Olympiad problems

1995 North Macedonia National Olympiad, 2

Let $ a, $ $ b $, and $ c $ be sides in a triangle, a $ h_a, $ $ h_b $, and $ h_c $ are the corresponding altitudes. Prove that $h ^ 2_a + h ^ 2_b + h ^ 2_c \leq \frac{3}{4} (a ^ 2 + b ^ 2 + c ^ 2). $ When is the equation valid?

1985 Federal Competition For Advanced Students, P2, 2

Tags: inequalities

For $ n \in \mathbb{N}$, let $ f(n)\equal{}1^n\plus{}2^{n\minus{}1}\plus{}3^{n\minus{}2}\plus{}...\plus{}n^1$. Determine the minimum value of: $ \frac{f(n\plus{}1)}{f(n)}.$

2022 Cyprus JBMO TST, 3

Tags: inequalities , am-gm , cauchy-schwarz inequality

If $a,b,c$ are positive real numbers with $abc=1$, prove that (a) \[2\left(\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\right) \geqslant \frac{9}{ab+bc+ca}\] (b)\[2\left(\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\right) \geqslant \frac{9}{a^2 b+b^2 c+c^2 a}\]

1995 Israel Mathematical Olympiad, 3

Tags: inequalities , sum , algebra

If $k$ and $n$ are positive integers, prove the inequality $$\frac{1}{kn} +\frac{1}{kn+1} +...+\frac{1}{(k+1)n-1} \ge n \left(\sqrt[n]{\frac{k+1}{k}}-1\right)$$

2000 USA Team Selection Test, 3

Tags: inequalities , number theory , fractional part

Let $p$ be a prime number. For integers $r, s$ such that $rs(r^2 - s^2)$ is not divisible by $p$, let $f(r, s)$ denote the number of integers $n \in \{1, 2, \ldots, p - 1\}$ such that $\{rn/p\}$ and $\{sn/p\}$ are either both less than $1/2$ or both greater than $1/2$. Prove that there exists $N > 0$ such that for $p \geq N$ and all $r, s$, \[ \left\lceil \frac{p-1}{3} \right\rceil \le f(r, s) \le \left\lfloor \frac{2(p-1)}{3} \right\rfloor. \]

2008 National Olympiad First Round, 29

Tags: geometry , parallelogram , inequalities , asymptote

$[AB]$ and $[CD]$ are not parallel in the convex quadrilateral $ABCD$. Let $E$ and $F$ be the midpoints of $[AD]$ and $[BC]$, respectively. If $|CD|=12$, $|AB|=22$, and $|EF|=x$, what is the sum of integer values of $x$? $ \textbf{(A)}\ 110 \qquad\textbf{(B)}\ 114 \qquad\textbf{(C)}\ 118 \qquad\textbf{(D)}\ 121 \qquad\textbf{(E)}\ \text{None of the above} $

2010 Contests, 3

Tags: inequalities , n-variable inequality

Let $x_1, \ldots , x_{100}$ be nonnegative real numbers such that $x_i + x_{i+1} + x_{i+2} \leq 1$ for all $i = 1, \ldots , 100$ (we put $x_{101 } = x_1, x_{102} = x_2).$ Find the maximal possible value of the sum $S = \sum^{100}_{i=1} x_i x_{i+2}.$ [i]Proposed by Sergei Berlov, Ilya Bogdanov, Russia[/i]

1986 IMO Longlists, 30

Tags: inequalities , geometry , 3d geometry , icosahedron , tetrahedron , octahedron , dodecahedron

Prove that a convex polyhedron all of whose faces are equilateral triangles has at most $30$ edges.

2002 Taiwan National Olympiad, 4

Tags: inequalities

Let $0<x_{1},x_{2},x_{3},x_{4}\leq\frac{1}{2}$ are real numbers. Prove that $\frac{x_{1}x_{2}x_{3}x_{4}}{(1-x_{1})(1-x_{2})(1-x_{3})(1-x_{4})}\leq\frac{x_{1}^{4}+x_{2}^{4}+x_{3}^{4}+x_{4}^{4}}{(1-x_{1})^{4}+(1-x_{2})^{4}+(1-x_{3})^{4}+(1-x_{4})^{4}}$.

1989 Tournament Of Towns, (230) 4

Tags: number theory , inequalities , combinatorics

Given the natural number N, consider triples of different positive integers $(a, b, c)$ such that $a + b + c = N$. Take the largest possible system of these triples such that no two triples of the system have any common elements. Denote the number of triples of this system by $K(N)$. Prove that: (a) $K(N) >\frac{N}{6}-1$ (b) $K(N) <\frac{2N}{9}$ (L.D. Kurliandchik, Leningrad)

2018 Costa Rica - Final Round, A1

Tags: algebra , inequalities , min

If $x \in R-\{-7\}$, determine the smallest value of the expression $$\frac{2x^2 + 98}{(x + 7)^2}$$

1984 IMO Longlists, 19

Tags: function , inequalities

Let $ABC$ be an isosceles triangle with right angle at point $A$. Find the minimum of the function $F$ given by \[F(M) = BM +CM-\sqrt{3}AM\]

2024 VJIMC, 1

Tags: inequalities , function , calculus , integration , derivative

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuously differentiable function. Prove that \[\left\vert f(1)-\int_0^1 f(x) dx\right\vert \le \frac{1}{2} \max_{x \in [0,1]} \vert f'(x)\vert.\]

1970 IMO, 2

Tags: inequalities , algebra , decimal representation , digit

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 $A'B<AB'$.

