Olympiad problems

MathLinks Contest 3rd, 3

Tags: inequalities

Let $n \ge 3$ be an integer. Find the minimal value of the real number $k_n$ such that for all positive numbers $x_1, x_2, ..., x_n$ with product $1$, we have $$\frac{1}{\sqrt{1 + k_nx_1}}+\frac{1}{\sqrt{1 + k_nx_2}}+ ... + \frac{1}{\sqrt{1 + k_nx_n}} \le n - 1.$$

2000 Brazil Team Selection Test, Problem 4

Tags: inequalities , relatively prime , number theory

[b]Problem:[/b]For a positive integer $ n$,let $ V(n; b)$ be the number of decompositions of $ n$ into a product of one or more positive integers greater than $ b$. For example,$ 36 \equal{} 6.6 \equal{}4.9 \equal{} 3.12 \equal{} 3 .3. 4$, so that $ V(36; 2) \equal{} 5$.Prove that for all positive integers $ n$; b it holds that $ V(n;b)<\frac{n}{b}$. :)

Today's calculation of integrals, 863

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

For $0<t\leq 1$, let $F(t)=\frac{1}{t}\int_0^{\frac{\pi}{2}t} |\cos 2x|\ dx.$ (1) Find $\lim_{t\rightarrow 0} F(t).$ (2) Find the range of $t$ such that $F(t)\geq 1.$

1997 Brazil Team Selection Test, Problem 5

Tags: geometry , geometric inequality , inequalities , geometric inequalities , triangle , area

Let $ABC$ be an acute-angled triangle with incenter $I$. Consider the point $A_1$ on $AI$ different from $A$, such that the midpoint of $AA_1$ lies on the circumscribed circle of $ABC$. Points $B_1$ and $C_1$ are defined similarly. (a) Prove that $S_{A_1B_1C_1}=(4R+r)p$, where $p$ is the semi-perimeter, $R$ is the circumradius and $r$ is the inradius of $ABC$. (b) Prove that $S_{A_1B_1C_1}\ge9S_{ABC}$.

2010 China Western Mathematical Olympiad, 5

Tags: inequalities , algebra

Let $k$ be an integer and $k > 1$. Define a sequence $\{a_n\}$ as follows: $a_0 = 0$, $a_1 = 1$, and $a_{n+1} = ka_n + a_{n-1}$ for $n = 1,2,...$. Determine, with proof, all possible $k$ for which there exist non-negative integers $l,m (l \not= m)$ and positive integers $p,q$ such that $a_l + ka_p = a_m + ka_q$.

1986 National High School Mathematics League, 3

Tags: inequalities

For real numbers $a,b,c$, if $$a^2-bc-8a+7=b^2+c^2+bc-6a-6=0,$$ then the range value of $a$ is $\text{(A)}(-\infty,+\infty)\qquad\text{(B)}(-\infty,1]\cup[9,+\infty)\qquad\text{(C)}(0,7)\qquad\text{(D)}[1,9]$

1998 Irish Math Olympiad, 1

Tags: integration , inequalities

Prove that if $ x \not\equal{} 0$ is a real number, then: $ x^8\minus{}x^5\minus{}\frac{1}{x}\plus{}\frac{1}{x^4} \ge 0$.

2009 Princeton University Math Competition, 4

Tags: geometry , 3d geometry , tetrahedron , inequalities

Tetrahedron $ABCD$ has sides of lengths, in increasing order, $7, 13, 18, 27, 36, 41$. If $AB=41$, then what is the length of $CD$?

MathLinks Contest 5th, 5.2

Tags: inequalities , function , algebra

Prove or disprove the existence of a function $f : S \to R$ such that for all $x \ne y \in S$ we have $|f(x) - f(y)| \ge \frac{1}{x^2 + y^2}$, in each of the cases: a) $S = R$ b) $S = Q$.

2004 Regional Olympiad - Republic of Srpska, 2

Tags: inequalities

The positive real numbers $x,y,z$ satisfy $x+y+z=1$. Show that \[\sqrt{3xyz}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{1-x}+\frac{1}{1-y}+\frac{1}{1-z}\right)\geq4+ \frac{4xyz}{(1-x)(1-y)(1-z)}.\]

1993 Romania Team Selection Test, 1

Tags: inequalities

Find max. numbers $A$ wich is true ineq.: $\frac{x}{\sqrt{y^{2}+z^{2}}}+\frac{y}{\sqrt{x^{2}+z^{2}}}+\frac{z}{\sqrt{x^{2}+y^{2}}}\geq A$ $x,y,z$ are positve reals numberes! :wink:

2021 Bosnia and Herzegovina Team Selection Test, 1

Tags: algebra , inequalities , max

Let $x,y,z$ be real numbers from the interval $[0,1]$. Determine the maximum value of expression $$W=y\cdot \sqrt{1-x}+z\cdot\sqrt{1-y}+x\cdot\sqrt{1-z}$$

