Found problems: 6530
2008 Macedonia National Olympiad, 2
Positive numbers $ a$, $ b$, $ c$ are such that $ \left(a \plus{} b\right)\left(b \plus{} c\right)\left(c \plus{} a\right) \equal{} 8$. Prove the inequality
\[ \frac {a \plus{} b \plus{} c}{3}\ge\sqrt [27]{\frac {a^3 \plus{} b^3 \plus{} c^3}{3}}
\]
2003 AMC 10, 7
How many non-congruent triangles with perimeter $ 7$ have integer side lengths?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 5$
2006 Croatia Team Selection Test, 4
Find all natural solutions of $3^{x}= 2^{x}y+1.$
2021 SYMO, Q2
Let $n\geq 3$ be a fixed positive integer. Determine the minimum possible value of \[\sum_{1\leq i<j<k\leq n} \max(x_ix_j + x_k, x_jx_k + x_i, x_kx_i + x_j)^2\]over all non-negative reals $x_1,x_2,\dots,x_n$ satisfying $x_1+x_2+\dots+x_n=n$.
2007 Indonesia TST, 3
Let $a, b, c$ be positive reals such that $a + b + c = 1$ and $P(x) = 3^{2005}x^{2007 }- 3^{2005}x^{2006} - x^2$.
Prove that $P(a) + P(b) + P(c) \le -1$.
2012 Today's Calculation Of Integral, 840
Let $x,\ y$ be real numbers. For a function $f(t)=x\sin t+y\cos t$, draw the domain of the points $(x,\ y)$ for which the following inequality holds.
\[\left|\int_{-\pi}^{\pi} f(t)\cos t\ dt\right|\leq \int_{-\pi}^{\pi} \{f(t)\}^2dt.\]
2012 China National Olympiad, 3
Find the smallest positive integer $k$ such that, for any subset $A$ of $S=\{1,2,\ldots,2012\}$ with $|A|=k$, there exist three elements $x,y,z$ in $A$ such that $x=a+b$, $y=b+c$, $z=c+a$, where $a,b,c$ are in $S$ and are distinct integers.
[i]Proposed by Huawei Zhu[/i]
2022 Taiwan TST Round 3, A
Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$
1996 Vietnam Team Selection Test, 3
Find the minimum value of the expression:
\[f(a,b,c)= (a+b)^4+(b+c)^4+(c+a)^4 - \frac{4}{7} \cdot (a^4+b^4+c^4).\]
2014 Middle European Mathematical Olympiad, 2
Determine all functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[ xf(xy) + xyf(x) \ge f(x^2)f(y) + x^2y \]
holds for all $x,y \in \mathbb{R}$.
2009 AMC 12/AHSME, 10
In quadrilateral $ ABCD$, $ AB \equal{} 5$, $ BC \equal{} 17$, $ CD \equal{} 5$, $ DA \equal{} 9$, and $ BD$ is an integer. What is $ BD$?
[asy]unitsize(4mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=4;
pair C=(0,0), B=(17,0);
pair D=intersectionpoints(Circle(C,5),Circle(B,13))[0];
pair A=intersectionpoints(Circle(D,9),Circle(B,5))[0];
pair[] dotted={A,B,C,D};
draw(D--A--B--C--D--B);
dot(dotted);
label("$D$",D,NW);
label("$C$",C,W);
label("$B$",B,E);
label("$A$",A,NE);[/asy]$ \textbf{(A)}\ 11 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15$
2010 Bosnia And Herzegovina - Regional Olympiad, 1
Prove the inequality $$ \frac{y^2-x^2}{2x^2+1}+\frac{z^2-y^2}{2y^2+1}+\frac{x^2-z^2}{2z^2+1} \geq 0$$ where $x$, $y$ and $z$ are real numbers
2014 Junior Balkan Team Selection Tests - Romania, 3
Consider two integers $n \ge m \ge 4$ and $A = \{a_1, a_2, ..., a_m\}$ a subset of the set $\{1, 2, ..., n\}$ such that:
[i]for all $a, b \in A, a \ne b$, if $a + b \le n$, then $a + b \in A$.[/i]
Prove that $\frac{a_1 + a_2 + ... + a_m}{m} \ge \frac{n + 1}{2}$ .
