Found problems: 85335
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.
2024 Sharygin Geometry Olympiad, 5
Points $A', B', C'$ are the reflections of vertices $A, B, C$ about the opposite sidelines of triangle $ABC$. Prove that the circles $AB'C', A'BC',$ and $A'B'C$ have a common point.
2024 Mathematical Talent Reward Programme, 4
Two circles (centres $d$ apart) have radii $15,95$. The external tangents to the circles cut at $60$ degrees, find $d$.
$$(A) 40$$
$$(B) 80$$
$$(C) 120$$
$$(D) 160$$
1996 All-Russian Olympiad, 8
Goodnik writes 10 numbers on the board, then Nogoodnik writes 10 more numbers, all 20 of the numbers being positive and distinct. Can Goodnik choose his 10 numbers so that no matter what Nogoodnik writes, he can form 10 quadratic trinomials of the form $x^2 +px+q$, whose coeficients $p$ and $q$ run through all of the numbers written, such that the real roots of these trinomials comprise exactly 11 values?
[i]I. Rubanov[/i]
2006 Germany Team Selection Test, 3
Let $n$ be a positive integer, and let $b_{1}$, $b_{2}$, ..., $b_{n}$ be $n$ positive reals. Set $a_{1}=\frac{b_{1}}{b_{1}+b_{2}+...+b_{n}}$ and $a_{k}=\frac{b_{1}+b_{2}+...+b_{k}}{b_{1}+b_{2}+...+b_{k-1}}$ for every $k>1$. Prove the inequality
$a_{1}+a_{2}+...+a_{n}\leq\frac{1}{a_{1}}+\frac{1}{a_{2}}+...+\frac{1}{a_{n}}$.
2007 APMO, 4
Let $x; y$ and $z$ be positive real numbers such that $\sqrt{x}+\sqrt{y}+\sqrt{z}= 1$. Prove that $\frac{x^{2}+yz}{\sqrt{2x^{2}(y+z)}}+\frac{y^{2}+zx}{\sqrt{2y^{2}(z+x)}}+\frac{z^{2}+xy}{\sqrt{2z^{2}(x+y)}}\geq 1.$
2012 Turkey Junior National Olympiad, 2
In a convex quadrilateral $ABCD$, the diagonals are perpendicular to each other and they intersect at $E$. Let $P$ be a point on the side $AD$ which is different from $A$ such that $PE=EC.$ The circumcircle of triangle $BCD$ intersects the side $AD$ at $Q$ where $Q$ is also different from $A$. The circle, passing through $A$ and tangent to line $EP$ at $P$, intersects the line segment $AC$ at $R$. If the points $B, R, Q$ are concurrent then show that $\angle BCD=90^{\circ}$.
2005 All-Russian Olympiad, 4
Integers $x>2,\,y>1,\,z>0$ satisfy an equation $x^y+1=z^2$. Let $p$ be a number of different prime divisors of $x$, $q$ be a number of different prime divisors of $y$. Prove that $p\geq q+2$.
1967 IMO, 2
Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$