Found problems: 6530
1968 Yugoslav Team Selection Test, Problem 1
Given $6$ points in a plane, assume that each two of them are connected by a segment. Let $D$ be the length of the longest, and $d$ the length of the shortest of these segments. Prove that $\frac Dd\ge\sqrt3$.
1999 Vietnam Team Selection Test, 1
Let a sequence of positive reals $\{u_n\}^{\infty}_{n=1}$ be given. For every positive integer $n$, let $k_n$ be the least positive integer satisfying:
\[\sum^{k_n}_{i=1} \frac{1}{i} \geq \sum^n_{i=1} u_i.\]
Show that the sequence $\left\{\frac{k_{n+1}}{k_n}\right\}$ has finite limit if and only if $\{u_n\}$ does.
2006 China Team Selection Test, 1
Let $A$ be a non-empty subset of the set of all positive integers $N^*$. If any sufficient big positive integer can be expressed as the sum of $2$ elements in $A$(The two integers do not have to be different), then we call that $A$ is a divalent radical. For $x \geq 1$, let $A(x)$ be the set of all elements in $A$ that do not exceed $x$, prove that there exist a divalent radical $A$ and a constant number $C$ so that for every $x \geq 1$, there is always $\left| A(x) \right| \leq C \sqrt{x}$.
2015 International Zhautykov Olympiad, 3
The area of a convex pentagon $ABCDE$ is $S$, and the circumradii of the triangles $ABC$, $BCD$, $CDE$, $DEA$, $EAB$ are $R_1$, $R_2$, $R_3$, $R_4$, $R_5$. Prove the inequality
\[ R_1^4+R_2^4+R_3^4+R_4^4+R_5^4\geq {4\over 5\sin^2 108^\circ}S^2. \]
2011 IMO Shortlist, 7
Let $a,b$ and $c$ be positive real numbers satisfying $\min(a+b,b+c,c+a) > \sqrt{2}$ and $a^2+b^2+c^2=3.$ Prove that
\[\frac{a}{(b+c-a)^2} + \frac{b}{(c+a-b)^2} + \frac{c}{(a+b-c)^2} \geq \frac{3}{(abc)^2}.\]
[i]Proposed by Titu Andreescu, Saudi Arabia[/i]
2002 India IMO Training Camp, 15
Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality
\[
\frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots +
\frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}.
\]
2023 SAFEST Olympiad, 2
There are $n!$ empty baskets in a row, labelled $1, 2, . . . , n!$. Caesar
first puts a stone in every basket. Caesar then puts 2 stones in every second basket.
Caesar continues similarly until he has put $n$ stones into every nth basket. In
other words, for each $i = 1, 2, . . . , n,$ Caesar puts $i$ stones into the baskets labelled
$i, 2i, 3i, . . . , n!.$
Let $x_i$ be the number of stones in basket $i$ after all these steps. Show that
$n! \cdot n^2 \leq \sum_{i=1}^{n!} x_i^2 \leq n! \cdot n^2 \cdot \sum_{i=1}^{n} \frac{1}{i} $
2002 AMC 12/AHSME, 18
If $a,b,c$ are real numbers such that $a^2+2b=7$, $b^2+4c=-7$, and $c^2+6a=-14$, find $a^2+b^2+c^2$.
$\textbf{(A) }14\qquad\textbf{(B) }21\qquad\textbf{(C) }28\qquad\textbf{(D) }35\qquad\textbf{(E) }49$
2024 Myanmar IMO Training, 2
Let $a, b, c$ be positive real numbers satisfying
\[a+b+c = a^2 + b^2 + c^2.\]
Let
\[M = \max\left(\frac{2a^2}{b} + c, \frac{2b^2}{a} + c \right) \quad \text{ and } \quad N = \min(a^2 + b^2, c^2).\]
Find the minimum possible value of $M/N$.
2005 China Western Mathematical Olympiad, 7
If $a,b,c$ are positive reals such that $a+b+c=1$, prove that \[ 10(a^3+b^3+c^3)-9(a^5+b^5+c^5)\geq 1 . \]
1952 Miklós Schweitzer, 3
Prove:If $ a\equal{}p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_{n}^{\alpha_n}$ is a perfect number, then
$ 2<\prod_{i\equal{}1}^n\frac{p_i}{p_i\minus{}1}<4$ ;
if moreover, $ a$ is odd, then the upper bound $ 4$ may be reduced to $ 2\sqrt[3]{2}$.
2010 Saudi Arabia BMO TST, 3
Let $a > 0$ be a real number and let $f : R \to R$ be a function satisfying $$f(x_1) + f(x_2) \ge a f(x_1 + x_2), \forall x_1 ,x_2 \in R.$$ Prove that $$f(x_1) + f(x_2) +(x_3) \ge \frac{3a^2}{a+2} f(x_1+ x_2 + x_3), \forall x_1 ,x_2,x_3 \in R$$.
