Found problems: 2529
For each real number $a$ with $0 \leq a \leq 1$, let numbers $x$ and $y$ be chosen independently at random from the intervals $[0, a]$ and $[0, 1]$, respectively, and let $P(a)$ be the probability that
$$\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1.$$
What is the maximum value of $P(a)?$
$\textbf{(A)}\ \frac{7}{12} \qquad\textbf{(B)}\ 2 - \sqrt{2} \qquad\textbf{(C)}\ \frac{1+\sqrt{2}}{4} \qquad\textbf{(D)}\ \frac{\sqrt{5}-1}{2} \qquad\textbf{(E)}\ \frac{5}{8}$
Show that for all positive real numbers $a, b$ and $c$, the inequality $(a+b)(a+c)\ge 2\sqrt{abc(a+b+c)}$ is true.
Find all real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying,$$\sqrt{x_{1}-1^{2}}+2 \sqrt{x_{2}-2^{2}}+\cdots+n \sqrt{x_{n}-n^{2}}=\frac{1}{2}\left(x_{1}+x_{2}+\cdots+x_{n}\right)$$
Prove that \[ \left(\frac{6}{5}\right)^{\sqrt{3}}>\left(\frac{5}{4}\right)^{\sqrt{2}}. \]
A deck of $n > 1$ cards is given. A positive integer is written on each card. The deck has the property that the arithmetic mean of the numbers on each pair of cards is also the geometric mean of the numbers on some collection of one or more cards.
For which $n$ does it follow that the numbers on the cards are all equal?
[i]Proposed by Oleg Košik, Estonia[/i]
The real numbers $x,y,z, m, n$ are positive, such that $m + n \ge 2$. Prove that
$x\sqrt{yz(x + my)(x + nz)} + y\sqrt{xz(y + mx)(y + nz)} + z\sqrt{xy(z + mx)(x + ny) }\le \frac{3(m + n)}{8}
(x + y)(y + z)(z + x)$
Find all $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for all distinct $x,y,z$
$f(x)^2-f(y)f(z)=f(x^y)f(y)f(z)[f(y^z)-f(z^x)]$
Let $I$ be an incenter of a triangle $ABC, A_1,B_1,C_1$ be intersection points of the circumcircle of the triangle $ABC$ and the lines $AI, BI, Cl$ respectively. Prove that
a) $\frac{AI}{IA_1}+ \frac{BI}{IB_1}+ \frac{CI}{IC_1}\ge 3$
b) $AI \cdot BI \cdot CI \le I_1A_1\cdot I_2B_1 \cdot I_1C_1$
(D. Pirshtuk)
If $a,b,c$ are three positive real numbers such that $a+b+c=3$, prove that
$ {\frac{a^{2}+3b^{2}}{ab^{2}(4-ab)}}+{\frac{b^{2}+3c^{2}}{bc^{2}(4-ab)}}+{\frac{c^{2}+3a^{2}}{ca^{2}(4-ca)}}\geq 4 $
Find all positive integers $a$, $b$ and $c$ such that $ab$ is a square, and
\[a+b+c-3\sqrt[3]{abc}=1.\]
[i]Proposed by usjl[/i]
Let $x,y,z$ be three positive reals such that $xyz\geq 1$. Prove that
\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]
[i]Hojoo Lee, Korea[/i]
Let $a, b,c$ be positive real numbers satisfying the condition $$(x + y + z) \left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right)= 10$$ Find the greatest value and the least value of
$$T = (x^2 + y^2 + z^2) \left(\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2}\right)$$
Trần Nam Dũng
Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$, $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that $z_{i+j}^2 \geq x_iy_j$ for all $1\le i,j \leq n$.
