Olympiad problems

2013 IFYM, Sozopol, 4

Tags: inequalities

Let $a,b,c$ be real numbers for which $a+b+c+d=19$ and $a^2+b^2+c^2+d^2=91$. Find the maximal value of $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}$.

2003 Baltic Way, 2

Tags: number theory , quadratic , inequalities , algebra

Prove that any real solution of $x^3+px+q=0$, where $p,q$ are real numbers, satisfies the inequality $4qx\le p^2$.

2019 Iran Team Selection Test, 6

Tags: inequalities

$x,y$ and $z$ are real numbers such that $x+y+z=xy+yz+zx$. Prove that $$\frac{x}{\sqrt{x^4+x^2+1}}+\frac{y}{\sqrt{y^4+y^2+1}}+\frac{z}{\sqrt{z^4+z^2+1}}\geq \frac{-1}{\sqrt{3}}.$$ [i]Proposed by Navid Safaei[/i]

2018 Poland - Second Round, 2

Tags: divisor , inequalities , combinatorics , number theory

Let $n$ be a positive integer, which gives remainder $4$ of dividing by $8$. Numbers $1 = k_1 < k_2 < ... < k_m = n$ are all positive diivisors of $n$. Show that if $i \in \{ 1, 2, ..., m - 1 \}$ isn't divisible by $3$, then $k_{i + 1} \le 2k_{i}$.

2008 Federal Competition For Advanced Students, P1, 3

Tags: algebra , inequalities , sequence , recurrence relation

Let $p > 1$ be a natural number. Consider the set $F_p$ of all non-constant sequences of non-negative integers that satisfy the recursive relation $a_{n+1} = (p+1)a_n - pa_{n-1}$ for all $n > 0$. Show that there exists a sequence ($a_n$) in $F_p$ with the property that for every other sequence ($b_n$) in $F_p$, the inequality $a_n \le b_n$ holds for all $n$.

2003 Federal Competition For Advanced Students, Part 2, 2

Tags: geometry , rotation , geometric transformation , inequalities

Let $a, b, c$ be nonzero real numbers for which there exist $\alpha, \beta, \gamma \in\{-1, 1\}$ with $\alpha a + \beta b + \gamma c = 0$. What is the smallest possible value of \[\left( \frac{a^3+b^3+c^3}{abc}\right)^2 ?\]

1982 IMO Longlists, 48

Tags: inequalities , algebra , combinatorial geometry , complex number , induction

Given a finite sequence of complex numbers $c_1, c_2, \ldots , c_n$, show that there exists an integer $k$ ($1 \leq k \leq n$) such that for every finite sequence $a_1, a_2, \ldots, a_n$ of real numbers with $1 \geq a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$, the following inequality holds: \[\left| \sum_{m=1}^n a_mc_m \right| \leq \left| \sum_{m=1}^k c_m \right|.\]

2011 Balkan MO, 2

Tags: inequalities , three variable inequality , algebra

Given real numbers $x,y,z$ such that $x+y+z=0$, show that \[\dfrac{x(x+2)}{2x^2+1}+\dfrac{y(y+2)}{2y^2+1}+\dfrac{z(z+2)}{2z^2+1}\ge 0\] When does equality hold?

1981 Bundeswettbewerb Mathematik, 4

Tags: logarithm , induction , number theory , floor function , inequalities , ceiling function

Let $X$ be a non empty subset of $\mathbb{N} = \{1,2,\ldots \}$. Suppose that for all $x \in X$, $4x \in X$ and $\lfloor \sqrt{x} \rfloor \in X$. Prove that $X=\mathbb{N}$.

2021/2022 Tournament of Towns, P6

Tags: inequalities , algebra

Prove that for any positive integers $a_1, a_2, \ldots , a_n$ the following inequality holds true: \[\left\lfloor\frac{a_1^2}{a_2}\right\rfloor+\left\lfloor\frac{a_2^2}{a_3}\right\rfloor+\cdots+\left\lfloor\frac{a_n^2}{a_1}\right\rfloor\geqslant a_1+a_2+\cdots+a_n.\] [i]Maxim Didin[/i]

2012 China Northern MO, 2

Tags: inequalities , algebra

Positive integers $x_1,x_2,...,x_n$ ($n \in N_+$) satisfy $x_1^2 +x_2^2+...+x_n^2=111$, find the maximum possible value of $S =\frac{x_1 +x_2+...+x_n}{n}$.

2012 AIME Problems, 6

Tags: geometry , triangle inequality , inequalities , reflection , geometric transformation

Let $z = a + bi$ be the complex number with $|z| = 5$ and $b > 0$ such that the distance between $(1 + 2i)z^3$ and $z^5$ is maximized, and let $z^4 = c + di$. Find $c+d$.

1995 Polish MO Finals, 1

Tags: combinatorics , inequalities

How many subsets of $\{1, 2, ... , 2n\}$ do not contain two numbers with sum $2n+1$?

