ at $N$ and $C_1B_1$ intersect $KA_1$ at $M$. Find the length of $MN$.
2006 Iran Team Selection Test, 4
Let $n$ be a fixed natural number.
Find all $n$ tuples of natural pairwise distinct and coprime numbers like $a_1,a_2,\ldots,a_n$ such that for $1\leq i\leq n$ we have
\[ a_1+a_2+\ldots+a_n|a_1^i+a_2^i+\ldots+a_n^i \]
Kvant 2024, M2786
There are $100$ white points on a circle. Asya and Borya play the following game: they alternate, starting with Asya, coloring a white point in green or blue. Asya wants to obtain as much as possible pairs of adjacent points of distinct colors, while Borya wants these pairs to be as less as possible. What is the maximal number of such pairs Asya can guarantee to obtain, no matter how Borya plays.
2005 Korea - Final Round, 1
Find all natural numbers that can be expressed in a unique way as a sum of five or less perfect squares.
2023 LMT Fall, 7
How many $2$-digit factors does $555555$ have?
2012 Balkan MO Shortlist, A1
Prove that
\[\sum_{cyc}(x+y)\sqrt{(z+x)(z+y)} \geq 4(xy+yz+zx),\]
for all positive real numbers $x,y$ and $z$.
II Soros Olympiad 1995 - 96 (Russia), 10.2
Find the smallest value that the expression can take
$$|a-1|+|b-2|+c-3|+|3a+2b+c|$$
($a$, $b$ and $c$ are arbitrary numbers).
2016 China National Olympiad, 6
Let $G$ be a complete directed graph with $100$ vertices such that for any two vertices $x,y$ one can find a directed path from $x$ to $y$.
a) Show that for any such $G$, one can find a $m$ such that for any two vertices $x,y$ one can find a directed path of length $m$ from $x$ to $y$ (Vertices can be repeated in the path)
b) For any graph $G$ with the properties above, define $m(G)$ to be smallest possible $m$ as defined in part a). Find the minimim value of $m(G)$ over all such possible $G$'s.
2002 CentroAmerican, 4
Let $ ABC$ be a triangle, $ D$ be the midpoint of $ BC$, $ E$ be a point on segment $ AC$ such that $ BE\equal{}2AD$ and $ F$ is the intersection point of $ AD$ with $ BE$. If $ \angle DAC\equal{}60^{\circ}$, find the measure of the angle $ FEA$.
VMEO II 2005, 8
If a,b,c>0, prove that:
\[ \frac{1}{a\sqrt{(a+b)}}+\frac{1}{b\sqrt{(b+c)}}+\frac{1}{c\sqrt{(c+a)}} \geq \frac{3}{\sqrt{2abc}} \]
thank u for ur help :oops:
1985 Brazil National Olympiad, 1
$a, b, c, d$ are integers with $ad \ne bc$. Show that $1/((ax+b)(cx+d))$ can be written in the form $ r/(ax+b) + s/(cx+d)$. Find the sum $1/1\cdot 4 + 1/4\cdot 7 + 1/7\cdot 10 + ... + 1/2998 \cdot 3001$.
2010 Morocco TST, 2
Let $a$, $b$, $c$ be positive real numbers with $abc \leq a+b+c$. Show that \[ a^2 + b^2 + c^2 \geq \sqrt 3 abc. \]
[i]Cristinel Mortici, Romania[/i]
2009 Moldova National Olympiad, 10.4
Let the isosceles triangle $ABC$ with $| AB | = | AC |$. The point $M$ is the midpoint of the base $[BC]$, the point $N$ is the orthogonal projection of the point $M$ on the line $AC$, and the point $P$ is located on the segment $(MC)$ such that $| MP | = | P C | \sin^2 C$. Prove that the lines $AP$ and $BN$ are perpendicular.
2013 China National Olympiad, 2
Find all nonempty sets $S$ of integers such that $3m-2n \in S$ for all (not necessarily distinct) $m,n \in S$.
2009 Poland - Second Round, 3
Disjoint circles $ o_1, o_2$, with centers $ I_1, I_2$ respectively, are tangent to the line $ k$ at $ A_1, A_2$ respectively and they lie on the same side of this line. Point $ C$ lies on segment $ I_1I_2$ and $ \angle A_1CA_2 \equal{} 90^{\circ}$. Let $ B_1$ be the second intersection of $ A_1C$ with $ o_1$, and let $ B_2$ be the second intersection of $ A_2C$ with $ o_2$. Prove that $ B_1B_2$ is tangent to the circles $ o_1, o_2$.
1993 National High School Mathematics League, 7
Equation $(1-\text{i})x^2+(\lambda+\text{i})x+(1+\text{i}\lambda)=0(\lambda\in\mathbb{R})$ has two imaginary roots, then the range value of $\lambda$ is________.
2023 IMC, 8
Let $T$ be a tree with $n$ vertices; that is, a connected simple graph on $n$ vertices that contains no cycle. For every pair $u$, $v$ of vertices, let $d(u,v)$ denote the distance between $u$ and $v$, that is, the number of edges in the shortest path in $T$ that connects $u$ with $v$.
Consider the sums
\[W(T)=\sum_{\substack{\{u,v\}\subseteq V(T)\\ u\neq v}}d(u,v) \quad \text{and} \quad H(T)=\sum_{\substack{\{u,v\}\subseteq V(T)\\ u\neq v}}\frac{1}{d(u,v)}\]
Prove that
\[W(T)\cdot H(T)\geq \frac{(n-1)^3(n+2)}{4}.\]
2016 Math Prize for Girls Problems, 18
Let $T = \{ 1, 2, 3, \dots, 14, 15 \}$. Say that a subset $S$ of $T$ is [i]handy[/i] if the sum of all the elements of $S$ is a multiple of $5$. For example, the empty set is handy (because its sum is 0) and $T$ itself is handy (because its sum is 120). Compute the number of handy subsets of $T$.
1976 Swedish Mathematical Competition, 6
Show that there are only finitely many integral solutions to
\[
3^m - 1 = 2^n
\]
and find them.
2021 Tuymaada Olympiad, 1
Quadratic trinomials $F$ and $G$ satisfy
$F(F(x)) > F(G(x)) > G(G(x))$
for all real $x$. Prove that $F(x) > G(x)$ for all real $x$.
1989 Irish Math Olympiad, 1
A quadrilateral $ABCD$ is inscribed, as shown, in a square of area one unit. Prove that $$2\le |AB|^2+|BC|^2+|CD|^2+|DA|^2\le 4$$
[asy]
size(6cm);
draw((0,0)--(10,0));
draw((10,0)--(10,10));
draw((0,10)--(10,10));
draw((0,0)--(0,10));
dot((0,8.5)); dot((3.5,10)); dot((10,3.5)); dot((3.5,0));
label("$D$",(0,8.5),W);
label("$A$",(3.5,10),NE);
label("$B$",(10,3.5),E);
label("$C$",(3.5,0),S);
draw((0,8.5)--(3.5,10));
draw((3.5,10)--(10,3.5));
draw((10,3.5)--(3.5,0));
draw((3.5,0)--(0,8.5));
[/asy]
2012 Olympic Revenge, 4
Say that two sets of positive integers $S, T$ are $\emph{k-equivalent}$ if the sum of the $i$th powers of elements of $S$ equals the sum of the $i$th powers of elements of $T$, for each $i= 1, 2, \ldots, k$. Given $k$, prove that there are infinitely many numbers $N$ such that $\{1,2,\ldots,N^{k+1}\}$ can be divided into $N$ subsets, all of which are $k$-equivalent to each other.