Found problems: 6530
2012 Estonia Team Selection Test, 5
Let $x, y, z$ be positive real numbers whose sum is $2012$. Find the maximum value of $$ \frac{(x^2 + y^2 + z^2)(x^3 + y^3 + z^3)}{(x^4 + y^4 + z^4)}$$
OIFMAT II 2012, 5
Let $ n \in N $. Let's define $ S_n = \{1, ..., n \} $. Let $ x_1 <x_2 <\cdots <x_n $ be any real. Determine the largest possible number of pairs $ (i, j) \in S_n \times S_n $ with $ i \not = j $, for which it is true that $ 1 <| x_i-x_j | <2 $ and justify why said value cannot be higher.
1896 Eotvos Mathematical Competition, 1
If $k$ is the number of distinct prime divisors of a natural number $n$, prove that log $n \geq k$ log $2$.
2011 Korea - Final Round, 1
Find the maximal value of the following expression, if $a,b,c$ are nonnegative and $a+b+c=1$.
\[ \frac{1}{a^2 -4a+9} + \frac {1}{b^2 -4b+9} + \frac{1}{c^2 -4c+9} \]
1964 German National Olympiad, 1
Prove that for all positive, entirely rational numbers $a$ and $b$ always holds
$$\frac{a + b}{2} \ge \sqrt[a+b]{a^b \cdot b^a}.$$ When does the equal sign hold?
2017 Nordic, 2
Let $a, b, \alpha, \beta$ be real numbers such that $0 \leq a, b \leq 1$, and $0 \leq \alpha, \beta \leq \frac{\pi}{2}$. Show that if \[ ab\cos(\alpha - \beta) \leq \sqrt{(1-a^2)(1-b^2)}, \] then \[ a\cos\alpha + b\sin\beta \leq 1 + ab\sin(\beta - \alpha). \]
2009 AMC 12/AHSME, 13
A ship sails $ 10$ miles in a straight line from $ A$ to $ B$, turns through an angle between $ 45^{\circ}$ and $ 60^{\circ}$, and then sails another $ 20$ miles to $ C$. Let $ AC$ be measured in miles. Which of the following intervals contains $ AC^2$?
[asy]unitsize(2mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
dotfactor=4;
pair B=(0,0), A=(-10,0), C=20*dir(50);
draw(A--B--C);
draw(A--C,linetype("4 4"));
dot(A);
dot(B);
dot(C);
label("$10$",midpoint(A--B),S);
label("$20$",midpoint(B--C),SE);
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,NE);[/asy]$ \textbf{(A)}\ [400,500] \qquad \textbf{(B)}\ [500,600] \qquad \textbf{(C)}\ [600,700] \qquad \textbf{(D)}\ [700,800]$
$ \textbf{(E)}\ [800,900]$
2014 Contests, 2
Let $x_1,x_2,\ldots,x_n $ be real numbers, where $n\ge 2$ is a given integer, and let $\lfloor{x_1}\rfloor,\lfloor{x_2}\rfloor,\ldots,\lfloor{x_n}\rfloor $ be a permutation of $1,2,\ldots,n$.
Find the maximum and minimum of $\sum\limits_{i=1}^{n-1}\lfloor{x_{i+1}-x_i}\rfloor$ (here $\lfloor x\rfloor $ is the largest integer not greater than $x$).
2011 Dutch BxMO TST, 4
Let $n \ge 2$ be an integer. Let $a$ be the greatest positive integer such that $2^a | 5^n - 3^n$.
Let $b$ be the greatest positive integer such that $2^b \le n$. Prove that $a \le b + 3$.
the 16th XMO, 1
Let $a_1,a_2,\ldots ,a_n\ge 0.$ For all $1\le k\le n$ define
$$b_k:=\min_{1\le i<j\le n,j-i\le 2}|2a_k-a_i-a_j|.$$
Here the index mod $n.$ Find the maximum value of $\frac{b_1+b_2+\cdots +b_n}{a_1+a_2+\cdots +a_n}.$
[i]Proposed by Zheng Wang[/i]
1990 IMO Longlists, 9
Let $\{ a_1, a_2, \ldots, a_n\} = \{1, 2, \ldots, n\}$. Prove that
\[\frac 12 +\frac 23 +\cdots+\frac{n-1}{n} \leq \frac{a_1}{a_2} + \frac{a_2}{a_3} +\cdots+\frac{a_{n-1}}{a_n}.\]
2018 Brazil National Olympiad, 4
Esmeralda writes $2n$ real numbers $x_1, x_2, \dots , x_{2n}$, all belonging to the interval $[0, 1]$, around a circle and multiplies all the pairs of numbers neighboring to each other, obtaining, in the counterclockwise direction, the products $p_1 = x_1x_2$, $p_2 = x_2x_3$, $\dots$ , $p_{2n} = x_{2n}x_1$. She adds the products with even indices and subtracts the products with odd indices. What is the maximum possible number Esmeralda can get?