2012 Iran Team Selection Test, 3

Tags: analytic geometry , geometry , geometric transformation , complex number , limit , reflection , inequalities

The pentagon $ABCDE$ is inscirbed in a circle $w$. Suppose that $w_a,w_b,w_c,w_d,w_e$ are reflections of $w$ with respect to sides $AB,BC,CD,DE,EA$ respectively. Let $A'$ be the second intersection point of $w_a,w_e$ and define $B',C',D',E'$ similarly. Prove that \[2\le \frac{S_{A'B'C'D'E'}}{S_{ABCDE}}\le 3,\] where $S_X$ denotes the surface of figure $X$. [i]Proposed by Morteza Saghafian, Ali khezeli[/i]

2018 Stars of Mathematics, 3

Tags: algebra , inequalities , function , floor function

Given a positive integer $n$, determine the largest integer $M$ satisfying $$\lfloor \sqrt{a_1}\rfloor + ... + \lfloor \sqrt{a_n} \rfloor \ge \lfloor\sqrt{ a_1 + ... + a_n +M \cdot min(a_1,..., a_n)}\rfloor $$ for all non-negative integers $a_1,...., a_n$. S. Berlov, A. Khrabrov

1994 All-Russian Olympiad Regional Round, 10.7

Tags: geometry , inequalities

In a convex pentagon $ ABCDE$ side $ AB$ is perpendicular to $ CD$ and side $ BC$ is perpendicular to $ DE$. Prove that if $ AB \equal{} AE \equal{} ED \equal{} 1$, then $ BC \plus{} CD < 1$.

2005 China National Olympiad, 1

Tags: inequalities , trigonometry

Suppose $\theta_{i}\in(-\frac{\pi}{2},\frac{\pi}{2}), i = 1,2,3,4$. Prove that, there exist $x\in \mathbb{R}$, satisfying two inequalities \begin{eqnarray*} \cos^2\theta_1\cos^2\theta_2-(\sin\theta\sin\theta_2-x)^2 &\geq& 0, \\ \cos^2\theta_3\cos^2\theta_4-(\sin\theta_3\sin\theta_4-x)^2 & \geq & 0 \end{eqnarray*} if and only if \[ \sum^4_{i=1}\sin^2\theta_i\leq2(1+\prod^4_{i=1}\sin\theta_i + \prod^4_{i=1}\cos\theta_i). \]

2010 Victor Vâlcovici, 2

Tags: am-gm , trigonometric inequality , inequalities

$ \sum_{cyc}\frac{1}{\left(\text{tg} y+\text{tg} z\right) \text{cos}^2 x} \ge 3, $ for any $ x,y,z\in (0,\pi/2) $ [i]Carmen[/i] and [i]Viorel Botea[/i]

1996 Irish Math Olympiad, 2

Tags: inequalities

Show that for every positive integer $ n$, $ 2^{\frac{1}{2}} \cdot 4^{\frac{1}{4}} \cdot 8^{\frac{1}{8}} \cdot ... \cdot (2^n)^{\frac{1}{2^n}}<4$.

2008 Bosnia Herzegovina Team Selection Test, 1

Tags: derivative , geometry , trigonometry , inequalities , calculus

Prove that in an isosceles triangle $ \triangle ABC$ with $ AC\equal{}BC\equal{}b$ following inequality holds $ b> \pi r$, where $ r$ is inradius.

1992 Dutch Mathematical Olympiad, 4

Tags: inequalities

For every positive integer $ n$, we define $ n?$ as $ 1?\equal{}1$ and $ n?\equal{}\frac{n}{(n\minus{}1)?}$ for $ n \ge 2$. Prove that $ \sqrt{1992}<1992?<\frac{4}{3} \sqrt{1992}.$

2014 Kazakhstan National Olympiad, 3

Tags: inequalities , number theory

Prove that, for all $n\in\mathbb{N}$, on $ [n-4\sqrt{n}, n+4\sqrt{n}]$ exists natural number $k=x^3+y^3$ where $x$, $y$ are nonnegative integers.

2017 Taiwan TST Round 3, 1

Tags: inequalities

There are $m$ real numbers $x_i \geq 0$ ($i=1,2,...,m$), $n \geq 2$, $\sum_{i=1}^{m} x_i=S$. Prove that\\ \[ \sum_{i=1}^{m} \sqrt[n]{\frac{x_i}{S-x_i}} \geq 2, \] The equation holds if and only if there are exactly two of $x_i$ are equal(not equal to $0$), and the rest are equal to $0$.

2002 Romania National Olympiad, 1

Tags: inequalities

Let $ab+bc+ca=1$. Show that \[\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\ge\sqrt{3}+\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\]

2010 Junior Balkan Team Selection Tests - Romania, 1

Tags: sum , combinatorics , inequalities

We consider on a circle a finite number of real numbers with the sum strictly greater than $0$. Of all the sums that have as terms numbers on consecutive positions on the circle, let $S$ be the largest sum and $s$ the smallest sum. Show that $S + s> 0$.