Found problems: 85335
Let $a, b, c, d$ be positive real numbers satisfying $a+b+c+d =1$. Show that \[\frac{2}{(a+b)(c+d)} \leq \frac{1}{\sqrt{ab}}+ \frac{1}{\sqrt{cd}}\mbox{.}\]
[i](Swiss Mathematical Olympiad 2011, Final round, problem 6)[/i]
A natural number is called a [i]prime power[/i] if that number can be expressed as $p^n$ for some prime $p$ and natural number $n$.
Determine the largest possible $n$ such that there exists a sequence of prime powers $a_1, a_2, \dots, a_n$ such that $a_i = a_{i - 1} + a_{i - 2}$ for all $3 \le i \le n$.
Let $a$, $b$, $c$ be the solutions to $x^3+3x^2-1=0$. Define $S_n=a^n+b^n+c^n$. Given that there are integers $0\le i,j,k\le36$ such that $S_n\equiv i^n+j^n+k^n\pmod{37}$ for all positive integer $n$, determine the product $ijk$.
At a certain pizzeria, there are five different toppings available and a pizza can be ordered with any (possibly empty) subset of them on it. In how many ways can one order an unordered pair of pizzas such that at most one topping appears on both pizzas and at least one topping appears on neither?
Let $a_1 < a_2 < \cdots <a_n$ be pairwise coprime positive integers with $a_1$ being prime and $a_1 \ge n + 2$. On the segment $I = [0, a_1 a_2 \cdots a_n ]$ of the real line, mark all integers that are divisible by at least one of the numbers $a_1 , \ldots , a_n$ . These points split $I$ into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by $a_1$.
[i]Proposed by Serbia[/i]
Let $ f : \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable, $ 2 \pi$-periodic even function. Prove that if \[ f''(x)\plus{}f(x)\equal{}\frac{1}{f(x\plus{} 3 \pi /2 )}\] holds for every $ x$, then $ f$ is $ \pi /2$-periodic.
[i]Z. Szabo, J. Terjeki[/i]
The Ministry of Education selects $2010$ students from $5$ regions of Thailand to participate in a debate tournament, where each pair of students will debate in one of the three topics: politics, economics, and societal problems. Show that there are $3$ students who were born in the same month, come from the same region, are of the same gender , and whose pairwise debates are on the same topic.
The set $\{1, 2, \cdots, 49\}$ is divided into three subsets. Prove that at least one of these subsets contains three different numbers $a, b, c$ such that $a + b = c$.
Let $ ABCD$ be a cyclic convex quadrilateral. Let $ E $ be the intersection of lines $ AB, CD $. $ P $ is the intersection of line passing $ B $ and perpendicular to $ AC $, and line passing $ C $ and perpendicular to $ BD$. $ Q $ is the intersection of line passing $ D $ and perpendicular to $ AC $, and line passing $ A $ and perpendicular to $ BD $. Prove that three points $ E, P, Q $ are collinear.
Points $M,\ N,\ K,\ L$ are taken on the sides $AB,\ BC,\ CD,\ DA$ of a rhombus
$ABCD,$ respectively, in such a way that $MN\parallel LK$ and the distance between $MN$ and $KL$ is equal to the height of $ABCD.$ Show that the circumcircles of the triangles $ALM$ and $NCK$ intersect each other, while those of $LDK$ and $MBN$ do not.
On a $30\times 30$ board both rows $ 1$ to $30$ and columns are numbered, in addition, to each box is assigned the number $ij$, where the box is in row $i$ and column $j$.
$N$ columns and $m$ rows are chosen, where $1 <n$ and $m <30$, and the cells that are simultaneously in any of the rows and in any of the selected columns are painted blue. They paint the others red .
(a) Prove that the sum of the numbers in the blue boxes cannot be prime.
(b) Can the sum of the numbers in the red cells be prime?
Denote $S$ as the set of prime divisors of all integers of form $2^{n^2+1} - 3^n, n \in Z^+$.
Prove that $S$ and $P-S$ both contain infinitely many elements (where $P$ is set of prime numbers).
