Olympiad problems

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Found problems: 426

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1989 USAMO, 3

Tags: algebra , polynomial , inequalities unsolved , inequalities

Let $P(z)= z^n + c_1 z^{n-1} + c_2 z^{n-2} + \cdots + c_n$ be a polynomial in the complex variable $z$, with real coefficients $c_k$. Suppose that $|P(i)| < 1$. Prove that there exist real numbers $a$ and $b$ such that $P(a + bi) = 0$ and $(a^2 + b^2 + 1)^2 < 4 b^2 + 1$.

2010 Tournament Of Towns, 3

Tags: invariant , inequalities unsolved

Each of $999$ numbers placed in a circular way is either $1$ or $-1$. (Both values appear). Consider the total sum of the products of every $10$ consecutive numbers. $(a)$ Find the minimal possible value of this sum. $(b)$ Find the maximal possible value of this sum.

2012 Kazakhstan National Olympiad, 3

Tags: inequalities , inequalities unsolved , Kazakhstan , Kanat Satylkhanov

Let $ a,b,c,d>0$ for which the following conditions:: $a)$ $(a-c)(b-d)=-4$ $b)$ $\frac{a+c}{2}\geq\frac{a^{2}+b^{2}+c^{2}+d^{2}}{a+b+c+d}$ Find the minimum of expression $a+c$

2007 Hong Kong TST, 3

Tags: inequalities , trigonometry , inequalities unsolved

[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url] Problem 3 Let $A$, $B$ and $C$ be real numbers such that (i) $\sin A \cos B+|\cos A \sin B|=\sin A |\cos A|+|\sin B|\cos B$, (ii) $\tan C$ and $\cot C$ are defined. Find the minimum value of $(\tan C-\sin A)^{2}+(\cot C-\cos B)^{2}$.

2007 Olympic Revenge, 2

Tags: inequalities , inequalities unsolved

Let $a, b, c \in \mathbb{R}$ with $abc = 1$. Prove that \[a^{2}+b^{2}+c^{2}+{1\over a^{2}}+{1\over b^{2}}+{1\over c^{2}}+2\left(a+b+c+{1\over a}+{1\over b}+{1\over c}\right) \geq 6+2\left({b\over a}+{c\over b}+{a\over c}+{c\over a}+{c\over b}+{b\over c}\right)\]

1995 China National Olympiad, 3

Tags: inequalities , inequalities unsolved , Sums and Products

Find the minimun value of $\sum_{i=1}^{10} \sum_{j=1}^{10} \sum_{k=1}^{10}|k(x+y-10i)(3x-6y-36j)(19x+95y-95k)|$ , where $x,y$ are integers.

2011 China Team Selection Test, 3

Tags: inequalities , inequalities unsolved

Let $n$ be a positive integer. Find the largest real number $\lambda$ such that for all positive real numbers $x_1,x_2,\cdots,x_{2n}$ satisfying the inequality \[\frac{1}{2n}\sum_{i=1}^{2n}(x_i+2)^n\geq \prod_{i=1}^{2n} x_i,\] the following inequality also holds \[\frac{1}{2n}\sum_{i=1}^{2n}(x_i+1)^n\geq \lambda\prod_{i=1}^{2n} x_i.\]

2013 Moldova Team Selection Test, 4

Tags: inequalities , inequalities unsolved

Prove that for any positive real numbers $a_i,b_i,c_i$ with $i=1,2,3$, $(a_1^3+b_1^3+c_1^3+1)(a_2^3+b_2^3+c_2^3+1)(a_3^3+b_3^3+c_3^3+1)\geq \frac{3}{4} (a_1+b_1+c_1)(a_2+b_2+c_2)(a_3+b_3+c_3)$

2003 All-Russian Olympiad, 2

Tags: inequalities , inequalities unsolved

Let $a, b, c$ be positive numbers with the sum $1$. Prove the inequality \[\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c} \geq \frac{2}{1+a}+\frac{2}{1+b}+\frac{2}{1+c}.\]

2003 Greece National Olympiad, 1

Tags: inequalities , function , calculus , derivative , inequalities unsolved

If $a, b, c, d$ are positive numbers satisfying $a^3 + b^3 +3ab = c + d = 1,$ prove that \[\left(a+\frac{1}{a}\right)^3+\left(b+\frac{1}{b}\right)^3+\left(c+\frac{1}{c}\right)^3+\left(d+\frac{1}{d}\right)^3\geq 40.\]

2003 China Girls Math Olympiad, 6

Tags: inequalities , inequalities unsolved

Let $ n \geq 2$ be an integer. Find the largest real number $ \lambda$ such that the inequality \[ a^2_n \geq \lambda \sum^{n\minus{}1}_{i\equal{}1} a_i \plus{} 2 \cdot a_n.\] holds for any positive integers $ a_1, a_2, \ldots a_n$ satisfying $ a_1 < a_2 < \ldots < a_n.$

2000 Canada National Olympiad, 5

Tags: inequalities unsolved , inequalities

Suppose that the real numbers $a_1, a_2, \ldots, a_{100}$ satisfy \begin{eqnarray*} 0 \leq a_{100} \leq a_{99} \leq \cdots \leq a_2 &\leq& a_1 , \\ a_1+a_2 & \leq & 100 \\ a_3+a_4+\cdots+a_{100} &\leq & 100. \end{eqnarray*} Determine the maximum possible value of $a_1^2 + a_2^2 + \cdots + a_{100}^2$, and find all possible sequences $a_1, a_2, \ldots , a_{100}$ which achieve this maximum.

