Found problems: 6530
2023 USA TSTST, 2
Let $n\ge m\ge 1$ be integers. Prove that
\[\sum_{k=m}^n \left (\frac 1{k^2}+\frac 1{k^3}\right) \ge m\cdot \left(\sum_{k=m}^n \frac 1{k^2}\right)^2.\]
[i]Raymond Feng and Luke Robitaille[/i]
2000 Iran MO (3rd Round), 2
Suppose that $a, b, c$ are real numbers such that for all positive numbers
$x_1,x_2,\dots,x_n$ we have
$(\frac{1}{n}\sum_{i=1}^nx_i)^a(\frac{1}{n}\sum_{i=1}^nx_i^2)^b(\frac{1}{n}\sum_{i=1}^nx_i^3)^c\ge 1$
Prove that vector $(a, b, c)$ is a nonnegative linear combination of vectors
$(-2,1,0)$ and $(-1,2,-1)$.
2016 Azerbaijan Junior Mathematical Olympiad, 6
For all reals $x,y,z$ prove that $$\sqrt {x^2+\frac {1}{y^2}}+ \sqrt {y^2+\frac {1}{z^2}}+ \sqrt {z^2+\frac {1}{x^2}}\geq 3\sqrt {2}. $$
2021 Iran Team Selection Test, 3
Prove there exist two relatively prime polynomials $P(x),Q(x)$ having integer coefficients and a real number $u>0$ such that if for positive integers $a,b,c,d$ we have:
$$|\frac{a}{c}-1|^{2021} \le \frac{u}{|d||c|^{1010}}$$
$$| (\frac{a}{c})^{2020}-\frac{b}{d}| \le \frac{u}{|d||c|^{1010}}$$
Then we have :
$$bP(\frac{a}{c})=dQ(\frac{a}{c})$$
(Two polynomials are relatively prime if they don't have a common root)
Proposed by [i]Navid Safaii[/i] and [i]Alireza Haghi[/i]
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]
2015 Sharygin Geometry Olympiad, 1
Circles $\alpha$ and $\beta$ pass through point $C$. The tangent to $\alpha$ at this point meets $\beta$ at point $B$, and the tangent to $\beta$ at $C$ meets $\alpha$ at point $A$ so that $A$ and $B$ are distinct from $C$ and angle $ACB$ is obtuse. Line $AB$ meets $\alpha$ and $\beta$ for the second time at points $N$ and $M$ respectively. Prove that $2MN < AB$.
(D. Mukhin)
2020 Lusophon Mathematical Olympiad, 3
Let $ABC$ be a triangle and on the sides we draw, externally, the squares $BADE, CBFG$ and $ACHI$. Determine the greatest positive real constant $k$ such that, for any triangle $\triangle ABC$, the following inequality is true:
$[DEFGHI]\geq k\cdot [ABC]$
Note: $[X]$ denotes the area of polygon $X$.
2011 Saudi Arabia Pre-TST, 4.3
Let $x_1,x_2,...,x_n$ be positive real numbers for which $$\frac{1}{1+x_1}+\frac{1}{1+x_2}+...+\frac{1}{1+x_n}=1$$
Prove that $x_1x_2...x_n \ge (n -1)^n$.
2013 NZMOC Camp Selection Problems, 10
Find the largest possible real number $C$ such that for all pairs $(x, y)$ of real numbers with $x \ne y$ and $xy = 2$, $$\frac{((x + y)^2- 6))(x-y)^2 + 8))}{(x-y)^2} \ge C.$$ Also determine for which pairs $(x, y)$ equality holds.
2008 Singapore Team Selection Test, 2
Let $ x_1, x_2,\ldots , x_n$ be positive real numbers such that $ x_1x_2\cdots x_n \equal{} 1$. Prove that
\[\sum_{i \equal{} 1}^n \frac {1}{n \minus{} 1 \plus{} x_i}\le 1.\]
2009 Junior Balkan Team Selection Tests - Romania, 4
Let $a,b,c > 0$ be real numbers with the sum equal to $3$. Show that:
$$\frac{a+3}{3a+bc}+\frac{b+3}{3b+ca}+\frac{c+3}{3c+ab} \ge 3$$
2012 Ukraine Team Selection Test, 1
Let $a, b, c$ be positive reals. Prove that $\sqrt{2a^2+bc}+\sqrt{2b^2+ac}+\sqrt{2c^2+ab}\ge 3 \sqrt{ab+bc+ca}$
2009 AMC 12/AHSME, 24
The [i]tower function of twos[/i] is defined recursively as follows: $ T(1) \equal{} 2$ and $ T(n \plus{} 1) \equal{} 2^{T(n)}$ for $ n\ge1$. Let $ A \equal{} (T(2009))^{T(2009)}$ and $ B \equal{} (T(2009))^A$. What is the largest integer $ k$ such that
\[ \underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ times}}
\]is defined?
