Olympiad problems

2005 Uzbekistan National Olympiad, 1

Given a,b c are lenth of a triangle (If ABC is a triangle then AC=b, BC=a, AC=b) and $a+b+c=2$. Prove that $1+abc<ab+bc+ca\leq \frac{28}{27}+abc$

2025 Kyiv City MO Round 1, Problem 5

Tags: algebra , inequality

Determine the largest possible constant $ C $ such that for any positive real numbers $ x, y, z $, which are the sides of a triangle, the following inequality holds: \[ \frac{xy}{x^2 + y^2 + xz} + \frac{yz}{y^2 + z^2 + yx} + \frac{zx}{z^2 + x^2 + zy} \geq C. \] [i]Proposed by Vadym Solomka[/i]

2005 Bosnia and Herzegovina Team Selection Test, 2

Tags: algebra , inequality , three variable inequality

If $a_1$, $a_2$ and $a_3$ are nonnegative real numbers for which $a_1+a_2+a_3=1$, then prove the inequality $a_1\sqrt{a_2}+a_2\sqrt{a_3}+a_3\sqrt{a_1}\leq \frac{1}{\sqrt{3}}$

2021 Federal Competition For Advanced Students, P1, 1

Tags: inequality , algebra

Let $a,b,c\geq 0$ and $a+b+c=1.$ Prove that$$\frac{a}{2a+1}+\frac{b}{3b+1}+\frac{c}{6c+1}\leq \frac{1}{2}.$$ [size=50](Marian Dinca)[/size]

2012 Junior Balkan MO, 1

Tags: algebra , inequality

Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that \[\frac {a}{b} + \frac {a}{c} + \frac {c}{b} + \frac {c}{a} + \frac {b}{c} + \frac {b}{a} + 6 \geq 2\sqrt{2}\left (\sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\right ).\] When does equality hold?

2000 Moldova National Olympiad, Problem 6

Tags: inequality , inequalities

Assuming that real numbers $x$ and $y$ satisfy $y\left(1+x^2\right)=x\left(\sqrt{1-4y^2}-1\right)$, find the maximum value of $xy$.

1983 Czech and Slovak Olympiad III A, 5

Tags: inequality , algebra , inequalities

Find all pair $(x,y)$ of positive integers satisfying $$\left|\frac{x}{y}-\sqrt2\right|<\frac{1}{y^3}.$$

2015 Indonesia MO Shortlist, A3

Tags: inequality , inequalities

Let $a,b,c$ positive reals such that $a^2+b^2+c^2=1$. Prove that $$\frac{a+b}{\sqrt{ab+1}}+\frac{b+c}{\sqrt{bc+1}}+\frac{c+a}{\sqrt{ac+1}}\le 3$$

2011 Peru IMO TST, 2

Tags: circumcircle , inequality , polygon , geometry

Let $A_1A_2 \ldots A_n$ be a convex polygon. Point $P$ inside this polygon is chosen so that its projections $P_1, \ldots , P_n$ onto lines $A_1A_2, \ldots , A_nA_1$ respectively lie on the sides of the polygon. Prove that for arbitrary points $X_1, \ldots , X_n$ on sides $A_1A_2, \ldots , A_nA_1$ respectively, \[\max \left\{ \frac{X_1X_2}{P_1P_2}, \ldots, \frac{X_nX_1}{P_nP_1} \right\} \geq 1.\] [i]Proposed by Nairi Sedrakyan, Armenia[/i]

1967 IMO Longlists, 58

Tags: geometry , function , algebra , inequality , linear function

A linear binomial $l(z) = Az + B$ with complex coefficients $A$ and $B$ is given. It is known that the maximal value of $|l(z)|$ on the segment $-1 \leq x \leq 1$ $(y = 0)$ of the real line in the complex plane $z = x + iy$ is equal to $M.$ Prove that for every $z$ \[|l(z)| \leq M \rho,\] where $\rho$ is the sum of distances from the point $P=z$ to the points $Q_1: z = 1$ and $Q_3: z = -1.$

2001 All-Russian Olympiad, 2

Tags: algebra , polynomial , function , inequality

The two polynomials $(x) =x^4+ax^3+bx^2+cx+d$ and $Q(x) = x^2+px+q$ take negative values on an interval $I$ of length greater than $2$, and nonnegative values outside of $I$. Prove that there exists $x_0 \in \mathbb R$ such that $P(x_0) < Q(x_0)$.

