Found problems: 85335
Determine an equation of third degree with integral coefficients having roots $\sin \frac{\pi}{14}, \sin \frac{5 \pi}{14}$ and $\sin \frac{-3 \pi}{14}.$
In a given triangle $ABC$, $O$ is its circumcenter, $D$ is the midpoint of $AB$ and $E$ is the centroid of the triangle $ACD$. Show that the lines $CD$ and $OE$ are perpendicular if and only if $AB=AC$.
Let $a,b,c\in\mathbb{N}$ prove that if there is a polynomial $P,Q,R\in\mathbb{C}[x]$, which have no common factors and satisfy $$P^a+Q^b=R^c$$ and $$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}>1.$$
[i](tatari/nightmare)[/i]
A set $S$ of $n-1$ natural numbers is given ($n\ge 3$). There exist at least at least two elements in this set whose difference is not divisible by $n$. Prove that it is possible to choose a non-empty subset of $S$ so that the sum of its elements is divisible by $n$.
Let $a$ and $b$ be positive integers and $K=\sqrt{\frac{a^2+b^2}2}$, $A=\frac{a+b}2$. If $\frac KA$ is a positive integer, prove that $a=b$.
Given a quadrilateral $ABCD$ inscribed in circle $\Gamma$.From a point P outside $\Gamma$, draw tangents $PA$ and $PB$ with $A$ and $B$ as touspoints. The line $PC$ intersects $\Gamma$ at point $D$. Draw a line through $B$ parallel to $PA$, this line intersects $AC$ and $AD$ at points $E$ and $F$ respectively. Prove that $BE = BF$.
Let $a,b,c,x,y,z,t$ be positive real numbers with $1\le x,y,z\le4$. Prove that
$$\frac x{(2a)^t}+\frac y{(2b)^t}+\frac z{(2c)^t}\ge\frac{y+z-x}{(b+c)^t}+\frac{z+x-y}{(c+a)^t}+\frac{x+y-z}{(a+b)^t}.$$
Determine the largest positive integer $n$ such that the following statement is true:
There exists $n$ real polynomials, $P_1(x),\ldots,P_n(x)$ such that the sum of any two of them have no real roots but the sum of any three does.
Let $n \geq 3$ be an integer. Two players play a game on an empty graph with $n + 1$ vertices, consisting of the vertices of a regular n-gon and its center. They alternately select a vertex of the n-gon and draw an edge (that has not been drawn) to an adjacent vertex on the n-gon or to the center of the n-gon. The player who first makes the graph connected wins. Between the player who goes first and the player who goes second, who has a winning strategy?
[i]Note: an empty graph is a graph with no edges.[/i]
Prove that there are infinitely many natural numbers $n$ such that we can divide $1,2,\ldots ,3n$ into three sequences $(a_n),(b_n)$ and $(c_n)$, with $n$ terms in each, satisfying the following conditions:
i) $a_1+b_1+c_1= a_2+b_2+c_2=\ldots =a_n+b_n+c_n$ and $a_1+b_1+c_1$ is divisible by $6$;
ii) $a_1+a_2+\ldots +a_n= b_1+b_2+\ldots +b_n=c_1+c_2+\ldots +c_n,$ and $a_1+a_2+\ldots +a_n$ is divisible by $6$.
The $\underline{\text{path number}}$ of a graph is the minimum number of paths we need to partition the vertices of a graph. Given a connected graph with the independence number $k > 1$, what is the maximum possible value for the path number in this graph? Find the answer in terms of $k$.
The independence number of a graph $\textbf{G}$ is the maximum possible number $k$, such that there exist $k$ pairwise non-adjacent vertices in $\textbf{G}$.
Let $ABCD$ be a rectangle with sides $AB,BC,CD$ and $DA$. Let $K,L$ be the midpoints of the sides $BC,DA$ respectivily. The perpendicular from $B$ to $AK$ hits $CL$ at $M$. Find
$$\frac{[ABKM]}{[ABCL]}$$
Determine all the natural numbers $n$ such that exactly one fifth of the natural numbers $1,2,...,n$ are divisors of $n$.
Given $n$ points in the plane, show that we can always find three which give an angle $\le \pi / n$.
Find all real roots of the equation \[ \sqrt{x^2-p}+2\sqrt{x^2-1}=x \] where $p$ is a real parameter.
For how many ordered pairs of positive integers $ (x,y)$ is $ x \plus{} 2y \equal{} 100$?
$ \textbf{(A)}\ 33 \qquad \textbf{(B)}\ 49 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 99 \qquad \textbf{(E)}\ 100$
Consider all sums that add up to $2015$. In each sum, the addends are consecutive positive integers, and all sums have less than $10$ addends. How many such sums are there?
$P$ and $Q$ are points on the longest side $AB$ of triangle $ABC$ such that $AQ = AC$ and $BP = BC$. Prove that the circumcentre of triangle $CPQ$ coincides with the incentre of triangle $ABC$.
Compute the number of quadruples $(a,b,c,d)$ of positive integers satisfying $$12a+21b+28c+84d=2024.$$
Let $A$ be a real $n\times n$ matrix satisfying
$$A+A^{\text T}=I,$$where $A^{\text T}$ denotes the transpose of $A$ and $I$ the $n\times n$ identity matrix. Show that $\det A>0$.
Let $ABC$ be a triangle with $AB = AC$. Suppose that the bisector of $\angle ABC$ meets the side $AC$ at point $D$ such that $BC = BD+AD$. Find the measure of $\angle BAC$.
We are given $n$ mass points of equal mass in space. We define a sequence of points $O_1,O_2,O_3,\ldots $ as follows: $O_1$ is an arbitrary point (within the unit distance of at least one of the $n$ points); $O_2$ is the centre of gravity of all the $n$ given points that are inside the unit sphere centred at $O_1$;$O_3$ is the centre of gravity of all of the $n$ given points that are inside the unit sphere centred at $O_2$; etc. Prove that starting from some $m$, all points $O_m,O_{m+1},O_{m+2},\ldots$ coincide.
consider $n\geq 6$ points $x_1,x_2,\dots,x_n$ on the plane such that no three of them are colinear. We call graph with vertices $x_1,x_2,\dots,x_n$ a "road network" if it is connected, each edge is a line segment, and no two edges intersect each other at points other than the vertices. Prove that there are three road networks $G_1,G_2,G_3$ such that $G_i$ and $G_j$ don't have a common edge for $1\leq i,j\leq 3$.
Proposed by Morteza Saghafian
Prove that for any prime $p,$ there exists a positive integer $n$ such that
\[1^n+2^{n-1}+3^{n-2}+\cdots+n^1\equiv 2020\pmod{p}.\]
[i]Robin Son[/i]
For positive integer $n$ let $z_n=\sqrt{\frac{3}{n}}+i$, where $i=\sqrt{-1}$. Find $|z_1 \cdot z_2 \cdot z_3 \cdots z_{47}|$.