2018 Bulgaria JBMO TST, 2

Tags: inequalities

For all positive reals $a$ and $b$, show that $$\frac{a^2+b^2}{2a^5b^5} + \frac{81a^2b^2}{4} + 9ab > 18.$$

2018 Thailand TST, 3

Tags: inequalities

Let $n \geq 3$ be an integer. Let $a_1,a_2,\dots, a_n\in[0,1]$ satisfy $a_1 + a_2 + \cdots + a_n = 2$. Prove that $$\sqrt{1-\sqrt{a_1}}+\sqrt{1-\sqrt{a_2}}+\cdots+\sqrt{1-\sqrt{a_n}}\leq n-3+\sqrt{9-3\sqrt{6}}.$$

2020 Caucasus Mathematical Olympiad, 8

Tags: inequalities , algebra

Let real $a$, $b$, and $c$ satisfy $$abc+a+b+c=ab+bc+ca+5.$$ Find the least possible value of $a^2+b^2+c^2$.

2013 ELMO Shortlist, 4

Tags: inequalities

Positive reals $a$, $b$, and $c$ obey $\frac{a^2+b^2+c^2}{ab+bc+ca} = \frac{ab+bc+ca+1}{2}$. Prove that \[ \sqrt{a^2+b^2+c^2} \le 1 + \frac{\lvert a-b \rvert + \lvert b-c \rvert + \lvert c-a \rvert}{2}. \][i]Proposed by Evan Chen[/i]

2020 Turkey EGMO TST, 6

Tags: algebra , inequalities

$x,y,z$ are positive real numbers such that: $$xyz+x+y+z=6$$ $$xyz+2xy+yz+zx+z=10$$ Find the maximum value of: $$(xy+1)(yz+1)(zx+1)$$

2013 Harvard-MIT Mathematics Tournament, 13

Tags: hmmt , inequalities

Find the smallest positive integer $n$ such that $\dfrac{5^{n+1}+2^{n+1}}{5^n+2^n}>4.99$.

2019 Peru MO (ONEM), 2

Tags: algebra , inequalities

Find all the real numbers $k$ that have the following property: For any non-zero real numbers $a$ and $b$, it is true that at least one of the following numbers: $$a, b,\frac{5}{a^2}+\frac{6}{b^3}$$is less than or equal to $k$.

1992 Irish Math Olympiad, 5

Tags: inequality , inequalities

If, for $k=1,2,\dots ,n$, $a_k$ and $b_k$ are positive real numbers, prove that $$\sqrt[n]{a_1a_2\cdots a_n}+\sqrt[n]{b_1b_2\cdots b_n}\le \sqrt[n]{(a_1+b_1)(a_2+b_2)\cdots (a_n+b_n)};$$ and that equality holds if, and only if, $$\frac{a_1}{b_1}=\frac{a_2}{b_2}=\cdots =\frac{a_n}{b_n}.$$

2018 Iran Team Selection Test, 2

Tags: inequalities

Determine the least real number $k$ such that the inequality $$\left(\frac{2a}{a-b}\right)^2+\left(\frac{2b}{b-c}\right)^2+\left(\frac{2c}{c-a}\right)^2+k \geq 4\left(\frac{2a}{a-b}+\frac{2b}{b-c}+\frac{2c}{c-a}\right)$$ holds for all real numbers $a,b,c$. [i]Proposed by Mohammad Jafari[/i]

2020 Mediterranean Mathematics Olympiad, 3

Tags: algebra , inequalities

Prove that all postive real numbers $a,b,c$ with $a+b+c=4$ satisfy the inequality $$\frac{ab}{\sqrt[4]{3c^2+16}}+ \frac{bc}{\sqrt[4]{3a^2+16}}+ \frac{ca}{\sqrt[4]{3b^2+16}} \le\frac43 \sqrt[4]{12}$$

2020 IMO Shortlist, A7

Tags: inequalities , n-variable inequality

Let $n$ and $k$ be positive integers. Prove that for $a_1, \dots, a_n \in [1,2^k]$ one has \[ \sum_{i = 1}^n \frac{a_i}{\sqrt{a_1^2 + \dots + a_i^2}} \le 4 \sqrt{kn}. \]

2007 Alexandru Myller, 2

Tags: geometry , discrete geometry , plane geometry , inequalities , angle

$ n $ lines meet at a point. Each one of the $ 2n $ disjoint angles formed around this point by these lines has either $ 7^{\circ} $ or $ 17^{\circ} . $ [b]a)[/b] Find $ n. $ [b]b)[/b] Prove that among these lines there are at least two perpendicular ones.

2006 Romania National Olympiad, 4

Tags: inequalities , calculus , function , geometry , trigonometry , vector , derivative

Let $a,b,c \in \left[ \frac 12, 1 \right]$. Prove that \[ 2 \leq \frac{ a+b}{1+c} + \frac{ b+c}{1+a} + \frac{ c+a}{1+b} \leq 3 . \] [i]selected by Mircea Lascu[/i]