2016 VJIMC, 1

Tags: inequality , inequalities

Let $a,b,c$ be positive real numbers such that $a + b + c = 1$. Show that $$\left(\frac{1}{a} + \frac{1}{bc}\right)\left(\frac{1}{b} + \frac{1}{ca}\right)\left(\frac{1}{c} + \frac{1}{ab}\right) \geq 1728$$

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 \[ \sqrt[n]{a_1 a_2 \dots a_n} + \frac{1}{\sqrt[n]{a_1 a_2 \dots a_n}} \ge m. \]

2025 239 Open Mathematical Olympiad, 4

Tags: algebra , inequality

Positive numbers $a$, $b$ and $c$ are such that $a^2+b^2+c^2+abc=4$. Prove that \[\sqrt{2-a}+\sqrt{2-b}+\sqrt{2-c}\geqslant 2+\sqrt{(2-a)(2-b)(2-c)}.\]

2024 Azerbaijan National Mathematical Olympiad, 2

Tags: algebra , inequality

Find all the real number triples $(x, y, z)$ satisfying the following system of inequalities under the condition $0 < x, y, z < \sqrt{2}$: $$y\sqrt{4-x^2y^2}\ge \frac{2}{\sqrt{xz}}$$ $$x\sqrt{4-x^2z^2}\ge \frac{2}{\sqrt{yz}}$$ $$z\sqrt{4-y^2z^2}\ge \frac{2}{\sqrt{xy}}$$.

2023 JBMO Shortlist, A5

Tags: algebra , inequality

Let $a \geq b \geq 1 \geq c \geq 0$ be real numbers such that $a+b+c=3$. Show that $$3 \left( \frac{a}{b}+\frac{b}{a} \right ) \geq 4c^2+\frac{a^2}{b}+\frac{b^2}{a}$$

1979 IMO Longlists, 29

Tags: inequality , maximization , calculus , optimization , lagrange

Given real numbers $x_1, x_2, \dots , x_n \ (n \geq 2)$, with $x_i \geq \frac 1n \ (i = 1, 2, \dots, n)$ and with $x_1^2+x_2^2+\cdots+x_n^2 = 1$ , find whether the product $P = x_1x_2x_3 \cdots x_n$ has a greatest and/or least value and if so, give these values.

2007 Balkan MO Shortlist, N5

Tags: prime number , inequality , number theory

Let $p \geq 5$ be a prime and let \begin{align*} (p-1)^p +1 = \prod _{i=1}^n q_i^{\beta_i} \end{align*} where $q_i$ are primes. Prove, \begin{align*} \sum_{i=1}^n q_i \beta_i >p^2 \end{align*}

1991 All Soviet Union Mathematical Olympiad, 543

Tags: algebra , inequality , aze junior national mo

Show that $\frac{(x + y + z)^2}{3} \ge x\sqrt{yz} + y\sqrt{zx} + z\sqrt{xy}$ for all non-negative reals $x, y, z$.

2024 Brazil EGMO TST, 1

Tags: inequality , algebra , inequalities

Decide whether there exists a positive real number $ a < 1 $ such that, for any positive real numbers $ x $ and $ y $, the inequality \[ \frac{2xy^2}{x^2 + y^2} \leq (1 - a)x + ay \] holds true.

2016 Balkan MO Shortlist, A2

Tags: algebra , inequality

For all $x,y,z>0$ satisfying $\frac{x}{yz}+\frac{y}{zx}+\frac{z}{xy}\le x+y+z$, prove that $$\frac{1}{x^2+y+z}+\frac{1}{y^2+z+x}+\frac{1}{z^2+x+y} \le 1$$

2023 Belarusian National Olympiad, 10.3

Tags: algebra , inequalities , inequality

Let $a,b,c$ be positive real numbers, that satisfy $abc=1$. Prove the inequality: $$\frac{ab}{1+c}+\frac{bc}{1+a}+\frac{ca}{1+b} \geq \frac{27}{(a+b+c)(3+a+b+c)}$$

2001 China Team Selection Test, 2

Tags: trigonometry , inequality , algebra

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]

2024 Indonesia Regional, 1

Tags: inequality , inequalities , algebra

Given a real number $C\leqslant 2$. Prove that for every positive real number $x,y$ with $xy=1$, the following inequality holds: \[ \sqrt{\frac{x^2+y^2}{2}} + \frac{C}{x+y} \geqslant 1 + \frac{C}{2}.\] [i]Proposed by Fajar Yuliawan, Indonesia[/i]

2021 Winter Stars of Mathematics, 1

Tags: algebra , inequality

Let $a_1,a_2,a_3,a_4$ be positive real numbers satisfying \[\sum_{i<j}a_ia_j=1.\]Prove that \[\sum_{\text{sym}}\frac{a_1a_2}{1+a_3a_4}\geq\frac{6}{7}.\][i]* * *[/i]