Found problems: 6530
2001 National High School Mathematics League, 10
The solution to inequality $\left|\frac{1}{\log_{\frac{1}{2}}x}+2\right|>\frac{3}{2}$ is________(express answer with a set).
2024 Korea Summer Program Practice Test, 6
Find all possible values of $C\in \mathbb R$ such that there exists a real sequence $\{a_n\}_{n=1}^\infty$ such that
$$a_na_{n+1}^2\ge a_{n+2}^4 +C$$
for all $n\ge 1$.
2008 District Olympiad, 1
Prove that for an integer $ n>\equal{}1$ we have $ n(1\plus{}\frac{1}{2}\plus{}\frac{1}{3}\plus{}\dots\plus{}\frac{1}{n})\geq (n\plus{}1)(\frac{1}{2}\plus{}\frac{1}{3}\plus{}\dots\frac{1}{n\plus{}1})$
2009 IMO Shortlist, 2
Let $a$, $b$, $c$ be positive real numbers such that $\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} = a+b+c$. Prove that:
\[\frac{1}{(2a+b+c)^2}+\frac{1}{(a+2b+c)^2}+\frac{1}{(a+b+2c)^2}\leq \frac{3}{16}.\]
[i]Proposed by Juhan Aru, Estonia[/i]
2000 Croatia National Olympiad, Problem 3
Let $j$ and $k$ be integers. Prove that the inequality
$$\lfloor(j+k)\alpha\rfloor+\lfloor(j+k)\beta\rfloor\ge\lfloor j\alpha\rfloor+\lfloor j\beta\rfloor+\lfloor k(\alpha+\beta)\rfloor$$holds for all real numbers $\alpha,\beta$ if and only if $j=k$.
2010 Romanian Masters In Mathematics, 2
For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying
\[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\]
[i]Marko Radovanović, Serbia[/i]
2014 AMC 12/AHSME, 20
For how many positive integers $x$ is $\log_{10}{(x-40)} + \log_{10}{(60-x)} < 2$?
${ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 19\qquad\textbf{(D)}}\ 20\qquad\textbf{(E)}\ \text{infinitely many} $
1986 Czech And Slovak Olympiad IIIA, 6
Assume that $M \subset N$ has the property that every two numbers $m,n$ of $M$ satisfy $|m-n| \ge mn/25$.
Prove that the set $M$ contains no more than $9$ elements.
Decide whether there exists such set M.
2008 China Western Mathematical Olympiad, 2
Given $ x,y,z\in (0,1)$ satisfying that
$ \sqrt{\frac{1 \minus{} x}{yz}} \plus{} \sqrt{\frac{1 \minus{} y}{xz}} \plus{} \sqrt{\frac{1 \minus{} z}{xy}} \equal{} 2$.
Find the maximum value of $ xyz$.
2025 Kyiv City MO Round 1, Problem 5
Real numbers \( a, b, c \) satisfy the following conditions:
\[
1000 < |a| < 2000, \quad 1000 < |b| < 2000, \quad 1000 < |c| < 2000,
\]
and
\[
\frac{ab^2}{a+b} + \frac{bc^2}{b+c} + \frac{ca^2}{c+a} = 0.
\]
What are the possible values of the expression
\[
\frac{a}{b} + \frac{b}{c} + \frac{c}{a}?
\]
[i]Proposed by Vadym Solomka[/i]
2021 Brazil Team Selection Test, 4
[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x .
\]
[i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x .
\]
2022 Thailand Mathematical Olympiad, 2
Define a function $f:\mathbb{N}\times \mathbb{N}\to\{-1,1\}$ such that
$$f(m,n)=\begin{cases} 1 &\text{if }m,n\text{ have the same parity, and} \\ -1 &\text{if }m,n\text{ have different parity}\end{cases}$$
for every positive integers $m,n$. Determine the minimum possible value of
$$\sum_{1\leq i<j\leq 2565} ijf(x_i,x_j)$$
across all permutations $x_1,x_2,x_3,\dots,x_{2565}$ of $1,2,\dots,2565$.
2008 APMO, 5
Let $ a, b, c$ be integers satisfying $ 0 < a < c \minus{} 1$ and $ 1 < b < c$. For each $ k$, $ 0\leq k \leq a$, Let $ r_k,0 \leq r_k < c$
be the remainder of $ kb$ when divided by $ c$. Prove that the two sets $ \{r_0, r_1, r_2, \cdots , r_a\}$ and $ \{0, 1, 2, \cdots , a\}$ are different.
