Olympiad problems

2010 Saudi Arabia Pre-TST, 4.3

Tags: algebra , inequalities

Let $a, b, c$ be positive real numbers such that $abc = 8$. Prove that $$\frac{a-2}{a+1}+\frac{b-2}{b+1}+\frac{c-2}{c+1} \le 0$$

2013 China Girls Math Olympiad, 5

Tags: inequalities

For any given positive numbers $a_1,a_2,\ldots,a_n$, prove that there exist positive numbers $x_1,x_2,\ldots,x_n$ satisfying $\sum_{i=1}^n x_i=1$, such that for any positive numbers $y_1,y_2,\ldots,y_n$ with $\sum_{i=1}^n y_i=1$, the inequality $\sum_{i=1}^n \frac{a_ix_i}{x_i+y_i}\ge \frac{1}{2}\sum_{i=1}^n a_i$ holds.

2014 Contests, 1

Tags: inequalities , algebra

Prove that for positive reals $a$,$b$,$c$ so that $a+b+c+abc=4$, \[\left (1+\dfrac{a}{b}+ca \right )\left (1+\dfrac{b}{c}+ab \right)\left (1+\dfrac{c}{a}+bc \right) \ge 27\] holds.

MathLinks Contest 4th, 4.1

Tags: algebra , inequalities

Let $N_0$ be the set of all non-negative integers and let $f : N_0 \times N_0 \to [0, +\infty)$ be a function such that $f(a, b) = f(b, a)$ and $$f(a, b) = f(a + 1, b) + f(a, b + 1),$$ for all $a, b \in N_0$. Denote by $x_n = f(n, 0)$ for all $n \in N_0$. Prove that for all $n \in N_0$ the following inequality takes place $$2^n x_n \ge x_0.$$

2017 AMC 10, 3

Tags: inequalities

Real numbers $x$, $y$, and $z$ satisfy the inequalities $$0<x<1,\qquad-1<y<0,\qquad\text{and}\qquad1<z<2.$$ Which of the following numbers is nessecarily positive? $\textbf{(A) } y+x^2 \qquad \textbf{(B) } y+xz \qquad \textbf{(C) }y+y^2 \qquad \textbf{(D) }y+2y^2 \qquad\\ \textbf{(E) } y+z$

1997 China National Olympiad, 3

Tags: inequalities , induction

Let $(a_n)$ be a sequence of non-negative real numbers satisfying $a_{n+m}\le a_n+a_m$ for all non-negative integers $m,n$. Prove that if $n\ge m$ then $a_n\le ma_1+\left(\dfrac{n}{m}-1\right)a_m$ holds.

2013 Brazil Team Selection Test, 4

Tags: inequalities

Let $a, b, c$ be non-negative reals with $a + b + c \le 2$. prove that $$\sqrt{b^2+ac} + \sqrt{a^2+bc} + \sqrt{c^2+ab} \le 3$$

2017 Junior Balkan Team Selection Tests - Romania, 4

Tags: inequalities , algebra

Let $a, b, c, d$ be non-negative real numbers satisfying $a + b + c + d = 3$. Prove that $$\frac{a}{1 + 2b^3} + \frac{b}{1 + 2c^3} +\frac{c}{1 + 2d^3} +\frac{d}{1 + 2a^3} \ge \frac{a^2 + b^2 + c^2 + d^2}{3}$$ When does the equality hold?

2021 XVII International Zhautykov Olympiad, #4

Tags: geometry , inequalities , geometric inequality

Let there be an incircle of triangle $ABC$, and 3 circles each inscribed between incircle and angles of $ABC$. Let $r, r_1, r_2, r_3$ be radii of these circles ($r_1, r_2, r_3 < r$). Prove that $$r_1+r_2+r_3 \geq r$$

2022 Indonesia TST, A

Tags: algebra , inequality , asymmetric , inequalities , minimum

Let $a$ and $b$ be two positive reals such that the following inequality \[ ax^3 + by^2 \geq xy - 1 \] is satisfied for any positive reals $x, y \geq 1$. Determine the smallest possible value of $a^2 + b$. [i]Proposed by Fajar Yuliawan[/i]

IV Soros Olympiad 1997 - 98 (Russia), 11.3

Tags: inequalities , radical , algebra

Solve the inequality $$\sqrt{(x-2)^2(x-x^2)}<\sqrt{4x-1-(x^2-3x)^2}$$

2017 International Zhautykov Olympiad, 3

Tags: geometry , inequalities , tetrahedron , 3d geometry

Let $ABCD$ be the regular tetrahedron, and $M, N$ points in space. Prove that: $AM \cdot AN + BM \cdot BN + CM \cdot CN \geq DM \cdot DN$

