Olympiad problems

2007 District Olympiad, 2

Tags: calculus , integration , inequalities , function , derivative , probability , expected value

Let $f : \left[ 0, 1 \right] \to \mathbb R$ be a continuous function and $g : \left[ 0, 1 \right] \to \left( 0, \infty \right)$. Prove that if $f$ is increasing, then \[\int_{0}^{t}f(x) g(x) \, dx \cdot \int_{0}^{1}g(x) \, dx \leq \int_{0}^{t}g(x) \, dx \cdot \int_{0}^{1}f(x) g(x) \, dx .\]

1965 AMC 12/AHSME, 40

Tags: polynomial , inequalities , algebra

Let $ n$ be the number of integer values of $ x$ such that $ P \equal{} x^4 \plus{} 6x^3 \plus{} 11x^2 \plus{} 3x \plus{} 31$ is the square of an integer. Then $ n$ is: $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 0$

2004 IMO Shortlist, 8

Tags: inequalities , graph theory , extremal combinatorics

For a finite graph $G$, let $f(G)$ be the number of triangles and $g(G)$ the number of tetrahedra formed by edges of $G$. Find the least constant $c$ such that \[g(G)^3\le c\cdot f(G)^4\] for every graph $G$. [i]Proposed by Marcin Kuczma, Poland [/i]

2012 India IMO Training Camp, 2

Tags: modular arithmetic , inequalities , number theory

Let $0<x<y<z<p$ be integers where $p$ is a prime. Prove that the following statements are equivalent: $(a) x^3\equiv y^3\pmod p\text{ and }x^3\equiv z^3\pmod p$ $(b) y^2\equiv zx\pmod p\text{ and }z^2\equiv xy\pmod p$

2023 District Olympiad, P1

Tags: real analysis , integral , inequalities

Let $f:[-\pi/2,\pi/2]\to\mathbb{R}$ be a twice differentiable function which satisfies \[\left(f''(x)-f(x)\right)\cdot\tan(x)+2f'(x)\geqslant 1,\]for all $x\in(-\pi/2,\pi/2)$. Prove that \[\int_{-\pi/2}^{\pi/2}f(x)\cdot \sin(x) \ dx\geqslant \pi-2.\]

2014 IFYM, Sozopol, 6

Tags: inequalities , algebra

The positive real numbers $a,b,c$ are such that $21ab+2bc+8ca\leq 12$. Find the smallest value of $\frac{1}{a}+\frac{2}{b}+\frac{3}{c}$.

2015 Costa Rica - Final Round, 5

Tags: algebra , inequalities , functional inequality , functional

Let $f: N^+ \to N^+$ be a function that satisfies that $$kf(n) \le f (kn) \le kf(n)+ k- 1, \,\, \forall k,n \in N^+$$ Prove that $$f(a) + f(b) \le f (a + b) \le f(a) + f(b) + 1, \,\, \forall a, b \in N^+$$

2021 Moldova Team Selection Test, 9

Tags: inequalities , algebra

Positive real numbers $a$, $b$, $c$ satisfy $a+b+c=1$. Find the smallest possible value of $$E(a,b,c)=\frac{a^3}{1-a^2}+\frac{b^3}{1-b^2}+\frac{c^3}{1-c^2}.$$

II Soros Olympiad 1995 - 96 (Russia), 10.1

Tags: trigonometry , algebra , inequalities

Find the largest and smallest value of the function $$y=\sqrt{7+5\cos x}-\cos x.$$

2003 Irish Math Olympiad, 5

Tags: function , inequalities , algebra

show that thee is no function f definedonthe positive real numbes such that : $f(y) > (y-x)f(x)^2$

1966 Swedish Mathematical Competition, 2

Tags: inequalities , algebra

$a_1 + a_2 + ... + a_n = 0$, for some $k$ we have $a_j \le 0$ for $j \le k$ and $a_j \ge 0$ for $j > k$. If ai are not all $0$, show that $a_1 + 2a_2 + 3a_3 + ... + na_n > 0$.

2015 Iran MO (3rd round), 6

Tags: algebra , inequalities

$a_1,a_2,\dots ,a_n>0$ are positive real numbers such that $\sum_{i=1}^{n} \frac{1}{a_i}=n$ prove that: $\sum_{i<j} \left(\frac{a_i-a_j}{a_i+a_j}\right)^2\le\frac{n^2}{2}\left(1-\frac{n}{\sum_{i=1}^{n}a_i}\right)$

2009 Dutch IMO TST, 3

Tags: inequalities , 3-variable inequality , three variable inequality

Let $a, b$ and $c$ be positive reals such that $a + b + c \ge abc$. Prove that $a^2 + b^2 + c^2 \ge \sqrt3 abc$.