2019 China Team Selection Test, 3
Let $n$ be a given even number, $a_1,a_2,\cdots,a_n$ be non-negative real numbers such that $a_1+a_2+\cdots+a_n=1.$ Find the maximum possible value of $\sum_{1\le i<j\le n}\min\{(i-j)^2,(n+i-j)^2\}a_ia_j .$
2014 Baltic Way, 3
Positive real numbers $a, b, c$ satisfy $\frac{1}{a} +\frac{1}{b} +\frac{1}{c} = 3.$ Prove the inequality \[\frac{1}{\sqrt{a^3+ b}}+\frac{1}{\sqrt{b^3 + c}}+\frac{1}{\sqrt{c^3 + a}}\leq \frac{3}{\sqrt{2}}.\]
1984 Spain Mathematical Olympiad, 3
If $p$ and $q$ are positive numbers with $p+q = 1$,
knowing that any real numbers $x,y$ satisfy $(x-y)^2 \ge 0$, show that
$\frac{x+y}{2} \ge \sqrt{xy}$,
$\frac{x^2+y^2}{2} \ge \big(\frac{x+y}{2}\big)^2$,
$\big(p+\frac{1}{p}\big)^2+\big(q+\frac{1}{q}\big)^2 \ge \frac{25}{2}$
1997 Iran MO (3rd Round), 4
Let $x, y, z$ be real numbers greater than $1$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2$. Prove that
\[\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}\leq \sqrt{x+y+z}.\]
2012 Olympic Revenge, 5
Let $x_1,x_2,\ldots ,x_n$ positive real numbers. Prove that:
\[\sum_{cyc} \frac{1}{x_i^3+x_{i-1}x_ix_{i+1}} \le \sum_{cyc} \frac{1}{x_ix_{i+1}(x_i+x_{i+1})}\]
2005 Poland - Second Round, 3
Prove that if the real numbers $a,b,c$ lie in the interval $[0,1]$, then
\[\frac{a}{bc+1}+\frac{b}{ac+1}+\frac{c}{ab+1}\le 2\]
1997 India National Olympiad, 4
In a unit square one hundred segments are drawn from the centre to the sides dividing the square into one hundred parts (triangles and possibly quadruilaterals). If all parts have equal perimetr $p$, show that $\dfrac{14}{10} < p < \dfrac{15}{10}$.
2003 District Olympiad, 3
(a) If $\displaystyle ABC$ is a triangle and $\displaystyle M$ is a point from its plane, then prove that
\[ \displaystyle AM \sin A \leq BM \sin B + CM \sin C . \]
(b) Let $\displaystyle A_1,B_1,C_1$ be points on the sides $\displaystyle (BC),(CA),(AB)$ of the triangle $\displaystyle ABC$, such that the angles of $\triangle A_1 B_1 C_1$ are $\widehat{A_1} = \alpha, \widehat{B_1} = \beta, \widehat{C_1} = \gamma$. Prove that
\[ \displaystyle \sum A A_1 \sin \alpha \leq \sum BC \sin \alpha . \]
[i]Dan Ştefan Marinescu, Viorel Cornea[/i]
1992 Austrian-Polish Competition, 3
For all positive numbers $a, b, c$ prove the inequality $2\sqrt{bc + ca + ab} \le \sqrt{3} \sqrt[3]{(b + c)(c + a)(a + b)}$.
2012 All-Russian Olympiad, 4
The positive real numbers $a_1,\ldots ,a_n$ and $k$ are such that $a_1+\cdots +a_n=3k$, $a_1^2+\cdots +a_n^2=3k^2$ and $a_1^3+\cdots +a_n^3>3k^3+k$. Prove that the difference between some two of $a_1,\ldots,a_n$ is greater than $1$.
1994 National High School Mathematics League, 1
$a,b,c$ are real numbers. The sufficient and necessary condition of $\forall x\in\mathbb{R},a\sin x+b\cos x+c>0$ is
$\text{(A)}$ $a=b=0,c>0$
$\text{(B)}$ $\sqrt{a^2+b^2}=c$
$\text{(C)}$ $\sqrt{a^2+b^2}<c$
$\text{(D)}$ $\sqrt{a^2+b^2}>c$
2011 China Team Selection Test, 3
Let $n$ be a positive integer. Find the largest real number $\lambda$ such that for all positive real numbers $x_1,x_2,\cdots,x_{2n}$ satisfying the inequality
\[\frac{1}{2n}\sum_{i=1}^{2n}(x_i+2)^n\geq \prod_{i=1}^{2n} x_i,\]
the following inequality also holds
\[\frac{1}{2n}\sum_{i=1}^{2n}(x_i+1)^n\geq \lambda\prod_{i=1}^{2n} x_i.\]
2011 Greece National Olympiad, 3
Let $a,b,c$ be positive real numbers with sum $6$. Find the maximum value of
\[S = \sqrt[3]{{{a^2} + 2bc}} + \sqrt[3]{{{b^2} + 2ca}} + \sqrt[3]{{{c^2} + 2ab}}.\]