Olympiad problems

2011 Indonesia TST, 1

Let $a, b, c$ be the sides of a triangle with $abc = 1$. Prove that $$\frac{\sqrt{b + c -a}}{a}+\frac{\sqrt{c + a - b}}{b}+\frac{\sqrt{a + b - c}}{c} \ge a + b + c$$

2018 Vietnam National Olympiad, 3

Tags: inequalities , geometry , combinatorial geometry

An investor has two rectangular pieces of land of size $120\times 100$. a. On the first land, she want to build a house with a rectangular base of size $25\times 35$ and nines circular flower pots with diameter $5$ outside the house. Prove that even the flower pots positions are chosen arbitrary on the land, the remaining land is still sufficient to build the desired house. b. On the second land, she want to construct a polygonal fish pond such that the distance from an arbitrary point on the land, outside the pond, to the nearest pond edge is not over $5$. Prove that the perimeter of the pond is not smaller than $440-20\sqrt{2}$.

2016 Nigerian Senior MO Round 2, Problem 6

Tags: inequality , inequalities , algebra

Given that $a, b, c, d \in \mathbb{R}$, prove that $(ab+cd)^2 \leq (a^2+c^2)(b^2+d^2)$.

2017 Canada National Olympiad, 1

Tags: algebra , inequalities

For pairwise distinct nonnegative reals $a,b,c$, prove that $$\frac{a^2}{(b-c)^2}+\frac{b^2}{(c-a)^2}+\frac{c^2}{(b-a)^2}>2$$.

1981 Czech and Slovak Olympiad III A, 5

Tags: max , integer , inequalities , algebra

Let $n$ be a positive integer. Determine the maximum of the sum $x_1+\cdots+x_n$ where $x_1,\ldots,x_n$ are non-negative integers satisfying the condition \[x_1^3+\cdots+x_n^3\le7n.\]

2016 Irish Math Olympiad, 5

Tags: positive integer , algebra , inequalities

Let $a_1, a_2, ..., a_m$ be positive integers, none of which is equal to $10$, such that $a_1 + a_2 + ...+ a_m = 10m$. Prove that $(a_1a_2a_3 \cdot ...\cdot a_m)^{1/m} \le 3\sqrt{11}$.

1995 Hungary-Israel Binational, 3

Tags: inequalities , algebra , polynomial

The polynomial $ f(x)\equal{}ax^2\plus{}bx\plus{}c$ has real coefficients and satisfies $ \left|f(x)\right|\le 1$ for all $ x\in [0, 1]$. Find the maximal value of $ |a|\plus{}|b|\plus{}|c|$.

1998 Junior Balkan Team Selection Tests - Romania, 2

Tags: geometry , inequalities , perimeter , rectangle

Consider the rectangle $ ABCD $ and the points $ M,N,P,Q $ on the segments $ AB,BC,CD, $ respectively, $ DA, $ excluding its extremities. Denote with $ p_{\square} , A_{\square} $ the perimeter, respectively, the area of $ \square. $ Prove that: [b]a)[/b] $ p_{MNPQ}\ge AC+BD. $ [b]b)[/b] $ p_{MNPQ} =AC+BD\implies A_{MNPQ}\le \frac{A_{ABCD}}{2} . $ [b]c)[/b] $ p_{MNPQ} =AC+BD\implies MP^2 +NQ^2\ge AC^2. $ [i]Dan Brânzei[/i] and [i]Gheorghe Iurea[/i]

2018 Taiwan TST Round 2, 1

Tags: inequalities

Given positive integers $a_1,a_2,\ldots, a_n$ with $a_1<a_2<\cdots<a_n)$, and a positive real $k$ with $k\geq 1$. Prove that \[\sum_{i=1}^{n}a_i^{2k+1}\geq \left(\sum_{i=1}^{n}a_i^k\right)^2.\]

2016 Spain Mathematical Olympiad, 6

Tags: jensen , esp , rearrangement inequality , inequalities

Let $n\geq 2$ an integer. Find the least value of $\gamma$ such that for any positive real numbers $x_1,x_2,...,x_n$ with $x_1+x_2+...+x_n=1$ and any real $y_1+y_2+...+y_n=1$ and $0\leq y_1,y_2,...,y_n\leq \frac{1}{2}$ the following inequality holds: $$x_1x_2...x_n\leq \gamma \left(x_1y_1+x_2y_2+...+x_ny_n\right)$$

1981 Romania Team Selection Tests, 3.

Tags: tetrahedron , geometry , 3d geometry , inequalities , algebra

Determine the lengths of the edges of a right tetrahedron of volume $a^3$ so that the sum of its edges' lengths is minumum.

2009 Hong Kong TST, 5

Tags: inequalities , quadratic , algebra , function

Let $ a,b,c$ be the three sides of a triangle. Determine all possible values of $ \frac {a^2 \plus{} b^2 \plus{} c^2}{ab \plus{} bc \plus{} ca}$

1990 All Soviet Union Mathematical Olympiad, 517

Tags: absolute value , algebra , inequalities , maximum

What is the largest possible value of $|...| |a_1 - a_2| - a_3| - ... - a_{1990}|$, where $a_1, a_2, ... , a_{1990}$ is a permutation of $1, 2, 3, ... , 1990$?