2004 Olympic Revenge, 2
If $a,b,c,x$ are positive reals, show that
$$\frac{a^{x+2}+1}{a^xbc+1}+\frac{b^{x+2}+1}{b^xac+1}+\frac{c^{x+2}+1}{c^xab+1}\geq 3$$
2024 IFYM, Sozopol, 2
For arbitrary real numbers \( x_1,x_2,\ldots,x_n \), prove that
\[
\left(\max_{1\leq i \leq n}x_i \right)^2 + 4\sum_{i=1}^{n-1}\left(\max_{1\leq j \leq i}x_j\right)\left(x_{i+1}-x_i\right) \leq 4x_n^2.
\]
2015 China Girls Math Olympiad, 7
Let $x_1,x_2,\cdots,x_n \in(0,1)$ , $n\geq2$. Prove that$$\frac{\sqrt{1-x_1}}{x_1}+\frac{\sqrt{1-x_2}}{x_2}+\cdots+\frac{\sqrt{1-x_n}}{x_n}<\frac{\sqrt{n-1}}{x_1 x_2 \cdots x_n}.$$
1989 National High School Mathematics League, 13
$a_1,a_2,\cdots,a_n$ are positive numbers, satisfying that $a_1a_2\cdots a_n=1$.
Prove that $(2+a_1)(2+a_2)\cdots(2+a_n)\geq3^n$
2012 Mathcenter Contest + Longlist, 6
Let $a,b,c>0$ and $abc=1$. Prove that $$\frac{a}{b^2(c+a)(a+b)}+\frac{b}{c^2(a+b)(b+c)}+\frac{c}{a^2(c+a)(a+b)}\ge \frac{3}{4}.$$
[i](Zhuge Liang)[/i]
2017 European Mathematical Cup, 4
The real numbers $x,y,z$ satisfy $x^2+y^2+z^2=3.$ Prove that the inequality
$x^3-(y^2+yz+z^2)x+yz(y+z)\le 3\sqrt{3}.$
and find all triples $(x,y,z)$ for which equality holds.
2005 Kyiv Mathematical Festival, 1
Prove that there exists a positive integer $ n$ such that for every $ x\ge0$ the inequality $ (x\minus{}1)(x^{2005}\minus{}2005x^{n\plus{}1}\plus{}2005x^n\minus{}1)\ge0$ holds.
1969 Bulgaria National Olympiad, Problem 2
Prove that
$$S_n=\frac1{1^2}+\frac1{2^2}+\ldots+\frac1{n^2}<2$$for every $n\in\mathbb N$.
2009 Miklós Schweitzer, 9
Let $ P\subseteq \mathbb{R}^m$ be a non-empty compact convex set and $ f: P\rightarrow \mathbb{R}_{ \plus{} }$ be a concave function. Prove, that for every $ \xi\in \mathbb{R}^m$
\[ \int_{P}\langle \xi,x \rangle f(x)dx\leq \left[\frac {m \plus{} 1}{m \plus{} 2}\sup_{x\in P}{\langle\xi,x\rangle} \plus{} \frac {1}{m \plus{} 2}\inf_{x\in P}{\langle\xi,x\rangle}\right] \cdot\int_{P}f(x)dx.\]
2008 Tuymaada Olympiad, 8
A convex hexagon is given. Let $ s$ be the sum of the lengths of the three segments connecting the midpoints of its opposite sides. Prove that there is a point in the hexagon such that the sum of its distances to the lines containing the sides of the hexagon does not exceed $ s.$
[i]Author: N. Sedrakyan[/i]
2013 Miklós Schweitzer, 12
There are ${n}$ tokens in a pack. Some of them (at least one, but not all) are white and the rest are black. All tokens are extracted randomly from the pack, one by one, without putting them back. Let ${X_i}$ be the ratio of white tokens in the pack before the ${i^{\text{th}}}$ extraction and let
\[ \displaystyle T =\max \{ |X_i-X_j| : 1 \leq i \leq j \leq n\}.\]
Prove that ${\Bbb{E}(T) \leq H(\Bbb{E}(X_1))},$ where ${H(x)=-x\ln x -(1-x)\ln(1-x)}.$
[i]Proposed by Tamás Móri[/i]
1970 IMO Longlists, 31
Prove that for any triangle with sides $a, b, c$ and area $P$ the following inequality holds:
\[P \leq \frac{\sqrt 3}{4} (abc)^{2/3}.\]
Find all triangles for which equality holds.