Olympiad problems

1974 IMO Longlists, 31

Tags: inequalities

Let $y^{\alpha}=\sum_{i=1}^n x_i^{\alpha}$ where $\alpha \neq 0, y > 0, x_i > 0$ are real numbers, and let $\lambda \neq \alpha$ be a real number. Prove that $y^{\lambda} > \sum_{i=1}^n x_i^{\lambda}$ if $\alpha (\lambda - \alpha) > 0,$ and $y^{\lambda} < \sum_{i=1}^n x_i^{\lambda}$ if $\alpha (\lambda - \alpha) < 0.$

2008 Balkan MO Shortlist, N2

Tags: inequalities , algebra , polynomial , modular arithmetic , induction , number theory , relatively prime

Let $ c$ be a positive integer. The sequence $ a_1,a_2,\ldots$ is defined as follows $ a_1\equal{}c$, $ a_{n\plus{}1}\equal{}a_n^2\plus{}a_n\plus{}c^3$ for all positive integers $ n$. Find all $ c$ so that there are integers $ k\ge1$ and $ m\ge2$ so that $ a_k^2\plus{}c^3$ is the $ m$th power of some integer.

2024 Czech-Polish-Slovak Junior Match, 4

Tags: integer , inequalities

Let $a,b,c$ be integers satisfying $a+b+c=1$ and $ab+bc+ca<abc$. Show that $ab+bc+ca<2abc$.

2005 Iran MO (3rd Round), 3

Tags: inequalities , inradius , circumcircle , trigonometry , geometry

Prove that in acute-angled traingle ABC if $r$ is inradius and $R$ is radius of circumcircle then: \[a^2+b^2+c^2\geq 4(R+r)^2\]

1999 Mongolian Mathematical Olympiad, Problem 2

Tags: inequality , inequalities

Let $a,b,c$ be the real numbers with $a\ge\frac85b>0$ and $a\ge c>0$. Prove the inequality $$\frac45\left(\frac1a+\frac1b\right)+\frac2c\ge\frac{27}2\cdot\frac1{a+b+c}.$$

2017 Saudi Arabia BMO TST, 1

Tags: inequalities , algebra

Let $a, b, c$ be positive real numbers. Prove that $$\frac{a(b^2 + c^2)}{(b + c)(a^2 + bc)} + \frac{b(c^2 + a^2)}{(c + a)(b^2 + ca)} + \frac{c(a^2 + b^2)}{(a + b)(c^2 + ab)} \ge \frac32$$

2012 Turkey Junior National Olympiad, 3

Tags: inequalities

Let $a, b, c$ be positive real numbers satisfying $a^3+b^3+c^3=a^4+b^4+c^4$. Show that \[ \frac{a}{a^2+b^3+c^3}+\frac{b}{a^3+b^2+c^3}+\frac{c}{a^3+b^3+c^2} \geq 1 \]

2011 Vietnam Team Selection Test, 3

Tags: inequalities

Let $n$ be a positive integer $\geq 3.$ There are $n$ real numbers $x_1,x_2,\cdots x_n$ that satisfy: \[\left\{\begin{aligned}&\ x_1\ge x_2\ge\cdots \ge x_n;\\& \ x_1+x_2+\cdots+x_n=0;\\& \ x_1^2+x_2^2+\cdots+x_n^2=n(n-1).\end{aligned}\right.\] Find the maximum and minimum value of the sum $S=x_1+x_2.$

2013 ELMO Shortlist, 9

Tags: function , logarithm , inequalities

Let $a, b, c$ be positive reals, and let $\sqrt[2013]{\frac{3}{a^{2013}+b^{2013}+c^{2013}}}=P$. Prove that \[\prod_{\text{cyc}}\left(\frac{(2P+\frac{1}{2a+b})(2P+\frac{1}{a+2b})}{(2P+\frac{1}{a+b+c})^2}\right)\ge \prod_{\text{cyc}}\left(\frac{(P+\frac{1}{4a+b+c})(P+\frac{1}{3b+3c})}{(P+\frac{1}{3a+2b+c})(P+\frac{1}{3a+b+2c})}\right).\][i]Proposed by David Stoner[/i]

1964 IMO, 2

Tags: geometry , trigonometry , triangle inequality , algebra , inequalities

Suppose $a,b,c$ are the sides of a triangle. Prove that \[ a^2(b+c-a)+b^2(a+c-b)+c^2(a+b-c) \leq 3abc \]

2023 Turkey EGMO TST, 4

Tags: inequalities , algebra , polynomial

Let $n$ be a positive integer and $P,Q$ be polynomials with real coefficients with $P(x)=x^nQ(\frac{1}{x})$ and $P(x) \geq Q(x)$ for all real numbers $x$. Prove that $P(x)=Q(x)$ for all real number $x$.

