Olympiad problems

2011 Czech and Slovak Olympiad III A, 3

Suppose that $x$, $y$, $z$ are real numbers satisfying \[x+y+z=12,\qquad\text{and}\qquad x^2+y^2+z^2=54.\] Prove that:[list](a) Each of the numbers $xy$, $yz$, $zx$ is at least $9$, but at most $25$. (b) One of the numbers $x$, $y$, $z$ is at most $3$, and another one is at least $5$.[/list]

2015 Middle European Mathematical Olympiad, 1

Tags: inequalities , fraction , muirhed inequality

Prove that for all positive real numbers $a$, $b$, $c$ such that $abc=1$ the following inequality holds: $$\frac{a}{2b+c^2}+\frac{b}{2c+a^2}+\frac{c}{2a+b^2}\le \frac{a^2+b^2+c^2}3.$$

2011 239 Open Mathematical Olympiad, 3

Tags: inequalities

Positive reals $a,b,c,d$ satisfy $a+b+c+d=4$. Prove that $\sum_{cyc}\frac{a}{a^3 + 4} \le \frac{4}{5}$

2006 Petru Moroșan-Trident, 3

Tags: inequalities , am-gm

Let a ,b and c be positive real numbers such that $a^2+b^2+c^2=3$. Prove that for whatever positive real numbers x y and z, the inequality below holds. $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\ge \sqrt{xy}+\sqrt{yz}+\sqrt{zx}$ At first I noticed $\sqrt{xy}+\sqrt{yz}+\sqrt{zx}\le \sqrt{x+y+z}\sqrt{x+y+z}=x+y+z$, so perhaps the next move is to prove $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\ge x+y+z$, but I don't see how to do that, the best thing that I can do with the LHS of this inequality is to prove it by AM-GM in the way that $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\ge 3\left(\frac{xyz}{abc}\right)^{\frac{1}{3}}\ge 3(xyz)^{\frac{1}{3}}$, but this isn't going to be helpful...

1981 Miklós Schweitzer, 6

Tags: integration , function , inequalities , trigonometry , real analysis

Let $ f$ be a strictly increasing, continuous function mapping $ I=[0,1]$ onto itself. Prove that the following inequality holds for all pairs $ x,y \in I$: \[ 1-\cos (xy) \leq \int_0^xf(t) \sin (tf(t))dt + \int_0^y f^{-1}(t) \sin (tf^{-1}(t)) dt .\] [i]Zs. Pales[/i]

2004 South East Mathematical Olympiad, 5

Tags: trigonometry , inequalities

For $\theta\in[0, \dfrac{\pi}{2}]$, the following inequality $\sqrt{2}(2a+3)\cos(\theta-\dfrac{\pi}{4})+\dfrac{6}{\sin\theta+\cos\theta}-2\sin2\theta<3a+6$ is always true. Determine the range of $a$.

2001 Romania National Olympiad, 1

Tags: inequalities

Determine all real numbers $a$ and $b$ such that $a+b\in\mathbb{Z}$ and $a^2+b^2=2$.

2019 Jozsef Wildt International Math Competition, W. 59

Tags: tetrahedron , geometry , inequalities , 3d geometry , circumradius , inradius

In the any $[ABCD]$ tetrahedron we denote with $\alpha$, $\beta$, $\gamma$ the measures, in radians, of the angles of the three pairs of opposite edges and with $r$, $R$ the lengths of the rays of the sphere inscribed and respectively circumscribed the tetrahedron. Demonstrate inequality$$\left(\frac{3r}{R}\right)^3\leq \sin \frac{\alpha +\beta +\gamma}{3}$$(A refinement of inequality $R \geq 3r$).

2019 ELMO Shortlist, A1

Tags: inequalities , algebra , inequality

Let $a$, $b$, $c$ be positive reals such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Show that $$a^abc+b^bca+c^cab\ge 27bc+27ca+27ab.$$ [i]Proposed by Milan Haiman[/i]

2009 Kosovo National Mathematical Olympiad, 3

Tags: inequalities , algebra

Let $a,b$ and $c$ be the sides of a triangle, prove that $\frac {a}{b+c}+\frac {b}{c+a}+\frac {c}{a+b}<2$.

2018 Federal Competition For Advanced Students, P1, 1

Tags: inequalities

Let $\alpha$ be an arbitrary positive real number. Determine for this number $\alpha$ the greatest real number $C$ such that the inequality$$\left(1+\frac{\alpha}{x^2}\right)\left(1+\frac{\alpha}{y^2}\right)\left(1+\frac{\alpha}{z^2}\right)\geq C\left(\frac{x}{z}+\frac{z}{x}+2\right)$$ is valid for all positive real numbers $x, y$ and $z$ satisfying $xy + yz + zx =\alpha.$ When does equality occur? [i](Proposed by Walther Janous)[/i]

2021 Latvia Baltic Way TST, P4

Tags: inequalities

Determine the smallest positive constant $k$ such that no matter what $3$ lattice points we choose the following inequality holds: $$ L_{\max} - L_{\min} \ge \frac{1}{\sqrt{k} \cdot L_{max}} $$ where $L_{\max}$, $L_{\min}$ is the maximal and minimal distance between chosen points.