$ \textbf{(A)}\ 2009\qquad \textbf{(B)}\ 2010\qquad \textbf{(C)}\ 2011\qquad \textbf{(D)}\ 2012\qquad \textbf{(E)}\ 2013$
2007 Indonesia TST, 3
For each real number $ x$< let $ \lfloor x \rfloor$ be the integer satisfying $ \lfloor x \rfloor \le x < \lfloor x \rfloor \plus{}1$ and let $ \{x\}\equal{}x\minus{}\lfloor x \rfloor$. Let $ c$ be a real number such that \[ \{n\sqrt{3}\}>\dfrac{c}{n\sqrt{3}}\] for all positive integers $ n$. Prove that $ c \le 1$.
2007 Bulgaria Team Selection Test, 4
Let $G$ is a graph and $x$ is a vertex of $G$. Define the transformation $\varphi_{x}$ over $G$ as deleting all incident edges with respect of $x$ and drawing the edges $xy$ such that $y\in G$ and $y$ is not connected with $x$ with edge in the beginning of the transformation. A graph $H$ is called $G-$[i]attainable[/i] if there exists a sequece of such transformations which transforms $G$ in $H.$ Let $n\in\mathbb{N}$ and $4|n.$ Prove that for each graph $G$ with $4n$ vertices and $n$ edges there exists $G-$[i]attainable[/i] graph with at least $9n^{2}/4$ triangles.
2012 Romania Team Selection Test, 1
Let $\Delta ABC$ be a triangle. The internal bisectors of angles $\angle CAB$ and $\angle ABC$ intersect segments $BC$, respectively $AC$ in $D$, respectively $E$. Prove that \[DE\leq (3-2\sqrt{2})(AB+BC+CA).\]
2002 Taiwan National Olympiad, 3
Suppose $x,y,,a,b,c,d,e,f$ are real numbers satifying
i)$\max{(a,0)}+\max{(b,0)}<x+ay+bz<1+\min{(a,0)}+\min{(b,0)}$, and
ii)$\max{(c,0)}+\max{(d,0)}<cx+y+dz<1+\min{(c,0)}+\min{(d,0)}$, and
iii)$\max{(e,0)}+\max{(f,0)}<ex+fy+z<1+\min{(e,0)}+\min{(f,0)}$.
Prove that $0<x,y,z<1$.
1983 IMO Shortlist, 6
Suppose that ${x_1, x_2, \dots , x_n}$ are positive integers for which $x_1 + x_2 + \cdots+ x_n = 2(n + 1)$. Show that there exists an integer $r$ with $0 \leq r \leq n - 1$ for which the following $n - 1$ inequalities hold:
\[x_{r+1} + \cdots + x_{r+i} \leq 2i+ 1, \qquad \qquad \forall i, 1 \leq i \leq n - r; \]
\[x_{r+1} + \cdots + x_n + x_1 + \cdots+ x_i \leq 2(n - r + i) + 1, \qquad \qquad \forall i, 1 \leq i \leq r - 1.\]
Prove that if all the inequalities are strict, then $r$ is unique and that otherwise there are exactly two such $r.$
2001 China Second Round Olympiad, 2
If nonnegative reals $x_1, x_2, \ldots, x_n$ satisfy
\[
\sum_{i=1}^n x_i^2 + 2\sum_{1 \leq k < j \leq n} \sqrt{\frac{k}{j}}x_kx_j = 1
\]
what are the minimum and maximum values of $\sum_{i=1}^n x_i$?
2009 ISI B.Math Entrance Exam, 9
Let $f(x)=ax^2+bx+c$ where $a,b,c$ are real numbers. Suppose $f(-1),f(0),f(1) \in [-1,1]$. Prove that $|f(x)|\le \frac{3}{2}$ for all $x \in [-1,1]$.
1977 IMO Longlists, 6
Let $x_1, x_2, \ldots , x_n \ (n \geq 1)$ be real numbers such that $0 \leq x_j \leq \pi, \ j = 1, 2,\ldots, n.$ Prove that if $\sum_{j=1}^n (\cos x_j +1) $ is an odd integer, then $\sum_{j=1}^n \sin x_j \geq 1.$
1996 Kurschak Competition, 1
Prove that in a trapezoid with perpendicular diagonals, the product of the legs is at least as much as the product of the bases.
2011 Laurențiu Duican, 4
For $a, b, c>0,$ and $k\geq1,$ prove that
\[\frac{a^{k+1}}{b^k+c^k}+\frac{b^{k+1}}{c^k+a^k}+\frac{c^{k+1}}{a^k+b^k}\geq\frac{3}{2}\sqrt{\frac{a^{k+1}+b^{k+1}+c^{k+1}}{{a^{k-1}+b^{k-1}+c^{k-1}}}}\]
Author: MIHALY BENCZE
2009 Indonesia TST, 2
Find the value of real parameter $ a$ such that $ 2$ is the smallest integer solution of \[ \frac{x\plus{}\log_2 (2^x\minus{}3a)}{1\plus{}\log_2 a} >2.\]
2018 Korea Winter Program Practice Test, 2
Let $\Delta ABC$ be a triangle and $P$ be a point in its interior. Prove that \[ \frac{[BPC]}{PA^2}+\frac{[CPA]}{PB^2}+\frac{[APB]}{PC^2} \ge \frac{[ABC]}{R^2} \]
where $R$ is the radius of the circumcircle of $\Delta ABC$, and $[XYZ]$ is the area of $\Delta XYZ$.