Olympiad problems

Mathematical Minds 2023, P2

Tags: algebra , inequalities

Let $a,b,c$ be real numbers with sum equal to zero. Prove that \[ab^3+bc^3+ca^3\leqslant 0.\]

2000 Austrian-Polish Competition, 9

Tags: algebra , inequalities

If three nonnegative reals $a$, $b$, $c$ satisfy $a+b+c=1$, prove that $2 \leq \left(1-a^{2}\right)^{2}+\left(1-b^{2}\right)^{2}+\left(1-c^{2}\right)^{2}\leq \left(1+a\right)\left(1+b\right)\left(1+c\right)$.

2012 ELMO Shortlist, 2

Tags: inequalities

Let $a,b,c$ be three positive real numbers such that $ a \le b \le c$ and $a+b+c=1$. Prove that \[\frac{a+c}{\sqrt{a^2+c^2}}+\frac{b+c}{\sqrt{b^2+c^2}}+\frac{a+b}{\sqrt{a^2+b^2}} \le \frac{3\sqrt{6}(b+c)^2}{\sqrt{(a^2+b^2)(b^2+c^2)(c^2+a^2)}}.\] [i]Owen Goff.[/i]

2013 Bogdan Stan, 2

Tags: induction , boundedness , inequalities , sequence

Let $ \left( a_n \right) ,\left( b_n \right) $ be two sequences of real numbers from the interval $ (-1,1) $ having the property that $$ \max\left( \left| a_{n+1} -a_n \right| ,\left| b_{n+1} -b_n \right| \right) \le\frac{1}{(n+4)(n+5)} , $$ for any natural number. Prove that $ \left| a_nb_n -a_1b_1 \right|\le 1/2, $ for any natural number $ n. $ [i]Cristinel Mortici[/i]

1985 Kurschak Competition, 3

Tags: geometric transformation , reflection , circumcircle , homothety , inequalities , geometry

We reflected each vertex of a triangle on the opposite side. Prove that the area of the triangle formed by these three reflection points is smaller than the area of the initial triangle multiplied by five.

2014 Tajikistan Team Selection Test, 3

Tags: geometry , inequalities

Let $a$, $b$, $c$ be side length of a triangle. Prove the inequality \begin{align*} \sqrt{a^2 + ab + b^2} + \sqrt{b^2 + bc + c^2} + \sqrt{c^2 + ca + a^2} \leq \sqrt{5a^2 + 5b^2 + 5c^2 + 4ab + 4 bc + 4ca}.\end{align*}

2024 Indonesia MO, 7

Tags: algebra , polynomial , inequalities

Suppose $P(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1x + a_0$ where $a_0, a_1, \ldots, a_{n-1}$ are reals for $n\geq 1$ (monic $n$th-degree polynomial with real coefficients). If the inequality \[ 3(P(x)+P(y)) \geq P(x+y) \] holds for all reals $x,y$, determine the minimum possible value of $P(2024)$.

2013 JBMO TST - Macedonia, 3

Tags: inequalities

$ a,b,c>0 $ and $ abc=1 $. Prove that $\frac{1}{2}\ (\sqrt{a}\ +\sqrt{b}\ + \sqrt{c}\ ) +\frac{1}{1+a}\ + \frac{1}{1+b}\ + \frac{1}{1+c}\ge\ 3 $. ( The official problem is with $ abc = 1 $ but it can be proved without using it. )

1999 Mongolian Mathematical Olympiad, Problem 3

Tags: inequalities , trigonometry , rectangle , analytic geometry , geometry

I couldn't solve this problem and the only solution I was able to find was very unnatural (it was an official solution, I think) and I couldn't be satisfied with it, so I ask you if you can find some different solutions. The problem is really great one! If $M$ is the centroid of a triangle $ABC$, prove that the following inequality holds: \[\sin\angle CAM+\sin\angle CBM\leq\frac{2}{\sqrt3}.\] The equality occurs in a very strange case, I don't remember it.

2008 China Girls Math Olympiad, 2

Tags: polynomial , inequalities , algebra

Let $ \varphi(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d$ be a polynomial with real coefficients. Given that $ \varphi(x)$ has three positive real roots and that $ \varphi(0) < 0$, prove that \[ 2b^3 \plus{} 9a^2d \minus{} 7abc \leq 0. \]

2021 Kyiv City MO Round 1, 11.4

Tags: algebra , inequalities

For positive real numbers $a, b, c$ with sum $\frac{3}{2}$, find the smallest possible value of the following expression: $$\frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} + \frac{1}{abc}$$ [i]Proposed by Serhii Torba[/i]

2007 Romania Team Selection Test, 1

Tags: function , inequalities , domain , combinatorics , algebra

Let $\mathcal{F}$ be the set of all the functions $f : \mathcal{P}(S) \longrightarrow \mathbb{R}$ such that for all $X, Y \subseteq S$, we have $f(X \cap Y) = \min (f(X), f(Y))$, where $S$ is a finite set (and $\mathcal{P}(S)$ is the set of its subsets). Find \[\max_{f \in \mathcal{F}}| \textrm{Im}(f) |. \]

