Found problems: 6530
2020 Argentina National Olympiad, 5
Determine the highest possible value of:
$$S = a_1a_2a_3 + a_4a_5a_6 +... + a_{2017}a_{2018}a_{2019} + a_{2020}$$
where $(a_1, a_2, a_3,..., a_{2020})$ is a permutation of $(1,2,3,..., 2020)$.
Clarification: In $S$, each term, except the last one, is the multiplication of three numbers.
2012 Bogdan Stan, 1
Let be three real numbers $ a,b,c\in [0,1] $ satisfying the condition $ ab+bc+ca=1. $ Prove that
$$ a^2+b^2+c^2\le 2, $$
and determine the cases in which equality is attained.
2014 India Regional Mathematical Olympiad, 2
Find all real $x,y$ such that
\[x^2 + 2y^2 + \frac{1}{2} \le x(2y+1) \]
2016 Saudi Arabia Pre-TST, 1.2
Let $a, b, c$ be positive numbers such that $a^2+b^2+c^2+abc = 4$.
Prove that $$\frac{a + b}{c} +\frac{b + c}{a} +\frac{c + a}{b} \ge a + b + c + \frac{1}{a} + \frac{1}{b} +\frac{1}{c}$$
2014 Hanoi Open Mathematics Competitions, 15
Let $a_1,a_2,...,a_9 \ge - 1$ and $a^3_1+a^3_2+...+a^3_9= 0$.
Determine the maximal value of $M = a_1 + a_2 + ... + a_9$.
2010 Indonesia MO, 3
A mathematical competition was attended by 120 participants from several contingents. At the closing ceremony, each participant gave 1 souvenir each to every other participants from the same contingent, and 1 souvenir to any person from every other contingents. It is known that there are 3840 souvenirs whom were exchanged.
Find the maximum possible contingents such that the above condition still holds?
[i]Raymond Christopher Sitorus, Singapore[/i]
1993 All-Russian Olympiad Regional Round, 11.8
There are $ 1993$ towns in a country, and at least $ 93$ roads going out of each town. It's known that every town can be reached from any other town. Prove that this can always be done with no more than $ 62$ transfers.
2011 Iran MO (3rd Round), 2
For nonnegative real numbers $x,y,z$ and $t$ we know that $|x-y|+|y-z|+|z-t|+|t-x|=4$.
Find the minimum of $x^2+y^2+z^2+t^2$.
[i]proposed by Mohammadmahdi Yazdi, Mohammad Ahmadi[/i]
1990 Tournament Of Towns, (251) 5
Find the number of pairs $(m, n)$ of positive integers, both of which are $\le 1000$, such that $\frac{m}{n+1}< \sqrt2 < \frac{m+1}{n}$
(recalling that $ \sqrt2 = 1.414213..$.).
(D. Fomin, Leningrad)
2024 Moldova Team Selection Test, 7
Prove that $a=2$ is the greatest real number for which the inequality:
$$
\frac{x_1}{x_n+x_2}+\frac{x_2}{x_1+x_3}+\dots+\frac{x_n}{x_{n-1}+x_1} \ge a
$$
holds true for any $n \ge 4$ and any positive real numbers $x_1, x_2,\dots,x_n$.
2014 Austria Beginners' Competition, 3
Let $a, b, c$ and $d$ be real numbers with $a < b < c < d$.
Sort the numbers $x = a \cdot b + c \cdot d, y = b \cdot c + a \cdot d$ and $z = c \cdot a + b \cdot d$ in ascending\order and prove the correctness of your result.
(R. Henner, Vienna)
1987 IMO Longlists, 4
Let $a_1, a_2, a_3, b_1, b_2, b_3$ be positive real numbers. Prove that
\[(a_1b_2 + a_2b_1 + a_1b_3 + a_3b_1 + a_2b_3 + a_3b_2)^2 \geq 4(a_1a_2 + a_2a_3 + a_3a_1)(b_1b_2 + b_2b_3 + b_3b_1)\]
and show that the two sides of the inequality are equal if and only if $\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3}.$
2011 AMC 12/AHSME, 22
Let $T_1$ be a triangle with sides $2011, 2012,$ and $2013$. For $n \ge 1$, if $T_n=\triangle ABC$ and $D,E,$ and $F$ are the points of tangency of the incircle of $\triangle ABC$ to the sides $AB,BC$ and $AC$, respectively, then $T_{n+1}$ is a triangle with side lengths $AD,BE,$ and $CF$, if it exists. What is the perimeter of the last triangle in the sequence $(T_n)$?