2002 China Team Selection Test, 1

Tags: inequalities , inequalities unsolved

Given $ n \geq 3$, $ n$ is a integer. Prove that: \[ (2^n \minus{} 2) \cdot \sqrt{2i\minus{}1} \geq \left( \sum_{j\equal{}0}^{i\minus{}1}C_n^j \plus{} C_{n\minus{}1}^{i\minus{}1} \right) \cdot \sqrt{n}\] where if $ n$ is even, then $ \displaystyle 1 \leq i \leq \frac{n}{2}$; if $ n$ is odd, then $ \displaystyle 1 \leq i \leq \frac{n\minus{}1}{2}$.

2006 CHKMO, 3

Tags: inequalities , function , inequalities unsolved , algebra

Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=1$. Prove that\[ 6(a^3+b^3+c^3+d^3)\ge(a^2+b^2+c^2+d^2)+\frac{1}{8} \]

2001 USA Team Selection Test, 1

Tags: inequalities , inequalities unsolved

Let $\{ a_n\}_{n \ge 0}$ be a sequence of real numbers such that $a_{n+1} \ge a_n^2 + \frac{1}{5}$ for all $n \ge 0$. Prove that $\sqrt{a_{n+5}} \ge a_{n-5}^2$ for all $n \ge 5$.

2010 Singapore Senior Math Olympiad, 3

Tags: inequalities , inequalities unsolved

Given $a_1\ge 1$ and $a_{k+1}\ge a_k+1$ for all $k\ge 1,2,\dots,n$, show that $a_1^3+a_2^3+\dots+a_n^3\ge (a_1+a_2+\dots+a_n)^2$

2010 Germany Team Selection Test, 2

Tags: inequalities , inequalities unsolved

Prove or disprove that $\forall a,b,c,d \in \mathbb{R}^+$ we have the following inequality: \[3 \leq \frac{4a+b}{a+4b} + \frac{4b+c}{b+4c} + \frac{4c+a}{c+4a} < \frac{33}{4}\]

1991 Vietnam National Olympiad, 2

Tags: geometry , circumcircle , inequalities unsolved , inequalities

Let $G$ be centroid and $R$ the circunradius of a triangle $ABC$. The extensions of $GA,GB,GC$ meet the circuncircle again at $D,E,F$. Prove that: $\frac{3}{R} \leq \frac{1}{GD} + \frac{1}{GE} + \frac{1}{GF} \leq \sqrt{3} \leq \frac{1}{AB} + \frac{1}{BC} + \frac{1}{CA}$

2001 Vietnam Team Selection Test, 1

Tags: inequalities , inequalities unsolved , algebra , inequalities proposed

Let’s consider the real numbers $a, b, c$ satisfying the condition \[21 \cdot a \cdot b + 2 \cdot b \cdot c + 8 \cdot c \cdot a \leq 12.\] Find the minimal value of the expression \[P(a, b, c) = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.\]

2003 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: inequalities , inequalities unsolved

Let $x$, $y$, and $z$ be positive real numbers satisfying $x^2+y^2+z^2=1$. Prove that \[x^2yz+xy^2z+xyz^2\le\frac{1}{3}.\]

2012 China Second Round Olympiad, 3

Tags: inequalities unsolved , inequalities

Suppose that $x,y,z\in [0,1]$. Find the maximal value of the expression \[\sqrt{|x-y|}+\sqrt{|y-z|}+\sqrt{|z-x|}.\]

2011 JBMO Shortlist, 1

Tags: inequalities , inequalities unsolved

Let $a,b,c$ be positive real numbers such that $abc = 1$. Prove that: $\displaystyle\prod(a^5+a^4+a^3+a^2+a+1)\geq 8(a^2+a+1)(b^2+b+1)(c^2+c+1)$

1993 Romania Team Selection Test, 1

Tags: inequalities unsolved , inequalities

Find max. numbers $A$ wich is true ineq.: $\frac{x}{\sqrt{y^{2}+z^{2}}}+\frac{y}{\sqrt{x^{2}+z^{2}}}+\frac{z}{\sqrt{x^{2}+y^{2}}}\geq A$ $x,y,z$ are positve reals numberes! :wink:

2004 Greece Junior Math Olympiad, 3

Tags: inequalities unsolved , inequalities

x,y,z positive real numbers such that $x^2+y^2+z^2=25$ Find the min price of $A=\frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}$

2014 Iran MO (2nd Round), 3

Tags: inequalities , quadratics , function , algebra , quadratic formula , inequalities unsolved

Let $ x,y,z $ be three non-negative real numbers such that \[x^2+y^2+z^2=2(xy+yz+zx). \] Prove that \[\dfrac{x+y+z}{3} \ge \sqrt[3]{2xyz}.\]

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