2018 Bulgaria JBMO TST, 3

Tags: inequalities

Prove for all positive real numbers $m,n,p,q$ that $$\frac{m}{t+n+p+q} + \frac{n}{t+p+q+m} + \frac{p}{t+q+m+n} + \frac{q}{t+m+n+p} \geq \frac{4}{5},$$ where $t=\frac{m+n+p+q}{2}.$

1997 Estonia Team Selection Test, 2

Tags: inequalities , n-variable inequality

Prove that for all positive real numbers $a_1,a_2,\cdots a_n$ \[\frac{1}{\frac{1}{1+a_1}+\frac{1}{1+a_2}+\cdots +\frac{1}{1+a_n}}-\frac{1}{\frac{1}{a_1}+\frac{1}{a_2}+\cdots +\frac{1}{a_n}}\geq \frac{1}{n}\] When does the inequality hold?

2004 Romania National Olympiad, 3

Tags: function , inequalities , calculus , algebra , logarithm

Let $n>2,n \in \mathbb{N}$ and $a>0,a \in \mathbb{R}$ such that $2^a + \log_2 a = n^2$. Prove that: \[ 2 \cdot \log_2 n>a>2 \cdot \log_2 n -\frac{1}{n} . \] [i]Radu Gologan[/i]

2011 Junior Balkan Team Selection Tests - Moldova, 2

Tags: inequalities , algebra

The real numbers $a, b, x$ satisfy the inequalities $| a + x + b | \le 1, | 4a + 2x + b | \le1, | 9a + 6x + 4b | \le 1$. Prove that $| x | \le15$.

1996 Spain Mathematical Olympiad, 3

Tags: function , algebra , polynomial , inequalities , triangle inequality

Consider the functions $ f(x) = ax^{2} + bx + c $ , $ g(x) = cx^{2} + bx + a $, where a, b, c are real numbers. Given that $ |f(-1)| \leq 1 $, $ |f(0)| \leq 1 $, $ |f(1)| \leq 1 $, prove that $ |f(x)| \leq \frac{5}{4} $ and $ |g(x)| \leq 2 $ for $ -1 \leq x \leq 1 $.

2010 Today's Calculation Of Integral, 525

Tags: calculus , integration , geometry , vector , inequalities , quadratic , trigonometry

Let $ a,\ b$ be real numbers satisfying $ \int_0^1 (ax\plus{}b)^2dx\equal{}1$. Determine the values of $ a,\ b$ for which $ \int_0^1 3x(ax\plus{}b)\ dx$ is maximized.

1975 Spain Mathematical Olympiad, 4

Tags: algebra , inequalities , sum , product

Prove that if the product of $n$ real and positive numbers is equal to $1$, its sum is greater than or equal to $n$.

2004 Cuba MO, 7

Tags: algebra , inequalities

For real numbers, $a,b,c$ with $bc \ne 0$ we have to $\frac{1-c^2}{bc} \ge 0$. Prove that $$5( a^2+b^2+c^2 -bc^3) \ge ab.$$

2022 Kyiv City MO Round 1, Problem 1

Tags: inequalities , algebra

What's the smallest possible value of $$\frac{(x+y+|x-y|)^2}{xy}$$ over positive real numbers $x, y$?

2007 China Team Selection Test, 3

Tags: inequalities , combinatorial geometry , analytic geometry , geometry

Assume there are $ n\ge3$ points in the plane, Prove that there exist three points $ A,B,C$ satisfying $ 1\le\frac{AB}{AC}\le\frac{n\plus{}1}{n\minus{}1}.$

2009 Vietnam National Olympiad, 1

Tags: linear algebra , matrix , inequalities , algebra

[b]Problem 1.[/b]Find all $ (x,y)$ such that: \[ \{\begin{matrix} \displaystyle\dfrac {1}{\sqrt {1 + 2x^2}} + \dfrac {1}{\sqrt {1 + 2y^2}} & = & \displaystyle\dfrac {2}{\sqrt {1 + 2xy}} \\ \sqrt {x(1 - 2x)} + \sqrt {y(1 - 2y)} & = & \displaystyle\dfrac {2}{9} \end{matrix}\; \]

2010 International Zhautykov Olympiad, 3

Tags: incenter , trigonometry , inequalities , euler , geometry

Let $ABC$ arbitrary triangle ($AB \neq BC \neq AC \neq AB$) And O,I,H it's circum-center, incenter and ortocenter (point of intersection altitudes). Prove, that 1) $\angle OIH > 90^0$(2 points) 2)$\angle OIH >135^0$(7 points) balls for 1) and 2) not additive.

2005 Georgia Team Selection Test, 3

Tags: calculus , derivative , cauchy inequality , inequalities

Let $ x,y,z$ be positive real numbers,satisfying equality $ x^{2}\plus{}y^{2}\plus{}z^{2}\equal{}25$. Find the minimal possible value of the expression $ \frac{xy}{z} \plus{} \frac{yz}{x} \plus{} \frac{zx}{y}$.