Found problems: 592
JOM 2015, 3
Let $ a, b, c $ be positive real numbers greater or equal to $ 3 $. Prove that $$ 3(abc+b+2c)\ge 2(ab+2ac+3bc) $$ and determine all equality cases.
2001 Slovenia National Olympiad, Problem 2
Tina wrote a positive number on each of five pieces of paper. She did not say which numbers she wrote, but revealed their pairwise sums instead: $17,20,28,14,42,36,28,39,25,31$. Which numbers did she write?
2015 JBMO TST-Turkey, 4
Prove that
$$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c} \ge \dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}+2(a+b+c)$$
for the all $a,b,c$ positive real numbers satisfying $a^2+b^2+c^2+2abc \le 1$.
2015 IMC, 10
Let $n$ be a positive integer, and let $p(x)$ be a polynomial of
degree $n$ with integer coefficients. Prove that
$$
\max_{0\le x\le1} \big|p(x)\big| > \frac1{e^n}.
$$
Proposed by Géza Kós, Eötvös University, Budapest
2018 JBMO Shortlist, A6
For $a,b,c$ positive real numbers such that $ab+bc+ca=3$, prove:
$ \frac{a}{\sqrt{a^3+5}}+\frac{b}{\sqrt{b^3+5}}+\frac{c}{\sqrt{c^3+5}} \leq \frac{\sqrt{6}}{2}$
[i]Proposed by Dorlir Ahmeti, Albania[/i]
1987 Czech and Slovak Olympiad III A, 4
Given an integer $n\ge3$ consider positive integers $x_1,\ldots,x_n$ such that $x_1<x_2<\cdots<x_n<2x_1$. If $p$ is a prime and $r$ is a positive integer such that $p^r$ divides the product $x_1\cdots x_n$, prove that $$\frac{x_1\cdots x_n}{p^r}>n!.$$
Russian TST 2022, P3
Let $n\geqslant 3$ be an integer and $x_1>x_2>\cdots>x_n$ be real numbers. Suppose that $x_k>0\geqslant x_{k+1}$ for an index $k{}$. Prove that \[\sum_{i=1}^k\left(x_i^{n-2}\prod_{j\neq i}\frac{1}{x_i-x_j}\right)\geqslant 0.\]
2017 Azerbaijan Senior National Olympiad, A5
$a,b,c \in (0,1)$ and $x,y,z \in ( 0, \infty)$ reals satisfies the condition $a^x=bc,b^y=ca,c^z=ab$. Prove that \[ \dfrac{1}{2+x}+\dfrac{1}{2+y}+\dfrac{1}{2+z} \leq \dfrac{3}{4} \] \\
2020 India National Olympiad, 4
Let $n \geqslant 2$ be an integer and let $1<a_1 \le a_2 \le \dots \le a_n$ be $n$ real numbers such that $a_1+a_2+\dots+a_n=2n$. Prove that$$a_1a_2\dots a_{n-1}+a_1a_2\dots a_{n-2}+\dots+a_1a_2+a_1+2 \leqslant a_1a_2\dots a_n.$$
[i]Proposed by Kapil Pause[/i]
2018 Mexico National Olympiad, 4
Let $n\geq 2$ be an integer. For each $k$-tuple of positive integers $a_1, a_2, \ldots, a_k$ such that $a_1+a_2+\cdots +a_k=n$, consider the sums $S_i=1+2+\ldots +a_i$ for $1\leq i\leq k$. Determine, in terms of $n$, the maximum possible value of the product $S_1S_2\cdots S_k$.
[i]Proposed by Misael Pelayo[/i]
2018 Azerbaijan JBMO TST, 1
Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that:
$$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$
2018 Junior Balkan Team Selection Tests - Moldova, 3
Let $a,b,c \in\mathbb{R^*_+}$.Prove the inequality $\frac{a^2+4}{b+c}+\frac{b^2+9}{c+a}+\frac{c^2+16}{a+b}\ge9$.