2003 Cuba MO, 5

Tags: algebra , inequalities

Let $a_1, a_2, ..., a_9$ be non-negative real numbers such that $a_1 = a_9 = 0$ and at least one of the remaining terms is different from $0$. a) Prove that for some $i$ $(i = 2, ..., 8$) ,holds that $a_{i-1} + a_{i+1} < 2a_i.$ b) Will the previous statement be true, if we change the number $2$ for $1.9$ in the inequality?

PEN O Problems, 43

Tags: algebra , polynomial , induction , inequalities

Is it possible to find a set $A$ of eleven positive integers such that no six elements of $A$ have a sum which is divisible by $6$?

2017 Serbia Team Selection Test, 5

Tags: algebra , inequalities , sequence

Let $n \geq 2$ be a positive integer and $\{x_i\}_{i=0}^n$ a sequence such that not all of its elements are zero and there is a positive constant $C_n$ for which: (i) $x_1+ \dots +x_n=0$, and (ii) for each $i$ either $x_i\leq x_{i+1}$ or $x_i\leq x_{i+1} + C_n x_{i+2}$ (all indexes are assumed modulo $n$). Prove that a) $C_n\geq 2$, and b) $C_n=2$ if and only $2 \mid n$.

2024 Benelux, 1

Tags: algebra inequality , algebra , inequalities , sequence

Let $a_0,a_1,\dots,a_{2024}$ be real numbers such that $\left|a_{i+1}-a_i\right| \le 1$ for $i=0,1,\dots,2023$. a) Find the minimum possible value of $$a_0a_1+a_1a_2+\dots+a_{2023}a_{2024}$$ b) Does there exist a real number $C$ such that $$a_0a_1-a_1a_2+a_2a_3-a_3a_4+\dots+a_{2022}a_{2023}-a_{2023}a_{2024} \ge C$$ for all real numbers $a_0,a_1,\dots,a_2024$ such that $\left|a_{i+1}-a_i\right| \le 1$ for $i=0,1,\dots,2023$.

2008 Cuba MO, 1

Tags: algebra , inequalities , trinomial

Given a polynomial of degree $2$, $p(x) = ax^2 +bx+c$ define the function $$S(p) = (a -b)^2 + (b - c)^2 + (c - a)^2.$$ Determine the real number$ r$such that, for any polynomial $p(x)$ of degree $2$ with real roots, holds $S(p) \ge ra^2$

2017 Iran Team Selection Test, 1

Tags: algebra , inequalities

Let $a,b,c,d$ be positive real numbers with $a+b+c+d=2$. Prove the following inequality: $$\frac{(a+c)^{2}}{ad+bc}+\frac{(b+d)^{2}}{ac+bd}+4\geq 4\left ( \frac{a+b+1}{c+d+1}+\frac{c+d+1}{a+b+1} \right).$$ [i]Proposed by Mohammad Jafari[/i]

1989 Tournament Of Towns, (213) 1

Tags: algebra , inequalities

The positive numbers $a, b, c$ and $d$ satisfy $a\le b\le c\le d$ and $a + b + c + d \le 1$ . Prove that $a^2 + 3b^2 + 5c^2 + 7 d^2 \ge 1$ .

2017 Poland - Second Round, 3

Tags: inequalities , n-variable inequality

Let $x_1 \le x_2 \le \ldots \le x_{2n-1}$ be real numbers whose arithmetic mean equals $A$. Prove that $$2\sum_{i=1}^{2n-1}\left( x_{i}-A\right)^2 \ge \sum_{i=1}^{2n-1}\left( x_{i}-x_{n}\right)^2.$$

2006 Junior Balkan Team Selection Tests - Romania, 2

Tags: inequalities , function , algebra

Prove that for all positive real numbers $a,b,c$ the following inequality holds \[ \left( \frac ab + \frac bc + \frac ca \right)^2 \geq \frac 32 \cdot \left ( \frac{a+b}c + \frac{b+c}a + \frac{c+a} b \right) . \]

2012 AMC 12/AHSME, 23

Tags: algebra , polynomial , inequalities , triangle inequality

Consider all polynomials of a complex variable, $P(z)=4z^4+az^3+bz^2+cz+d$, where $a, b, c$ and $d$ are integers, $0 \le d \le c \le b \le a \le 4$, and the polynomial has a zero $z_0$ with $|z_0|=1$. What is the sum of all values $P(1)$ over all the polynomials with these properties? $ \textbf{(A)}\ 84\qquad\textbf{(B)}\ 92\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 108 \qquad\textbf{(E)}\ 120 $

2006 International Zhautykov Olympiad, 3

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

Let $ ABCDEF$ be a convex hexagon such that $ AD \equal{} BC \plus{} EF$, $ BE \equal{} AF \plus{} CD$, $ CF \equal{} DE \plus{} AB$. Prove that: \[ \frac {AB}{DE} \equal{} \frac {CD}{AF} \equal{} \frac {EF}{BC}. \]