1962 Putnam, A4

Tags: inequalities , derivative

Assume that $|f(x)|\leq 1$ and $|f''(x)|\leq 1$ for all $x$ on an interval of length at least $2.$ Show that $|f'(x)|\leq 2 $ on the interval.

2005 Kazakhstan National Olympiad, 2

Tags: inequalities

Prove that \[ab+bc+ca\ge 2(a+b+c)\] where $a,b,c$ are positive reals such that $a+b+c+2=abc$.

1988 Romania Team Selection Test, 4

Tags: induction , inequalities

Prove that for all positive integers $0<a_1<a_2<\cdots <a_n$ the following inequality holds: \[ (a_1+a_2+\cdots + a_n)^2 \leq a_1^3+a_2^3 + \cdots + a_n^3 . \] [i]Viorel Vajaitu[/i]

2019 Thailand TST, 3

Tags: inequalities

Find the maximal value of \[S = \sqrt[3]{\frac{a}{b+7}} + \sqrt[3]{\frac{b}{c+7}} + \sqrt[3]{\frac{c}{d+7}} + \sqrt[3]{\frac{d}{a+7}},\] where $a$, $b$, $c$, $d$ are nonnegative real numbers which satisfy $a+b+c+d = 100$. [i]Proposed by Evan Chen, Taiwan[/i]

2014 Hanoi Open Mathematics Competitions, 2

Tags: binomial , inequalities , combinatorics , combination

How many integers are there in $\{0,1, 2,..., 2014\}$ such that $C^x_{2014} \ge C^{999}{2014}$ ? (A): $15$, (B): $16$, (C): $17$, (D): $18$, (E) None of the above. Note: $C^{m}_{n}$ stands for $\binom {m}{n}$

2019 Saudi Arabia JBMO TST, 2

Tags: inequalities , algebra , sum

Let $a, b, c$ be positive real numbers. Prove that $$\frac{a^3}{a^2 + bc}+\frac{b^3}{b^2 + ca}+\frac{c^3}{c^2 + ab} \ge \frac{(a^2 + b^2 + c^2)(ab + bc + ca)}{a^3 + b^3 + c^3 + 3abc}$$

2012 Czech And Slovak Olympiad IIIA, 3

Tags: algebra , combinatorics , inequalities

Prove that there are two numbers $u$ and $v$, between any $101$ real numbers that apply $100 |u - v| \cdot |1 - uv| \le (1 + u^2)(1 + v^2)$

2014 USA TSTST, 3

Tags: algebra , polynomial , inequalities , induction , trigonometry , calculus , derivative

Find all polynomials $P(x)$ with real coefficients that satisfy \[P(x\sqrt{2})=P(x+\sqrt{1-x^2})\]for all real $x$ with $|x|\le 1$.

2009 Mathcenter Contest, 5

Tags: inequalities , algebra

Let $a$ and $b$ be real numbers, where $a \not= 0$ and $a \not= b$ and all the roots of the equation $ax^{3}-x^{2}+bx-1 = 0$ is a real and positive number. Find the smallest possible value of $P = \dfrac{5a^{2}-3ab+2}{a^{2}(b-a)}$. [i](Heir of Ramanujan)[/i]

1970 IMO Longlists, 10

Tags: inequalities , trigonometry

In $\triangle ABC$, prove that $1< \sum_{cyc}{\cos A}\le \frac{3}{2}$.

2020 Regional Olympiad of Mexico Southeast, 5

Tags: tangent , geometry , circumcircle , inequality , inequalities

Let $ABC$ an acute triangle with $\angle BAC\geq 60^\circ$ and $\Gamma$ it´s circumcircule. Let $P$ the intersection of the tangents to $\Gamma$ from $B$ and $C$. Let $\Omega$ the circumcircle of the triangle $BPC$. The bisector of $\angle BAC$ intersect $\Gamma$ again in $E$ and $\Omega$ in $D$, in the way that $E$ is between $A$ and $D$. Prove that $\frac{AE}{ED}\leq 2$ and determine when equality holds.

2005 Austrian-Polish Competition, 10

Tags: power , inequalities

Determine all pairs $(k,n)$ of non-negative integers such that the following inequality holds $\forall x,y>0$: \[1+ \frac{y^n}{x^k} \geq \frac{(1+y)^n}{(1+x)^k}.\]