Found problems: 85335
If possible, construct an equilateral triangle whose three vertices are on three given circles.
From a set of consecutive natural numbers one number is excluded so that the aritmetic mean of the remaining numbers is $ 50.55$. Find the initial set of numbers and the excluded number.
Find all triples $m$; $n$; $l$ of natural numbers such that
$m + n = gcd(m; n)^2$; $m + l = gcd(m; l)^2$; $n + l = gcd(n; l)^2$:
[i]S. Tokarev[/i]
Let $k$ be a constant such that exactly three real values of $x$ satisfy $$x-|x^2-4x+3| = k$$ The sum of all possible values of $k$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.
[i]Proposed by Andy Xu[/i]
There is convex $2009$-gon on the plane.
[b]a)[/b] Find the greatest number of vertices of $2009$-gon such that no two forms the side of the polygon.
[b]b)[/b] Find the greatest number of vertices of $2009$-gon such that among any three of them there is one that is not connected with other two by side.
Compute the sum of all positive integers less than $100$ that do not have consecutive $1$s in their binary representation.
Let $ABC$ be a triangle, and $A_1, B_1, C_1$ be points on the sides $BC, CA, AB$, respectively, such that $AA_1, BB_1, CC_1$ are the internal angle bisectors of $\triangle ABC$. The circumcircle $k' = (A_1B_1C_1)$ touches the side $BC$ at $A_1$. Let $B_2$ and $C_2$, respectively, be the second intersection points of $k'$ with lines $AC$ and $AB$. Prove that $|AB| = |AC|$ or $|AC_1| = |AB_2|$.
Prove that between every $27$ different positive integers , less than $100$, there exist some two which are[color=red] NOT [/color]relative prime.
[u]babis[/u]
Let $a, b$, and $c$ be positive real numbers satisfying $abc = 1$.
Show that $a + b + c \ge \sqrt{\frac13 (a + 2)(b + 2)(c + 2)}$
What conditions should real numbers $ a $, $ b $, $ c $, $ d $, $ e $, $ f $ meet in order for a polynomial of second degree $$ax^2 + 2bxy + cy^2 + 2dx + 2ey + f$$ was the product of two first degree polynomials with real coefficients ?
In the sixth, seventh, eighth, and ninth basketball games of the season, a player scored $ 23$, $ 14$, $ 11$, and $ 20$ points, respectively. Her points-per-game average was higher after nine games than it was after the first five games. If her average after ten games was greater than $ 18$, what is the least number of points she could have scored in the tenth game?
$ \textbf{(A)}\ 26\qquad
\textbf{(B)}\ 27\qquad
\textbf{(C)}\ 28\qquad
\textbf{(D)}\ 29\qquad
\textbf{(E)}\ 30$
You are traveling in a foreign country whose currency consists of five different-looking kinds of coins.
You have several of each coin in your pocket. You remember that the coins are worth $1, 2, 5, 10$, and $20$ florins, but you have no idea which coin is which and you don’t speak the local language. You find a vending machine where a single candy can be bought for $1$ florin: you insert any kind of coin, and receive $1$ candy plus any change owed. You can only buy one candy at a time, but you can buy as many as you want, one after the other.
What is the least number of candies that you must buy to ensure that you can determine the values of all the coins? Prove that your answer is correct.
Let \( n \) be a positive integer, and let \( A \) and \( B \) be \( n \times n \) matrices with real coefficients such that
\[
ABBA - BAAB = A - B.
\]
(a) Prove that \( \text{Tr}(A) = \text{Tr}(B) \) and that \( \text{Tr}(A^2) = \text{Tr}(B^2) \).
(b) If \(BA^2B= A^2B^2\) and \(AB^2A= B^2A^2\), prove that \( \det A = \det B \).
Note: \( \text{Tr}(X) \) denotes the trace of \( X \), which is the sum of the elements on its main diagonal, and \( \det X \) denotes the determinant of \( X \).
A natural number is written in each square of an $ m \times n$ chess board. The allowed move is to add an integer $ k$ to each of two adjacent numbers in such a way that non-negative numbers are obtained. (Two squares are adjacent if they have a common side.) Find a necessary and sufficient condition for it to be possible for all the numbers to be zero after finitely many operations.
The real-valued function $f$ is defined for positive integers, and the positive integer $a$ satisfies
$f(a) = f(1995), f(a+1) = f(1996), f(a+2) = f(1997), f(n + a) = \frac{f(n) - 1}{f(n) + 1}$ for all positive integers $n$.
(i) Show that $f(n+ 4a) = f(n)$ for all positive integers $n$.
(ii) Determine the smallest possible $a$.
There are three candidates in the Hundilaane forest governor elections: $A, B$ and $C$. For each of the $20$ forest dwellers, the names of all three candidates were written on the ballot paper in the order of their preference. Examination of the ballots revealed that $11$ forest dwellers prefer $A$, $12$ $B$ and $14$ $C$. Which of the candidates will be marked first on the largest number of ballot papers when it is known that each possible the order of the candidates appears on at least one ballot?
Calculate the lower bound of the areas of convex hexagons whose vertices all have integer coordinates.
A graph $G(V,E)$ is triangle-free, but adding any edges to the graph will form a triangle. It's given that $|V|=2019$, $|E|>2018$, find the minimum of $|E|$ .
Let $k$ be a positive integer. Show that there exist infinitely many positive integers $n$ such that $\frac{n^n-1}{n-1}$ has at least $k$ distinct prime divisors.
[i]Proposed by Adnan Sadik[/i]
Let $f : \mathbb{Z} \rightarrow \mathbb{Z}^+$ be a function, and define $h : \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}^+$ by $h(x, y) = \gcd (f(x), f(y))$. If $h(x, y)$ is a two-variable polynomial in $x$ and $y$, prove that it must be constant.
An equilateral triangle $BDE$ is constructed on the diagonal $BD$ of the square $ABCD$, and the point $C$ is located inside the triangle $BDE$. Let $M$ be the midpoint of $BE$. Find the angle between the lines $MC$ and $DE$.
(Dmitry Shvetsov)
Prove no three lattice points in the plane form an equilateral triangle.
Find a perfect set $ H \subset [0,1]$ of positive measure and a continuous function $ f$ defined on $ [0,1]$ such that for any twice differentiable function $ g$ defined on $ [0,1]$, the set $ \{ x \in H : \;f(x)\equal{}g(x)\ \}$ is finite.
[i]M. Laczkovich[/i]
in a triangle $ABC$, $I$ is the incenter. $BI$ and $CI$ cut the circumcircle of $ABC$ at $E$ and $F$ respectively. $M$ is the midpoint of $EF$. $C$ is a circle with diameter $EF$. $IM$ cuts $C$ at two points $L$ and $K$ and the arc $BC$ of circumcircle of $ABC$ (not containing $A$) at $D$. prove that $\frac{DL}{IL}=\frac{DK}{IK}$.(25 points)
The length of the middle line of a trapezoid is $4$ and the angles at one of the bases are $40^\circ$ and $50^\circ$. Determine the lengths of the bases if the distance between their midpoints is $1$.