2018 IFYM, Sozopol, 4

Tags: algebra , inequalities

The real numbers $a$, $b$, $c$ are such that $a+b+c+ab+bc+ca+abc \geq 7$. Prove that $\sqrt{a^2+b^2+2}+\sqrt{b^2+c^2+2}+\sqrt{c^2+a^2+2} \geq 6$

2025 All-Russian Olympiad Regional Round, 10.5

Tags: inequalities , geometry

The heights $BD$ and $CE$ of the acute-angled triangle $ABC$ intersect at point $H$, the heights of the triangle $ADE$ intersect at point $F$, point $M$ is the midpoint of side $BC$. Prove that $BH + CH \geqslant 2 FM$. [i]A. Kuznetsov[/i]

2019 Nigerian Senior MO Round 4, 1

Tags: function , inequalities , functional equation , number theory

Let $f: N \to N$ be a function satisfying (a) $1\le f(x)-x \le 2019$ $\forall x \in N$ (b) $f(f(x))\equiv x$ (mod $2019$) $\forall x \in N$ Show that $\exists x \in N$ such that $f^k(x)=x+2019 k, \forall k \in N$

2011 Junior Balkan Team Selection Tests - Moldova, 2

Tags: algebra , inequalities

The real numbers $a, b, x$ satisfy the inequalities $| a + x + b | \le 1, | 4a + 2x + b | \le1, | 9a + 6x + 4b | \le 1$. Prove that $| x | \le15$.

2010 Indonesia TST, 1

Tags: inequalities

Given $ a,b, c $ positive real numbers satisfying $ a+b+c=1 $. Prove that \[ \dfrac{1}{\sqrt{ab+bc+ca}}\ge \sqrt{\dfrac{2a}{3(b+c)}} +\sqrt{\dfrac{2b}{3(c+a)}} + \sqrt{\dfrac{2c}{3(a+b)}} \ge \sqrt{a} +\sqrt{b}+\sqrt{c} \]

2002 Federal Math Competition of S&M, Problem 2

Tags: geometry , inequalities , geometric inequality , area

Points $A_0,A_1,\ldots,A_{2k}$, in this order, divide a circumference into $2k+1$ equal arcs. Point $A_0$ is connected by chords to all the other points. These $2k$ chords divide the interior of the circle into $2k+1$ parts. These parts are alternately painted red and blue so that there are $k+1$ red and $k$ blue parts. Show that the blue area is larger than the red area.

2014 Belarus Team Selection Test, 3

Tags: inequalities , algebra

Given $a,b,c$ ,$(a, b,c \in (0,2)$), with $a + b + c = ab+bc+ca$, prove that $$\frac{a^2}{a^2-a+1}+\frac{b^2}{b^2-b+1}+\frac{c^2}{c^2-c+1} \le 3$$ (D. Pirshtuk)

2012 China Team Selection Test, 3

Tags: geometric transformation , geometry , inequalities , algebra , polynomial

Find the smallest possible value of a real number $c$ such that for any $2012$-degree monic polynomial \[P(x)=x^{2012}+a_{2011}x^{2011}+\ldots+a_1x+a_0\] with real coefficients, we can obtain a new polynomial $Q(x)$ by multiplying some of its coefficients by $-1$ such that every root $z$ of $Q(x)$ satisfies the inequality \[ \left\lvert \operatorname{Im} z \right\rvert \le c \left\lvert \operatorname{Re} z \right\rvert. \]

Today's calculation of integrals, 766

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

Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and \[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\] Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$

2016 Taiwan TST Round 3, 1

Tags: inequalities

Let $x,y,z$ be positive real numbers satisfying $x+y+z=1$. Find the smallest $k$ such that $\frac{x^2y^2}{1-z}+\frac{y^2z^2}{1-x}+\frac{z^2x^2}{1-y}\leq k-3xyz$.

2011 Balkan MO Shortlist, A1

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?

2018 Latvia Baltic Way TST, P1

Tags: inequalities , algebra

Let $p_1,p_2,...,p_n$ be $n\ge 2$ fixed positive real numbers. Let $x_1,x_2,...,x_n$ be nonnegative real numbers such that $$x_1p_1+x_2p_2+...+x_np_n=1.$$ Determine the [i](a)[/i] maximal; [i](b)[/i] minimal possible value of $x_1^2+x_2^2+...+x_n^2$.

1998 National Olympiad First Round, 36

Tags: reflection , geometry , geometric transformation , symmetry , inequalities

$ ABCD$ is a $ 4\times 4$ square. $ E$ is the midpoint of $ \left[AB\right]$. $ M$ is an arbitrary point on $ \left[AC\right]$. How many different points $ M$ are there such that $ \left|EM\right|\plus{}\left|MB\right|$ is an integer? $\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6$