2008 IberoAmerican Olympiad For University Students, 3

Tags: inequalities

Prove that $x+\frac{1}{x^x}<2$ for $0<x<1$.

1993 Austrian-Polish Competition, 6

Tags: algebra , inequalities

If $a,b \ge 0$ are real numbers, prove the inequality $$\left(\frac{\sqrt{a}+\sqrt{b}}{2}\right)^2\leq\frac{a+\sqrt[3] {a^2b}+\sqrt[3] {ab^2}+b}{4}\leq\frac{a+\sqrt{ab}+b}{3} \leq \sqrt{\left(\frac{a^{2/3}+b^{2/3}}{2}\right)^{3}}$$ For each of the inequalities, find the cases of equality.

2016 Junior Balkan Team Selection Tests - Romania, 3

Tags: inequalities

Let a,b,c be real numbers such that:$a\ge b\ge 1\ge c\ge 0$ and a+b+c=3. a)Prove that $2\le ab +bc+ca\le 3$ b)Prove that $\dfrac{24}{a^3+b^3+c^3}+\dfrac{25}{ab+bc+ca}\ge 14$. Find the equality cases

1998 German National Olympiad, 5

Tags: sum , sequence , recurrence relation , algebra , inequalities

A sequence ($a_n$) is given by $a_0 = 0, a_1 = 1$ and $a_{k+2} = a_{k+1} +a_k$ for all integers $k \ge 0$. Prove that the inequality $\sum_{k=0}^n \frac{a_k}{2^k}< 2$ holds for all positive integers $n$.

2020 Thailand TST, 4

Tags: inequalities , algebra

Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]

2014 China Team Selection Test, 4

Tags: inequalities , induction

For any real numbers sequence $\{x_n\}$ ,suppose that $\{y_n\}$ is a sequence such that: $y_1=x_1, y_{n+1}=x_{n+1}-(\sum\limits_{i = 1}^{n} {x^2_i})^{ \frac{1}{2}}$ ${(n \ge 1})$ . Find the smallest positive number $\lambda$ such that for any real numbers sequence $\{x_n\}$ and all positive integers $m$ , have $\frac{1}{m}\sum\limits_{i = 1}^{m} {x^2_i}\le\sum\limits_{i = 1}^{m} {\lambda^{m-i}y^2_i} .$ (High School Affiliated to Nanjing Normal University )

2009 BAMO, 5

Tags: inequalities

Let $\triangle ABC$ be an acute triangle with angles $\alpha, \beta,$ and $\gamma$. Prove that $$\frac{\cos(\beta-\gamma)}{cos\alpha}+\frac{\cos(\gamma-\alpha)}{\cos \beta}+\frac{\cos(\alpha-\beta)}{\cos \gamma} \geq \frac{3}{2}$$

2012 Brazil Team Selection Test, 5

Tags: inequalities , function

Let $ n $ be an integer greater than or equal to $ 2 $. Prove that if the real numbers $ a_1 , a_2 , \cdots , a_n $ satisfy $ a_1 ^2 + a_2 ^2 + \cdots + a_n ^ 2 = n $, then \[\sum_{1 \le i < j \le n} \frac{1}{n- a_i a_j} \le \frac{n}{2} \] must hold.

2011 Korea Junior Math Olympiad, 7

Tags: inequalities , convex function , n-variable inequality

For those real numbers $x_1 , x_2 , \ldots , x_{2011}$ where each of which satisfies $0 \le x_1 \le 1$ ($i = 1 , 2 , \ldots , 2011$), find the maximum of \[ x_1^3+x_2^3+ \cdots + x_{2011}^3 - \left( x_1x_2x_3 + x_2x_3x_4 + \cdots + x_{2011}x_1x_2 \right) \]

2013 India National Olympiad, 6

Tags: polynomial , logarithm , function , inequalities , algebra

Let $a,b,c,x,y,z$ be six positive real numbers satisfying $x+y+z=a+b+c$ and $xyz=abc.$ Further, suppose that $a\leq x<y<z\leq c$ and $a<b<c.$ Prove that $a=x,b=y$ and $c=z.$

2023 Greece JBMO TST, 3

Tags: algebra , inequalities

Let $a, b,$ and $c$ be positive real numbers such that $a^2 + b^2 + c^2 = 3$. Prove that $$\frac{a^2 + b^2}{2ab} + \frac{b^2 + c^2}{2bc} + \frac{c^2 + a^2}{2ca} + \frac{2(ab + bc + ca)}{3} \ge 5 $$ When equality holds?

1972 Spain Mathematical Olympiad, 2

Tags: inequalities , geometric inequalities , geometry , analytic geometry

A point moves on the sides of the triangle $ABC$, defined by the vertices $A(-1.8, 0)$, $B(3.2, 0)$, $C(0, 2.4)$ . Determine the positions of said point, in which the sum of their distance to the three vertices is absolute maximum or minimum. [img]https://cdn.artofproblemsolving.com/attachments/2/5/9e5bb48cbeefaa5f4c069532bf5605b9c1f5ea.png[/img]