KoMaL A Problems 2017/2018, A. 727
For any finite sequence $(x_1,\ldots,x_n)$, denote by $N(x_1,\ldots,x_n)$ the number of ordered index pairs $(i,j)$ for which $1 \le i<j\le n$ and $x_i=x_j$. Let $p$ be an odd prime, $1 \le n<p$, and let $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ be arbitrary residue classes modulo $p$. Prove that there exists a permutation $\pi$ of the indices $1,2,\ldots,n$ for which
\[N(a_1+b_{\pi(1)},a_2+b_{\pi(2)},\ldots,a_n+b_{\pi(n)})\le \min(N(a_1,a_2,\ldots,a_n),N(b_1,b_2,\ldots,b_n)).\]
1982 IMO, 3
Consider infinite sequences $\{x_n\}$ of positive reals such that $x_0=1$ and $x_0\ge x_1\ge x_2\ge\ldots$.
[b]a)[/b] Prove that for every such sequence there is an $n\ge1$ such that: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}\ge3.999. \]
[b]b)[/b] Find such a sequence such that for all $n$: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}<4. \]
1978 IMO Longlists, 16
Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$
1980 IMO Shortlist, 2
Define the numbers $a_0, a_1, \ldots, a_n$ in the following way:
\[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \]
Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]
2017 JBMO Shortlist, A1
Let $a, b, c$ be positive real numbers such that $a + b + c + ab + bc + ca + abc = 7$. Prove
that $\sqrt{a^2 + b^2 + 2 }+\sqrt{b^2 + c^2 + 2 }+\sqrt{c^2 + a^2 + 2 } \ge 6$ .
2014 China Western Mathematical Olympiad, 1
Let $x,y$ be positive real numbers .Find the minimum of $x+y+\frac{|x-1|}{y}+\frac{|y-1|}{x}$.
2006 Bosnia and Herzegovina Team Selection Test, 3
Prove that for every positive integer $n$ holds inequality $\{n\sqrt{7}\}>\frac{3\sqrt{7}}{14n}$, where $\{x\}$ is fractional part of $x$.
2014 Dutch Mathematical Olympiad, 4
A quadruple $(p, a, b, c)$ of positive integers is called a Leiden quadruple if
- $p$ is an odd prime number,
- $a, b$, and $c$ are distinct and
- $ab + 1, bc + 1$ and $ca + 1$ are divisible by $p$.
a) Prove that for every Leiden quadruple $(p, a, b, c)$ we have $p + 2 \le \frac{a+b+c}{3}$ .
b) Determine all numbers $p$ for which a Leiden quadruple $(p, a, b, c)$ exists with $p + 2 = \frac{a+b+c}{3} $
1969 IMO Longlists, 14
$(CZS 3)$ Let $a$ and $b$ be two positive real numbers. If $x$ is a real solution of the equation $x^2 + px + q = 0$ with real coefficients $p$ and $q$ such that $|p| \le a, |q| \le b,$ prove that $|x| \le \frac{1}{2}(a +\sqrt{a^2 + 4b})$ Conversely, if $x$ satisfies the above inequality, prove that there exist real numbers $p$ and
$q$ with $|p|\le a, |q|\le b$ such that $x$ is one of the roots of the equation $x^2+px+ q = 0.$
2005 Federal Math Competition of S&M, Problem 3
If $x,y,z$ are nonnegative numbers with $x+y+z=3$, prove that
$$\sqrt x+\sqrt y+\sqrt z\ge xy+yz+xz.$$
2022 Korea Winter Program Practice Test, 2
Let $n\ge 2$ be a positive integer. There are $n$ real coefficient polynomials $P_1(x),P_2(x),\cdots ,P_n(x)$ which is not all the same, and their leading coefficients are positive. Prove that
$$\deg(P_1^n+P_2^n+\cdots +P_n^n-nP_1P_2\cdots P_n)\ge (n-2)\max_{1\le i\le n}(\deg P_i)$$
and find when the equality holds.
1994 Korea National Olympiad, Problem 2
Let $ \alpha,\beta,\gamma$ be the angles of a triangle. Prove that
$csc^2\frac{\alpha}{2}+csc^2\frac{\beta}{2}+csc^2\frac{\gamma}{2} \ge 12$
and find the conditions for equality.
2023 JBMO Shortlist, A1
Prove that for all positive real numbers $a,b,c,d$,
$$\frac{2}{(a+b)(c+d)+(b+c)(a+d)} \leq \frac{1}{(a+c)(b+d)+4ac}+\frac{1}{(a+c)(b+d)+4bd}$$
and determine when equality occurs.