2013 Switzerland - Final Round, 8
Let $a, b, c > 0$ be real numbers. Show the following inequality:
$$a^2 \cdot \frac{a - b}{a + b}+ b^2\cdot \frac{b - c}{b + c}+ c^2\cdot \frac{c - a}{c + a}
\ge 0 .$$ When does equality holds?
1974 USAMO, 2
Prove that if $ a,b,$ and $ c$ are positive real numbers, then \[ a^ab^bc^c \ge (abc)^{(a\plus{}b\plus{}c)/3}.\]
2006 Bulgaria Team Selection Test, 2
Prove that if $a,b,c>0,$ then \[ \frac{ab}{3a+4b+5c}+\frac{bc}{3b+4c+5a}+\frac{ca}{3c+4a+5b}\le \frac{a+b+c}{12}. \]
[i] Nikolai Nikolov[/i]
2010 Sharygin Geometry Olympiad, 14
We have a convex quadrilateral $ABCD$ and a point $M$ on its side $AD$ such that $CM$ and $BM$ are parallel to $AB$ and $CD$ respectively. Prove that $S_{ABCD} \geq 3 S_{BCM}.$
[i]Remark.[/i] $S$ denotes the area function.
2000 Junior Balkan Team Selection Tests - Romania, 1
Let be a natural number $ n\ge 2, n $ real numbers $ b_1,b_2,\ldots ,b_n , $ and $ n-1 $ positive real numbers $ a_1,a_2,\ldots ,a_{n-1} $ such that $ a_1+a_2+\cdots +a_{n-1} =1. $ Prove the inequality
$$ b_1^2+\frac{b_2^2}{a_1} +\frac{b_3^2}{a_2} +\cdots +\frac{b_n^2}{a_{n-1}} \ge 2b_1\left( b_2+b_3+\cdots +b_n \right) , $$
and specify when equality is attained.
[i]Dumitru Acu[/i]
2024 Kyiv City MO Round 2, Problem 1
Prove that for any real numbers $x, y, z$ at least one of numbers $x^2 + y + \frac{1}{4}, y^2 + z + \frac{1}{4}, z^2 + x + \frac{1}{4}$ is nonnegative.
[i]Proposed by Oleksii Masalitin[/i]
2003 Iran MO (2nd round), 2
In a village, there are $n$ houses with $n>2$ and all of them are not collinear. We want to generate a water resource in the village. For doing this, point $A$ is [i]better[/i] than point $B$ if the sum of the distances from point $A$ to the houses is less than the sum of the distances from point $B$ to the houses. We call a point [i]ideal[/i] if there doesn’t exist any [i]better[/i] point than it. Prove that there exist at most $1$ [i]ideal[/i] point to generate the resource.
2006 Germany Team Selection Test, 2
In a room, there are $2005$ boxes, each of them containing one or several sorts of fruits, and of course an integer amount of each fruit.
[b]a)[/b] Show that we can find $669$ boxes, which altogether contain at least a third of all apples and at least a third of all bananas.
[b]b)[/b] Can we always find $669$ boxes, which altogether contain at least a third of all apples, at least a third of all bananas and at least a third of all pears?
1993 Miklós Schweitzer, 10
Let $U_1 , U_2 , U_3$ be iid random variables on [0,1], which in order of magnitude, $U_1^{\ast} \le U_2^{\ast} \leq U_3 ^ {\ast}$. Let $\alpha, p_1 , p_2 , p_3 \in [0,1]$ such that $P(U_j ^ {\ast} \ge p_j)= \alpha$ ( j = 1,2,3). Prove that
$$P \left( p_1 + (p_2-p_1) U_3^{\ast} + (p_3- p_2) U_2^{\ast} + (1-p_3) U_1^{\ast} \geq \frac{1}{2} \right) \geq 1-\alpha$$
1910 Eotvos Mathematical Competition, 1
If $a, b, c$ are real numbers such that $$a^2 + b^2 + c^2 = 1$$ prove the inequalities $$- \frac12 \le ab + bc + ca \le 1$$
2014 Contests, 3
$N$ in natural. There are natural numbers from $N^3$ to $N^3+N$ on the board. $a$ numbers was colored in red, $b$ numbers was colored in blue. Sum of red numbers in divisible by sum of blue numbers. Prove, that $b|a$
1996 IMO Shortlist, 1
Suppose that $a, b, c > 0$ such that $abc = 1$. Prove that \[ \frac{ab}{ab + a^5 + b^5} + \frac{bc}{bc + b^5 + c^5} + \frac{ca}{ca + c^5 + a^5} \leq 1. \]
1999 Canada National Olympiad, 1
Find all real solutions to the equation $4x^2 - 40 \lfloor x \rfloor + 51 = 0$.