2006 Victor Vâlcovici, 1

Tags: inequalities

Let be two nonnegative real numbers $ a,b, $ not both $ 0, $ satisfying $ \frac{1}{2a+b} +\frac{1}{a+2b} =1. $ Prove the following inequalities and explain the equality cases for [b]a),b).[/b] [b]a)[/b] $ 4/3\le a+b\le 3/2 $ [b]b)[/b] $ 8/9\le a^2+b^2\le 9/4 $ [b]c)[/b] $ ab<1/2 $ [i]Laurențiu Panaitopol[/i]

2015 Thailand TSTST, 2

Tags: inequalities

Let $a, b, c\in (0, 1)$ with $a + b + c = 1$. Prove that $$\frac{a^5+b^5}{a^3+b^3}+\frac{b^5+c^5}{b^3+c^3}+\frac{c^5+a^5}{c^3+a^3}\geq\frac{a}{8+b^3+c^3}+\frac{b}{8+c^3+a^3}+\frac{c}{8+a^3+b^3}.$$

1983 Polish MO Finals, 2

Tags: irrational , inequalities , algebra

Let be given an irrational number $a$ in the interval $(0,1)$ and a positive integer $N$. Prove that there exist positive integers $p,q,r,s$ such that $\frac{p}{q} < a <\frac{r}{s}, \frac{r}{s} -\frac{p}{q}<\frac{1}{N}$, and $rq- ps = 1$.

2000 Junior Balkan MO, 2

Tags: inequalities , induction , number theory , difference of squares , special factorizations , algebra

Find all positive integers $n\geq 1$ such that $n^2+3^n$ is the square of an integer. [i]Bulgaria[/i]

2009 Postal Coaching, 5

Tags: inequalities , number theory , least common multiple , lcm

For positive integers $n, k$ with $1 \le k \le n$, define $$L(n, k) = Lcm \,(n, n - 1, n -2, ..., n - k + 1)$$ Let $f(n)$ be the largest value of $k$ such that $L(n, 1) < L(n, 2) < ... < L(n, k)$. Prove that $f(n) < 3\sqrt{n}$ and $f(n) > k$ if $n > k! + k$.

2019 Federal Competition For Advanced Students, P2, 4

Tags: inequalities

Let $a, b, c$ be the positive real numbers such that $a+b+c+2=abc .$ Prove that $$(a+1)(b+1)(c+1)\geq 27.$$

2005 Junior Balkan Team Selection Tests - Romania, 4

Tags: inequalities , algebra

Let $a,b,c$ be positive numbers such that $a+b+c \geq \dfrac 1a + \dfrac 1b + \dfrac 1c$. Prove that \[ a+b+c \geq \frac 3{abc}. \]

2005 Georgia Team Selection Test, 10

Tags: algebra , 3-variable inequality , inequalities

Let $ a,b,c$ be positive numbers, satisfying $ abc\geq 1$. Prove that \[ a^{3} \plus{} b^{3} \plus{} c^{3} \geq ab \plus{} bc \plus{} ca.\]

2019 Tournament Of Towns, 2

Tags: triangle inequality , geometric inequality , geometry , inequalities , area of a triangle

Two acute triangles $ABC$ and $A_1B_1C_1$ are such that $B_1$ and $C_1$ lie on $BC$, and $A_1$ lies inside the triangle $ABC$. Let $S$ and $S_1$ be the areas of those triangles respectively. Prove that $\frac{S}{AB + AC}> \frac{S_1}{A_1B_1 + A_1C_1}$ (Nairi Sedrakyan, Ilya Bogdanov)

1970 IMO Longlists, 42

Tags: inequalities , decimal representation , digit , algebra

We have $0\le x_i<b$ for $i=0,1,\ldots,n$ and $x_n>0,x_{n-1}>0$. If $a>b$, and $x_nx_{n-1}\ldots x_0$ represents the number $A$ base $a$ and $B$ base $b$, whilst $x_{n-1}x_{n-2}\ldots x_0$ represents the number $A'$ base $a$ and $B'$ base $b$, prove that $A'B<AB'$.

2016 China Team Selection Test, 2

Tags: complex number , inequalities

Find the smallest positive number $\lambda $ , such that for any complex numbers ${z_1},{z_2},{z_3}\in\{z\in C\big| |z|<1\}$ ,if $z_1+z_2+z_3=0$, then $$\left|z_1z_2 +z_2z_3+z_3z_1\right|^2+\left|z_1z_2z_3\right|^2 <\lambda .$$

2003 Irish Math Olympiad, 4

Tags: inequalities

Given real positive a,b , find the larget real c such that $c\leq max(ax+\frac{1}{ax},bx+\frac{1}{bx})$ for all positive ral x. There is a solution here,,,, http://www.kalva.demon.co.uk/irish/soln/sol039.html but im wondering if there is a better one . Thank you.

2021 Cyprus JBMO TST, 1

Tags: algebra , am-gm , inequalities

Let $x,y,z$ be positive real numbers such that $x^2+y^2+z^2=3$. Prove that \[ xyz(x+y+z)+2021\geqslant 2024xyz\]