2010 Singapore Senior Math Olympiad, 3

Tags: inequalities

Given $a_1\ge 1$ and $a_{k+1}\ge a_k+1$ for all $k\ge 1,2,\dots,n$, show that $a_1^3+a_2^3+\dots+a_n^3\ge (a_1+a_2+\dots+a_n)^2$

2021 Taiwan TST Round 2, A

Tags: algebra , inequality , inequalities

[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x . \] [i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x . \]

2001 Romania Team Selection Test, 2

Tags: function , inequalities , algebra

Prove that there is no function $f:(0,\infty )\rightarrow (0,\infty)$ such that \[f(x+y)\ge f(x)+yf(f(x)) \] for every $x,y\in (0,\infty )$.

MathLinks Contest 4th, 7.1

Tags: inequalities , algebra

Let $a, b, c, d$ be positive reals such that $abcd = 1$. Prove that $$\frac{1}{a(b + 1)} +\frac{1}{b(c + 1)} +\frac{1}{c(d + 1)} +\frac{1}{d(a + 1)} \ge 2.$$

1972 Spain Mathematical Olympiad, 2

Tags: geometry , geometric inequalities , analytic geometry , inequalities

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]

2025 Kosovo National Mathematical Olympiad`, P4

Tags: absolute value , algebra , inequalities

Show that for any real numbers $a$ and $b$ different from $0$, the inequality $$\bigg \lvert \frac{a}{b} + \frac{b}{a}+ab \bigg \lvert \geq \lvert a+b+1 \rvert$$ holds. When is equality achieved?

2014 Estonia Team Selection Test, 2

Tags: inequalities , algebra

Let $a, b$ and $c$ be positive real numbers for which $a + b + c = 1$. Prove that $$\frac{a^2}{b^3 + c^4 + 1}+\frac{b^2}{c^3 + a^4 + 1}+\frac{c^2}{a^3 + b^4 + 1} > \frac{1}{5}$$

1987 IMO Longlists, 44

Tags: trigonometry , modular arithmetic , inequalities

Let $\theta_1,\theta_2,\cdots,\theta_n$ be $n$ real numbers such that $\sin \theta_1+\sin \theta_2+\cdots+\sin \theta_n=0$. Prove that \[|\sin \theta_1+2 \sin \theta_2+\cdots +n \sin \theta_n| \leq \left[ \frac{n^2}{4} \right]\]

1998 Israel National Olympiad, 5

Tags: inequalities , algebra

(a) Find two real numebrs $a,b$ such that $|ax+b-\sqrt{x}| \le \frac{1}{24}$ for $1 \le x \le 4$. (b) Prove that the constant $\frac{1}{24}$ cannot be replaced by a smaller one.

2011 N.N. Mihăileanu Individual, 3

Tags: algebra , inequalities

Find $ \inf_{z\in\mathbb{C}} \left( |z^2+z+1|+|z^2-z+1| \right) . $ [i]Gheorghe Andrei[/i] and [i]Doru Constantin Caragea[/i]

1972 Swedish Mathematical Competition, 6

Tags: algebra , inequalities

$a_1,a_2,a_3,\dots$ and $b_1,b_2,b_3,\dots$ are sequences of positive integers. Show that we can find $m < n$ such that $a_m \leq a_n$ and $b_m \leq b_n$.

2024 Moldova EGMO TST, 4

Tags: geometry , geometric inequality , inequalities

In the acute-angled triangle $ABC$, on the lines $BC$, $AC$, $AB$ we consider the points $D$, $E$ and, respectively, $F$, such that $AD\perp AC, BE\perp AB, CF\perp AC$. Let the point $A', B', C'$ be such that $\{A'\}=BC\cap EF, \{B'\}=AC\cap DF, \{C'\}=AB\cap DE$. Prove that the following inequality is true $$\frac{A'F}{A'E} \cdot \frac{B'D}{B'F} \cdot \frac{C'E}{C'D}\geq8$$

2008 Germany Team Selection Test, 1

Tags: inequalities , algebra

Determine $ Q \in \mathbb{R}$ which is so big that a sequence with non-negative reals elements $ a_1 ,a_2, \ldots$ which satisfies the following two conditions: [b](i)[/b] $ \forall m,n \geq 1$ we have $ a_{m \plus{} n} \leq 2 \left(a_m \plus{} a_n \right)$ [b](ii)[/b] $ \forall k \geq 0$ we have $ a_{2^k} \leq \frac {1}{(k \plus{} 1)^{2008}}$ such that for each sequence element we have the inequality $ a_n \leq Q.$

2010 Belarus Team Selection Test, 5.3

Tags: function , inequalities , algebra , functional inequality

Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that \[f\left(x-f(y)\right)>yf(x)+x\] [i]Proposed by Igor Voronovich, Belarus[/i]