2013 Kosovo National Mathematical Olympiad, 3
Find all numbers $x$ such that:
$1+2\cdot2^x+3\cdot3^x<6^x$
2011 Irish Math Olympiad, 4
Suppose that $x,y$ and $z$ are positive numbers such that $$1=2xyz+xy+yz+zx$$ Prove that
(i)
$$\frac{3}{4}\le xy+yz+zx<1$$
(ii)
$$xyz\le \frac{1}{8}$$
Using (i) or otherwise, deduce that $$x+y+z\ge \frac{3}{2}$$ and derive the case of equality.
2021 JBMO TST - Turkey, 4
Let $x,y,z$ be real numbers such that $$\left|\dfrac yz-xz\right|\leq 1\text{ and }\left|yz+\dfrac xz\right|\leq 1$$ Find the maximum value of the expression $$x^3+2y$$
2006 International Zhautykov Olympiad, 2
Let $ a,b,c,d$ be real numbers with sum 0. Prove the inequality:
\[ (ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd)^2 \plus{} 12\geq 6(abc \plus{} abd \plus{} acd \plus{} bcd).
\]
2020 Bulgaria Team Selection Test, 5
Given is a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $|f(x+y)-f(x)-f(y)|\leq 1$.
Prove the existence of an additive function $g:\mathbb{R}\rightarrow \mathbb{R}$ (that is $g(x+y)=g(x)+g(y)$) such that $|f(x)-g(x)|\leq 1$ for any $x \in \mathbb{R}$
2012 Korea National Olympiad, 4
$a,b,c$ are positive numbers such that $ a^2 + b^2 + c^2 = 2abc + 1 $. Find the maximum value of
\[ (a-2bc)(b-2ca)(c-2ab) \]
2010 South East Mathematical Olympiad, 1
$ABC$ is a triangle with a right angle at $C$. $M_1$ and $M_2$ are two arbitrary points inside $ABC$, and $M$ is the midpoint of $M_1M_2$. The extensions of $BM_1,BM$ and $BM_2$ intersect $AC$ at $N_1,N$ and $N_2$ respectively.
Prove that $\frac{M_1N_1}{BM_1}+\frac{M_2N_2}{BM_2}\geq 2\frac{MN}{BM}$
1999 IMO Shortlist, 2
The numbers from 1 to $n^2$ are randomly arranged in the cells of a $n \times n$ square ($n \geq 2$). For any pair of numbers situated on the same row or on the same column the ratio of the greater number to the smaller number is calculated. Let us call the [b]characteristic[/b] of the arrangement the smallest of these $n^2\left(n-1\right)$ fractions. What is the highest possible value of the characteristic ?
2020 Saint Petersburg Mathematical Olympiad, 3.
On the side $AD$ of the convex quadrilateral $ABCD$ with an acute angle at $B$, a point $E$ is marked.
It is known that $\angle CAD = \angle ADC=\angle ABE =\angle DBE$.
(Grade 9 version) Prove that $BE+CE<AD$.
(Grade 10 version) Prove that $\triangle BCE$ is isosceles.(Here the condition that $\angle B$ is acute is not necessary.)
2002 Korea - Final Round, 3
Let $p_n$ be the $n^{\mbox{th}}$ prime counting from the smallest prime $2$ in increasing order. For example, $p_1=2, p_2=3, p_3 =5, \cdots$
(a) For a given $n \ge 10$, let $r$ be the smallest integer satisfying
\[2\le r \le n-2, \quad n-r+1 < p_r\]
and define $N_s=(sp_1p_2\cdots p_{r-1})-1$ for $s=1,2,\ldots, p_r$. Prove that there exists $j, 1\le j \le p_r$, such that none of $p_1,p_2,\cdots, p_n$ divides $N_j$.
(b) Using the result of (a), find all positive integers $m$ for which
\[p_{m+1}^2 < p_1p_2\cdots p_m\]
1999 Canada National Olympiad, 5
Let $ x$, $ y$, and $ z$ be non-negative real numbers satisfying $ x \plus{} y \plus{} z \equal{} 1$. Show that
\[ x^2 y \plus{} y^2 z \plus{} z^2 x \leq \frac {4}{27}
\]
and find when equality occurs.
2010 Contests, 1
Let $a,b$ be real numbers. Prove the inequality
\[ 2(a^4+a^2b^2+b^4)\ge 3(a^3b+ab^3).\]