Olympiad problems

1986 IMO Longlists, 78

If $T$ and $T_1$ are two triangles with angles $x, y, z$ and $x_1, y_1, z_1$, respectively, prove the inequality \[\frac{\cos x_1}{\sin x}+\frac{\cos y_1}{\sin y}+\frac{\cos z_1}{\sin z} \leq \cot x+\cot y+\cot z.\]

2004 India IMO Training Camp, 1

Tags: geometry , trigonometry , inequalities

Prove that in any triangle $ABC$, \[ 0 < \cot { \left( \frac{A}{4} \right)} - \tan{ \left( \frac{B}{4} \right) } - \tan{ \left( \frac{C}{4} \right) } - 1 < 2 \cot { \left( \frac{A}{2} \right) }. \]

2022 VJIMC, 1

Tags: inequalities , calculus , function , integration

Determine whether there exists a differentiable function $f:[0,1]\to\mathbb R$ such that $$f(0)=f(1)=1,\qquad|f'(x)|\le2\text{ for all }x\in[0,1]\qquad\text{and}\qquad\left|\int^1_0f(x)dx\right|\le\frac12.$$

1985 IMO Longlists, 24

Tags: trigonometry , inequalities , algebra

Let $d \geq 1$ be an integer that is not the square of an integer. Prove that for every integer $n \geq 1,$ \[(n \sqrt d +1) \cdot | \sin(n \pi \sqrt d )| \geq 1\]

2013 IMC, 1

Tags: vector , linear algebra , inequalities , matrix

Let $\displaystyle{A}$ and $\displaystyle{B}$ be real symmetric matrixes with all eigenvalues strictly greater than $\displaystyle{1}$. Let $\displaystyle{\lambda }$ be a real eigenvalue of matrix $\displaystyle{{\rm A}{\rm B}}$. Prove that $\displaystyle{\left| \lambda \right| > 1}$. [i]Proposed by Pavel Kozhevnikov, MIPT, Moscow.[/i]

1951 Moscow Mathematical Olympiad, 188

Tags: inequalities , algebra

Prove that $x^{12} - x^9 + x^4 - x + 1 > 0$ for all $x$.

2018 Cyprus IMO TST, 3

Tags: inequalities

Find all triples $(\alpha, \beta, \gamma)$ of positive real numbers for which the expression $$K = \frac{\alpha+3 \gamma}{\alpha + 2\beta + \gamma} + \frac{4\beta}{\alpha+\beta+2\gamma} - \frac{8 \gamma}{\alpha+ \beta + 3\gamma}$$ obtains its minimum value.

2019 Junior Balkan Team Selection Tests - Romania, 2

Tags: inequalities , algebra

If $x, y$ and $z$ are real numbers such that $x^2 + y^2 + z^2 = 2$, prove that $x + y + z \le xyz + 2$.

2011 Albania National Olympiad, 3

Tags: inequalities , geometry , exterior angle , trigonometry

In a convex quadrilateral $ABCD$ ,$\angle ABC$ and $\angle BCD$ are $\geq 120^o$. Prove that $|AC|$ + $|BD| \geq |AB|+|BC|+|CD|$. (With $|XY|$ we understand the length of the segment $XY$).

2011 China Team Selection Test, 1

Tags: inequalities

Let $n\geq 3$ be an integer. Find the largest real number $M$ such that for any positive real numbers $x_1,x_2,\cdots,x_n$, there exists an arrangement $y_1,y_2,\cdots,y_n$ of real numbers satisfying \[\sum_{i=1}^n \frac{y_i^2}{y_{i+1}^2-y_{i+1}y_{i+2}+y_{i+2}^2}\geq M,\] where $y_{n+1}=y_1,y_{n+2}=y_2$.

2017 Hanoi Open Mathematics Competitions, 11

Tags: inequalities , geometric inequality , angle , equilateral , geometry

Let $ABC$ be an equilateral triangle, and let $P$ stand for an arbitrary point inside the triangle. Is it true that $| \angle PAB - \angle PAC| \ge | \angle PBC - \angle PCB|$ ?

