Olympiad problems

Ukrainian TYM Qualifying - geometry, VII.12

Let $a, b$, and $c$ be the lengths of the sides of an arbitrary triangle, and let $\alpha,\beta$, and $\gamma$ be the radian measures of its corresponding angles. Prove that $$ \frac{\pi}{3}\le \frac{\alpha a +\beta b + \gamma c}{a+b+c} < \frac{\pi}{2}.$$ Suggest spatial analogues of this inequality.

2022 Bulgarian Spring Math Competition, Problem 10.1

Tags: system of equations , Inequality , algebra

If $x, y, z \in \mathbb{R}$ are solutions to the system of equations $$\begin{cases} x - y + z - 1 = 0\\ xy + 2z^2 - 6z + 1 = 0\\ \end{cases}$$ what is the greatest value of $(x - 1)^2 + (y + 1)^2$?

1991 IMO Shortlist, 25

Tags: Sequence , Inequality , inequality system , algebra , IMO Shortlist

Suppose that $ n \geq 2$ and $ x_1, x_2, \ldots, x_n$ are real numbers between 0 and 1 (inclusive). Prove that for some index $ i$ between $ 1$ and $ n \minus{} 1$ the inequality \[ x_i (1 \minus{} x_{i\plus{}1}) \geq \frac{1}{4} x_1 (1 \minus{} x_{n})\]

2017 Iran MO (2nd Round), 4

Tags: algebra , 2-variable inequality , Inequality , inequalities

Let $x,y$ be two positive real numbers such that $x^4-y^4=x-y$. Prove that $$\frac{x-y}{x^6-y^6}\leq \frac{4}{3}(x+y).$$

1969 IMO Shortlist, 69

Tags: Inequality , three variable inequality , algebra , IMO Shortlist , IMO Longlist

$(YUG 1)$ Suppose that positive real numbers $x_1, x_2, x_3$ satisfy $x_1x_2x_3 > 1, x_1 + x_2 + x_3 <\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}$ Prove that: $(a)$ None of $x_1, x_2, x_3$ equals $1$. $(b)$ Exactly one of these numbers is less than $1.$

2017 JBMO Shortlist, A3

Tags: JBMO Shortlist , Inequality

let $a\le b\le c \le d$ show that: $$ab^3+bc^3+cd^3+da^3\ge a^2b^2+b^2c^2+c^2d^2+d^2a^2$$

2024 Korea Junior Math Olympiad (First Round), 19.

Tags: algebra , maximum value , Inequality

For all integers $ {a}_{0},{a}_{1}, \cdot\cdot\cdot {a}_{100}$, find the maximum of ${a}_{5}-2{a}_{40}+3{a}_{60}-4{a}_{95} $ $\bigstar$ 1) ${a}_{0}={a}_{100}=0$ 2) for all $i=0,1,\cdot \cdot \cdot 99, $ $|{a}_{i+1}-{a}_{i}|\le1$ 3) $ {a}_{10}={a}_{90} $

2021 Science ON all problems, 3

Tags: algebra , Inequality , symmetric sums , Newton Sums

Are there any real numbers $a,b,c$ such that $a+b+c=6$, $ab+bc+ca=9$ and $a^4+b^4+c^4=260$? What about if we let $a^4+b^4+c^4=210$? [i] (Andrei Bâra)[/i]

1969 IMO Longlists, 64

Tags: Inequality , factorial , algebra , IMO Shortlist , IMO Longlist

$(USS 1)$ Prove that for a natural number $n > 2, (n!)! > n[(n - 1)!]^{n!}.$

2012 Bosnia and Herzegovina Junior BMO TST, 4

Tags: Triangle , perimeter , Inequality , geometry , inequalities

If $a$, $b$ and $c$ are sides of triangle which perimeter equals $1$, prove that: $a^2+b^2+c^2+4abc<\frac{1}{2}$

2008 India Regional Mathematical Olympiad, 3

Tags: inequalities , Inequality , positive integers , real number

Prove that for every positive integer $n$ and a non-negative real number $a$, the following inequality holds: $$n(n+1)a+2n \geqslant 4\sqrt{a}(\sqrt{1}+\sqrt{2}+\dots+\sqrt{n}).$$

2001 Slovenia National Olympiad, Problem 2

Tags: Inequality , algebra

Tina wrote a positive number on each of five pieces of paper. She did not say which numbers she wrote, but revealed their pairwise sums instead: $17,20,28,14,42,36,28,39,25,31$. Which numbers did she write?