Let $M=\max\{z_2,\ldots,z_{2n}\}$. Prove that \[
\left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2
\ge
\left( \frac{x_1+\dots+x_n}{n} \right)
\left( \frac{y_1+\dots+y_n}{n} \right). \]
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[i]Edited by Orl.[/i]
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[i]Proposed by Reid Barton, USA[/i]
Let $x,y$ and $z$ be positive real numbers such that $xy+yz+xz=3xyz$. Prove that \[ x^2y+y^2z+z^2x \ge 2(x+y+z)-3 \] and determine when equality holds.
[i]UK - David Monk[/i]
Let $n$ and $k$ be positive integers. Prove that for $a_1, \dots, a_n \in [1,2^k]$ one has
\[ \sum_{i = 1}^n \frac{a_i}{\sqrt{a_1^2 + \dots + a_i^2}} \le 4 \sqrt{kn}. \]
A list of integers with average $89$ is split into two disjoint groups. The average of the integers in the first group is $73$ while the average of the integers in the second group is $111$. What is the smallest possible number of integers in the original list?
[i] Proposed by David Altizio [/i]
Find all pairs of positive integers $ (x,y)$ such that
\[ x^y \equal{} y^{x \minus{} y}.
\]
[i]Albania[/i]
Let $1<k\leq n$ be positive integers and $x_1 , x_2 , \ldots , x_k$ be positive real numbers such that $x_1 \cdot x_2 \cdot \ldots \cdot x_k = x_1 + x_2 + \ldots +x_k.$
a) Show that $x_{1}^{n-1} +x_{2}^{n-1} + \ldots +x_{k}^{n-1} \geq kn.$
b) Find all numbers $k,n$ and $x_1, x_2 ,\ldots , x_k$ for which equality holds.
Let $x,y$ and $z$ be positive real numbers. Show that $x^2+xy^2+xyz^2\ge 4xyz-4$.
Let $x,y$ and $z$ be positive real numbers such that $xy+yz+xz=3xyz$. Prove that \[ x^2y+y^2z+z^2x \ge 2(x+y+z)-3 \] and determine when equality holds.
[i]UK - David Monk[/i]
Find the least value of $k$ such that for all $a,b,c,d \in \mathbb{R}$ the inequality
\[ \begin{array} c \sqrt{(a^2+1)(b^2+1)(c^2+1)} +\sqrt{(b^2+1)(c^2+1)(d^2+1)} +\sqrt{(c^2+1)(d^2+1)(a^2+1)} +\sqrt{(d^2+1)(a^2+1)(b^2+1)} \\ \ \\ \ge 2( ab+bc+cd+da+ac+bd)-k \end{array}\]
holds.
Professor Oak is feeding his $100$ Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is $100$ kilograms. Professor Oak distributes $100$ kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of the bowl). The [i]dissatisfaction level[/i] of a Pokémon who received $N$ kilograms of food and whose bowl has a capacity of $C$ kilograms is equal to $\lvert N-C\rvert$.
Find the smallest real number $D$ such that, regardless of the capacities of the bowls, Professor Oak can distribute food in a way that the sum of the dissatisfaction levels over all the $100$ Pokémon is at most $D$.
[i]Oleksii Masalitin, Ukraine[/i]
A $100 \times 100$ table is given. At the beginning, every unit square has number $"0"$ written in them. Two players playing a game and the game stops after $200$ steps (each player plays $100$ steps).
In every step, one can choose a row or a column and add $1$ to the written number in all of it's squares $\pmod 3.$
First player is the winner if more than half of the squares ($5000$ squares) have the number $"1"$ written in them,
Second player is the winner if more than half of the squares ($5000$ squares) have the number $"0"$ written in them. Otherwise, the game is draw.
Assume that both players play at their best. What will be the result of the game ?
[i]Proposed by Mahyar Sefidgaran[/i]
Let $ a,b$ be $ 2$ distinct positive interger number such that $ (a^2\plus{}ab\plus{}b^2)|ab(a\plus{}b)$. Prove that: $ |a\minus{}b|>\sqrt [3] {ab}$.
we are given $n$ rectangles in the plane. Prove that between $4n$ right angles formed by these rectangles there are at least $[4\sqrt n]$ distinct right angles.