2014 Junior Balkan Team Selection Tests - Romania, 2

Tags: algebra , inequalities

Determine all real numbers $x, y, z \in (0, 1)$ that satisfy simultaneously the conditions: $(x^2 + y^2)\sqrt{1- z^2}\ge z$ $(y^2 + z^2)\sqrt{1- x^2}\ge x$ $(z^2 + x^2)\sqrt{1- y^2}\ge y$

2019 LIMIT Category B, Problem 12

Tags: inequalities

The system of inequalities $$a-b^2\ge\frac14$$$$b-c^2\ge\frac14$$$$c-d^2\ge\frac14$$$$d-a^2\ge\frac14$$where $a,b,c,d$ are real numbers has $\textbf{(A)}~\text{no solutions}$ $\textbf{(B)}~\text{exactly one solution}$ $\textbf{(C)}~\text{exactly two solutions}$ $\textbf{(D)}~\text{infinitely many solutions}$

2007 Today's Calculation Of Integral, 232

Tags: calculus , integration , trigonometry , calculus computations , inequalities

For $ f(x)\equal{}1\minus{}\sin x$, let $ g(x)\equal{}\int_0^x (x\minus{}t)f(t)\ dt.$ Show that $ g(x\plus{}y)\plus{}g(x\minus{}y)\geq 2g(x)$ for any real numbers $ x,\ y.$

2014 Estonia Team Selection Test, 3

Tags: max , area , inequalities , algebra

Three line segments, all of length $1$, form a connected figure in the plane. Any two different line segments can intersect only at their endpoints. Find the maximum area of the convex hull of the figure.

2007 Middle European Mathematical Olympiad, 1

Tags: inequalities

Let $ a,b,c,d$ be real numbers which satisfy $ \frac{1}{2}\leq a,b,c,d\leq 2$ and $ abcd\equal{}1$. Find the maximum value of \[ \left(a\plus{}\frac{1}{b}\right)\left(b\plus{}\frac{1}{c}\right)\left(c\plus{}\frac{1}{d}\right)\left(d\plus{}\frac{1}{a}\right).\]

1999 Romania Team Selection Test, 5

Tags: inequalities

Let $x_1,x_2,\ldots,x_n$ be distinct positive integers. Prove that \[ x_1^2+x_2^2 + \cdots + x_n^2 \geq \frac {2n+1}3 ( x_1+x_2+\cdots + x_n). \] [i]Laurentiu Panaitopol[/i]

2011 IFYM, Sozopol, 3

Tags: inequalities , maximum value , algebra

If $x$ and $y$ are real numbers, determine the greatest possible value of the expression $\frac{(x+1)(y+1)(xy+1)}{(x^2+1)(y^2+1)}$.

1965 Kurschak Competition, 1

Tags: diophantine , number theory , inequalities

What integers $a, b, c$ satisfy $a^2 + b^2 + c^2 + 3 < ab + 3b + 2c$ ?

2008 China Northern MO, 6

Tags: inequalities , trigonometry

Let $a, b, c$ be side lengths of a right triangle and $c$ be the length of the hypotenuse .Find the minimum value of $\frac{a^3+b^3+c^3}{abc}$.

2019 Jozsef Wildt International Math Competition, W. 69

Tags: angle bisector , triangle , geometric inequality , circumradius , inradius , inequalities , geometry

Denote $\overline{w_a}, \overline{w_b}, \overline{w_c}$ the external angle-bisectors in triangle $ABC$, prove that $$\sum \limits_{cyc} \frac{1}{w_a}\leq \sqrt{\frac{(s^2 - r^2 - 4Rr)(8R^2 - s^2 - r^2 - 2Rr)}{8s^2R^2r}}$$

1980 Dutch Mathematical Olympiad, 1

Tags: algebra , inequalities , polynomial

$f(x) = x^3-ax+1$ , $a \in R$ has three different zeros in $R$. Prove that for the zero $x_o$ with the smallest absolute value holds: $\frac{1}{a}< x_0 < \frac{2}{a}$

2008 Putnam, B3

Tags: geometry , symmetry , 3d geometry , sphere , inequalities , trigonometry

What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1?

2014 Contests, 1

Tags: inequalities

Suppose $x$, $y$, $z$ are positive numbers such that $x+y+z=1$. Prove that \[ \frac{(1+xy+yz+zx)(1+3x^3 + 3y^3 + 3z^3)}{9(x+y)(y+z)(z+x)} \ge \left( \frac{x \sqrt{1+x} }{\sqrt[4]{3+9x^2}} + \frac{y \sqrt{1+y} }{\sqrt[4]{3+9y^2}} + \frac{z \sqrt{1+z}}{\sqrt[4]{3+9z^2}} \right)^2. \]

2019 China Team Selection Test, 5

Tags: inequalities

Find all integer $n$ such that the following property holds: for any positive real numbers $a,b,c,x,y,z$, with $max(a,b,c,x,y,z)=a$ , $a+b+c=x+y+z$ and $abc=xyz$, the inequality $$a^n+b^n+c^n \ge x^n+y^n+z^n$$ holds.