2001 China Team Selection Test, 2

Tags: algebra , trigonometry , Inequality

Let $\theta_i \in \left ( 0,\frac{\pi}{4} \right ]$ for $i=1,2,3,4$. Prove that: $\tan \theta _1 \tan \theta _2 \tan \theta _3 \tan \theta _4 \le (\frac{\sin^8 \theta _1+\sin^8 \theta _2+\sin^8 \theta _3+\sin^8 \theta _4}{\cos^8 \theta _1+\cos^8 \theta _2+\cos^8 \theta _3+\cos^8 \theta _4})^\frac{1}{2}$ [hide=edit]@below, fixed now. There were some problems (weird characters) so aops couldn't send it.[/hide]

2023 China Western Mathematical Olympiad, 7

Tags: number theory , algebra , China , Inequality , number theory proposed

For positive integers $x, y, $ $r_x(y)$ to represent the smallest positive integer $ r $ such that $ r \equiv y(\text{mod x})$ .For any positive integers $a, b, n ,$ Prove that $$\sum_{i=1}^{n} r_b(a i)\leq \frac{n(a+b)}{2}$$

1977 IMO, 1

Tags: function , trigonometry , Trigonometric inequality , Inequality , IMO , IMO 1977

Let $a,b,A,B$ be given reals. We consider the function defined by \[ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). \] Prove that if for any real number $x$ we have $f(x) \geq 0$ then $a^2 + b^2 \leq 2$ and $A^2 + B^2 \leq 1.$

2021 Science ON all problems, 4

Tags: Inequality , inequalities , algebra

Consider positive real numbers $x,y,z$. Prove the inequality $$\frac 1x+\frac 1y+\frac 1z+\frac{9}{x+y+z}\ge 3\left (\left (\frac{1}{2x+y}+\frac{1}{x+2y}\right )+\left (\frac{1}{2y+z}+\frac{1}{y+2z}\right )+\left (\frac{1}{2z+x}+\frac{1}{x+2z}\right )\right ).$$ [i] (Vlad Robu \& Sergiu Novac)[/i]

2012 Israel National Olympiad, 3

Tags: algebra , Inequality , Symmetric inequality , inequalities

Let $a,b,c$ be real numbers such that $a^3(b+c)+b^3(a+c)+c^3(a+b)=0$. Prove that $ab+bc+ca\leq0$.

1994 All-Russian Olympiad, 5

Tags: least common multiple , Inequality , number theory

Prove that, for any natural numbers $k,m,n$: $[k,m] \cdot [m,n] \cdot [n,k] \ge [k,m,n]^2$

2015 Serbia National Math Olympiad, 5

Tags: Inequality , inequalities

Let $x,y,z$ be nonnegative positive integers. Prove $\frac{x-y}{xy+2y+1}+\frac{y-z}{zy+2z+1}+\frac{z-x}{xz+2x+1}\ge 0$

1999 All-Russian Olympiad, 7

Tags: Russia , High school olympiad , algebra , Inequality

Positive numbers $x,y$ satisfy $x^2+y^3 \ge x^3+y^4$. Prove that $x^3+y^3 \le 2$.

2020 Taiwan APMO Preliminary, P6

Tags: algebra , inequalities , Inequality

Let $a,b,c$ be positive reals. Find the minimum value of $$\dfrac{13a+13b+2c}{2a+2b}+\dfrac{24a-b+13c}{2b+2c}+\dfrac{(-a+24b+13c)}{2c+2a}$$. (1)What is the minimum value? (2)If the minimum value occurs when $(a,b,c)=(a_0,b_0,c_0)$,then find $\frac{b_0}{a_0}+\frac{c_0}{b_0}$.

2005 Federal Math Competition of S&M, Problem 3

Tags: inequalities , Inequality

If $x,y,z$ are nonnegative numbers with $x+y+z=3$, prove that $$\sqrt x+\sqrt y+\sqrt z\ge xy+yz+xz.$$

2001 Moldova National Olympiad, Problem 6

Tags: Inequality , Sequences , inequalities , algebra

Set $a_n=\frac{2n}{n^4+3n^2+4},n\in\mathbb N$. Prove that $\frac14\le a_1+a_2+\ldots+a_n\le\frac12$ for all $n$.

2020 Tuymaada Olympiad, 2

Tags: Inequality , inequalities , n-variable inequality

Given positive real numbers $a_1, a_2, \dots, a_n$. Let \[ m = \min \left( a_1 + \frac{1}{a_2}, a_2 + \frac{1}{a_3}, \dots, a_{n - 1} + \frac{1}{a_n} , a_n + \frac{1}{a_1} \right). \] Prove the inequality \[ \sqrt[n]{a_1 a_2 \dots a_n} + \frac{1}{\sqrt[n]{a_1 a_2 \dots a_n}} \ge m. \]

2020 Thailand Mathematical Olympiad, 8

Tags: Inequality , inequalities

For all positive real numbers $a,b,c$ with $a+b+c=3$, prove the inequality $$\frac{a^6}{c^2+2b^3} + \frac{b^6}{a^2+2c^3} + \frac{c^6}{b^2+2a^3} \geq 1.$$