$ \textbf{(A)}\ \frac{1509}{8} \qquad
\textbf{(B)}\ \frac{1509}{32} \qquad
\textbf{(C)}\ \frac{1509}{64} \qquad
\textbf{(D)}\ \frac{1509}{128} \qquad
\textbf{(E)}\ \frac{1509}{256} $
2009 Princeton University Math Competition, 2
Tetrahedron $ABCD$ has sides of lengths, in increasing order, $7, 13, 18, 27, 36, 41$. If $AB=41$, then what is the length of $CD$?
2007 ITAMO, 6
a) For each $n \ge 2$, find the maximum constant $c_{n}$ such that
$\frac 1{a_{1}+1}+\frac 1{a_{2}+1}+\ldots+\frac 1{a_{n}+1}\ge c_{n}$
for all positive reals $a_{1},a_{2},\ldots,a_{n}$ such that $a_{1}a_{2}\cdots a_{n}= 1$.
b) For each $n \ge 2$, find the maximum constant $d_{n}$ such that
$\frac 1{2a_{1}+1}+\frac 1{2a_{2}+1}+\ldots+\frac 1{2a_{n}+1}\ge d_{n}$
for all positive reals $a_{1},a_{2},\ldots,a_{n}$ such that $a_{1}a_{2}\cdots a_{n}= 1$.
2015 Hanoi Open Mathematics Competitions, 6
Let $a, b, c \in [-1, 1] $ such that $1 + 2abc \ge a^2 + b^2 + c^2$.
Prove that $1 + 2a^2b^2c^2 \ge a^4 + b^4 + c^4$.
2003 Korea - Final Round, 1
Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively.
Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.
2010 Contests, 2
Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that
\[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\]
[i]Proposed by Nazar Serdyuk, Ukraine[/i]
2004 Bosnia and Herzegovina Team Selection Test, 3
Let $a$, $b$ and $c$ be positive real numbers such that $abc=1$. Prove the inequality:
$\frac{ab}{a^5+b^5+ab} +\frac{bc}{b^5+c^5+bc}+\frac{ac}{c^5+a^5+ac}\leq 1$
2009 Today's Calculation Of Integral, 488
For $ 0\leq x <\frac{\pi}{2}$, prove the following inequality.
$ x\plus{}\ln (\cos x)\plus{}\int_0^1 \frac{t}{1\plus{}t^2}\ dt\leq \frac{\pi}{4}$
2007 Romania Team Selection Test, 2
Prove that for $n, p$ integers, $n \geq 4$ and $p \geq 4$, the proposition $\mathcal{P}(n, p)$
\[\sum_{i=1}^{n}\frac{1}{{x_{i}}^{p}}\geq \sum_{i=1}^{n}{x_{i}}^{p}\quad \textrm{for}\quad x_{i}\in \mathbb{R}, \quad x_{i}> 0 , \quad i=1,\ldots,n \ ,\quad \sum_{i=1}^{n}x_{i}= n,\] is false.
[i]Dan Schwarz[/i]
[hide="Remark"]In the competition, the students were informed (fact that doesn't actually relate to the problem's solution) that the propositions $\mathcal{P}(4, 3)$ are $\mathcal{P}(3, 4)$ true.[/hide]
2004 Nordic, 4
Let $a, b, c$ be the sides and $R$ be the circumradius of a triangle. Prove that
\[\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\ge\frac{1}{R^2}.\]
1972 Czech and Slovak Olympiad III A, 1
Show that the inequality \[\prod_{k=2}^n\left(1-\frac{1}{k^3}\right)>\frac12\] holds for every positive integer $n>1.$
1998 Czech and Slovak Match, 3
Let $ABCDEF$ be a convex hexagon such that $AB = BC, CD = DE, EF = FA$.
Prove that $\frac{BC}{BE} +\frac{DE}{DA} +\frac{FA}{FC} \ge \frac{3}{2}$ . When does equality occur?
2014 Taiwan TST Round 2, 1
Let $a_i > 0$ for $i=1,2,\dots,n$ and suppose $a_1 + a_2 + \dots + a_n = 1$. Prove that for any positive integer $k$,
\[ \left( a_1^k + \frac{1}{a_1^k} \right) \left( a_2^k + \frac{1}{a_2^k} \right) \dots \left( a_n^k + \frac{1}{a_n^k} \right) \ge \left( n^k + \frac{1}{n^k} \right)^n. \]