Found problems: 85335
Two $10$-digit integers are called neighbours if they differ in exactly one digit (for example, integers $1234567890$ and $1234507890$ are neighbours). Find the maximal number of elements in the set of $10$-digit integers with no two integers being neighbours.
Quadrilateral $ABCD$ with perpendicular diagonals $AC$ and $BD$ is inscribed in a circle. Altitude $DE$ in triangle $ABD$ intersects diagonal $AC$ in $F$. Prove that $FB = BC$
On a river with a current speed of \(3 \, \text{km/h}\), there are two harbors. Every Saturday, a cruise ship departs from Harbor 1 to Harbor 2, stays overnight, and returns to Harbor 1 on Sunday. On the ship live two snails, Romeo and Juliet. One Saturday, immediately after the ship departs, both snails start moving to meet each other and do so exactly when the ship arrives at Harbor 2. On the following Sunday, as the ship departs from Harbor 2, Romeo starts moving towards Juliet's house and reaches there exactly when the ship arrives back at Harbor 1. Given that Juliet moves half as fast as Romeo, determine the speed of the ship in still water.
[i]Proposed by Demetre Gelashvili, Georgia [/i]
The numerical sequence $x_1 , x_2 ,.. $ satisfies $x_1 = \frac12$ and $x_{k+1} =x^2_k+x_k$ for all natural integers $k$ . Find the integer part of the sum $\frac{1}{x_1+1}+\frac{1}{x_2+1}+...+\frac{1}{x_{100}+1}$
{A. Andjans, Riga)
We randomly choose 5 distinct positive integers less than or equal to 90. What is the floor of 10 times the expected value of the fourth largest number?
In triangle \( ABC \), let \( I \) be the \( A \)-excenter. Points \( X \) and \( Y \) are placed on line \( BC \) such that \( B \) is between \( X \) and \( C \), and \( C \) is between \( Y \) and \( B \). Moreover, \( B \) and \( C \) are the contact points of \( BC \) with the \( A \)-excircle of triangles \( BAY \) and \( AXC \), respectively. Let \( J \) be the \( A \)-excenter of triangle \( AXY \), and let \( H' \) be the reflection of the orthocenter of triangle \( ABC \) with respect to its circumcenter. Prove that \( I \), \( J \), and \( H' \) are collinear.
Proposed by Ali Nazarboland
Find $x^2$ given that $\tan^{-1}(x)+\tan^{-1}(3x)=\frac{\pi}{6}$ and $0<x<\frac{\pi}{6}$.
Let $ a$, $ b$, $ c$, $ d$ be positive real numbers such that $ abcd \equal{} 1$ and $ a \plus{} b \plus{} c \plus{} d > \dfrac{a}{b} \plus{} \dfrac{b}{c} \plus{} \dfrac{c}{d} \plus{} \dfrac{d}{a}$. Prove that
\[ a \plus{} b \plus{} c \plus{} d < \dfrac{b}{a} \plus{} \dfrac{c}{b} \plus{} \dfrac{d}{c} \plus{} \dfrac{a}{d}\]
[i]Proposed by Pavel Novotný, Slovakia[/i]
A point $A_1$ is marked inside an acute non-isosceles triangle $ABC$ such that $\angle A_1AB = \angle A_1BC$ and $\angle A_1AC=\angle A_1CB$. Points $B_1$ and $C_1$ are defined same way. Let $G$ be the gravity center if the triangle $ABC$.
Prove that the points $A_1,B_1,C_1,G$ are concyclic.
Let $a$, $b$, $c$ be real numbers. Prove that
\[ab + bc + ca + \max\{|a - b|, |b - c|, |c - a|\} \le 1 + \frac{1}{3} (a + b + c)^2.\]
Given an arbitrary natural number $ n, $ is there a multiple of $ n $ whose base $ 10 $ representation can be written only with the digits $ 0,2,7? $ Explain.
The base $A_{1}A_{2}\ldots A_{n}$ of the pyramid $VA_{1}A_{2}\ldots A_{n}$ is a regular polygon. Prove that if \[\angle VA_{1}A_{2}\equiv \angle VA_{2}A_{3}\equiv \cdots \equiv \angle VA_{n-1}A_{n}\equiv \angle VA_{n}A_{1},\] then the pyramid is regular.
How many positive perfect squares less than $2023$ are divisible by $5$?
$\textbf{(A) } 8 \qquad\textbf{(B) }9 \qquad\textbf{(C) }10 \qquad\textbf{(D) }11 \qquad\textbf{(E) } 12$
Is it possible to select 100 points on a circle so that there are exactly 1000 right triangles with the vertices at selected points?
[i]Sergey Dvoryaninov[/i]
Find all $f : \mathbb{R}\to \mathbb{R}$ , bounded in $(0,1)$ and satisfying:
$x^2 f(x) - y^2 f(y) = (x^2-y^2) f(x+y) -xy f(x-y)$
for all $x,y \in \mathbb{R}$
[i]Proposed by Nikolay Nikolov[/i]
Each two of $n$ students, who attended an activity, have different ages. It is given that each student shook hands with at least one student, who did not shake hands with anyone younger than the other. Find all possible values of $n$.
It is given positive integer $n$. Let $a_1, a_2,..., a_n$ be positive integers with sum $2S$, $S \in \mathbb{N}$. Positive integer $k$ is called separator if you can pick $k$ different indices $i_1, i_2,...,i_k$ from set $\{1,2,...,n\}$ such that $a_{i_1}+a_{i_2}+...+a_{i_k}=S$. Find, in terms of $n$, maximum number of separators
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$
Let $\alpha,\beta ,\gamma$ be the angles of $\triangle ABC$.
a) Show that $cos^2\alpha +cos^2\beta +cos^2 \gamma =1-2cos\alpha cos\beta cos\gamma$ .
b) Given that $cos\alpha : cos\beta : cos\gamma = 39 : 33 : 25$, find $sin\alpha : sin\beta : sin\gamma$ .
Let $n$ be a positive integer. Suppose we are given $2^n+1$ distinct sets, each containing finitely many objects. Place each set into one of two categories, the red sets and the blue sets, so that there is at least one set in each category. We define the [i]symmetric difference[/i] of two sets as the set of objects belonging to exactly one of the two sets. Prove that there are at least $2^n$ different sets which can be obtained as the symmetric difference of a red set and a blue set.
Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$.
Prove that Sisyphus cannot reach the aim in less than
\[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \]
turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )
Mark $4 \cdot k$ points in a circle and number them arbitrarily with numbers from $1$ to $4 \cdot k$. The chords cannot share common endpoints, also, the endpoints of these chords should be among the $4 \cdot k$ points.
[b]i.[/b] Prove that $2 \cdot k$ pairwisely non-intersecting chords can be drawn for each of whom its endpoints differ in at most $3 \cdot k - 1$.
[b]ii.[/b] Prove that the $3 \cdot k - 1$ cannot be improved.
For each real number $x,$ let $\lfloor x\rfloor$ denote the greatest integer that does not exceed $x.$ For how many positive integers $n$ is it true that $n<1000$ and that $\lfloor \log_2 n\rfloor$ is a positive even integer.
Let $a,b$ be positive integers . How many integers numbers can be written in the form $ap+bq$, where $p,q$ are nonnegative integers and $p+q\le{2005}$ ?
There are 20 buns with jam and 20 buns with treacle arranged in a row in random order. Alice and Bob take in turn a bun from any end of the row. Alice starts, and wants to finally obtain 10 buns of each type; Bob tries to prevent this. Is it true for any order of the buns that Alice can win no matter what are the actions of Bob?
[i]Alexandr Gribalko[/i]