A number of $N$ children are at a party and they sit in a circle to play a game of Pass and Parcel. Because the host has no other form of entertainment, the parcel has infinitely many layers. On turn $i$, starting with $i=1$, the following two things happen in order:
[b]$(1)$[/b] The parcel is passed $i^2$ positions clockwise; and
[b]$(2)$[/b] The child currently holding the parcel unwraps a layer and claims the prize inside.
For what values of $N$ will every chidren receive a prize?
$Patrick \ Winter \, United \ Kingdom$
Two circles $\omega_1(O)$ and $\omega_2$ intersect each other at $A,B$ ,and $O$ lies on $\omega_2$. Let $S$ be a point on $AB$ such that $OS\perp AB$. Line $OS$ intersects $\omega_2$ at $P$ (other than $O$). The bisector of $\hat{ASP}$ intersects $\omega_1$ at $L$ ($A$ and $L$ are on the same side of the line $OP$). Let $K$ be a point on $\omega_2$ such that $PS=PK$ ($A$ and $K$ are on the same side of the line $OP$). Prove that $SL=KL$.
[i]Proposed by Ali Zamani [/i]
Prove that the system of equations
$
\begin{cases}
\ a^2 - b = c^2 \\
\ b^2 - a = d^2 \\
\end{cases}
$
has no integer solutions $a, b, c, d$.
Find all the odd naturals whose indicator is the same as $1990$.
We clarify that, if a natural decomposes into prime factors in the form $\Pi_{j=1}^r p_j^{a_j}$, define the [i]indicator [/i] as : $\phi (n) = r\Pi_{j=1}^r p_j^{a_j-1} (p_j + 1)$.
[hide=official wording for first sentence]Encuentre todos los naturales impares cuyo indicador es el mismo que el de 1990.[/hide]
Let $ABC$ be a acute triangle where $\angle BAC =60$. Prove that if the Euler's line of $ABC$ intersects $AB,AC$ in $D,E$, then $ADE$ is equilateral.
Let $d$ be any positive integer not equal to 2, 5, or 13. Show that one can find distinct $a$ and $b$ in the set $\{2,5,13,d\}$ such that $ab - 1$ is not a perfect square.
Does there exist a set $ M$ of points in space such that every plane intersects $ M$ at a finite but nonzero number of points?
Each of the $11$ girls wants to mail each of the other a gift for Christmas. The packages contain no more than two gifts. If they have enough time, what is the smallest possible number of packages that they have to send?
Chess piece called [i]skew knight[/i], if placed on the black square, attacks all the gray squares.
[img]https://i.ibb.co/HdTDNjN/Kyiv-MO-2021-Round-1-11-2.png[/img]
What is the largest number of such knights that can be placed on the $8\times 8$ chessboard without them attacking each other?
[i]Proposed by Arsenii Nikolaiev[/i]
Let $p$ and $q$ be different primes, and $\alpha\in (0, 3)$ a real number. Prove that in sequence
$$\left[ \alpha \right] , \left[ 2\alpha \right] , \left[ 3\alpha \right] \dots$$
exists number less than $2pq$, divisible by $p$ or $q$.
There is a $40\%$ chance of rain on Saturday and a $30\%$ of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$.
A [i]special number[/i] is a positive integer $n$ for which there exists positive integers $a$, $b$, $c$, and $d$ with \[ n = \frac {a^3 + 2b^3} {c^3 + 2d^3}. \] Prove that
i) there are infinitely many special numbers;
ii) $2014$ is not a special number.
[i]Romania[/i]
In a certain country there are more than $101$ towns. The capital of this country is connected by direct air routes with $100$ towns, and each town, except for the capital, is connected by direct air routes with $10$ towns (if $A$ is connected with $B, B$ is connected with $A$). It is known that from any town it is possible to travel by air to any other town (possibly via other towns). Prove that it is possible to close down half of the air routes connected with the capital, and preserve the capability of travelling from any town to any other town within the country.
(IS Rubanov)