Found problems: 85335
An arbitrary triangle $ABC$ is given. Using ruler and compass construct three pairwise tangent circles $w_A$,$w_B$, $w_C$ with equal radii such that $A \in w_A, B \in w_B, C \in w_C$.
[i]Matsvei Zorka[/i]
In an acute scalene triangle $ABC$ with incenter $I$, the line $AI$ intersects the circumcircle again at $D$, and let $J$ be a point such that $D$ is the midpoint of $IJ$. Consider points $E$ and $F$ on line $BC$ such that $IE$ and $JF$ are perpendicular to $AI$. Consider points $G$ on $AE$ and $H$ on $AF$ such that $IG$ and $JH$ are perpendicular to $AE$ and $AF$, respectively. Prove that $BG=CH$.
Given is a positive integer $n \geq 2$ and three pairwise disjoint sets $A, B, C$, each of $n$ distinct real numbers. Denote by $a$ the number of triples $(x, y, z) \in A \times B \times C$ satisfying $x<y<z$ and let $b$ denote the number of triples $(x, y, z) \in A \times B \times C$ such that $x>y>z$. Prove that $n$ divides $a-b$.
Find the number of positive integers $n$ with exactly $1974$ factors such that no prime greater than $40$ divides $n$, and $n$ ends in one of the digits $1$, $3$, $7$, $9$. (Note that $1974 = 2 \cdot 3 \cdot 7 \cdot 47$.)
[i]Proposed by Yonah Borns-Weil[/i]
What is the product of the real roots of the equation \[x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}\,\,?\]
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ that are coprime to $n$. Find the sum of all $1<n<100$ such that $\phi(n)\mid n$.
Find all integers of the form $2^n$ (where $n$ is a natural number) such that after deleting the first digit of its decimal representation we again get a power of $2$.
We are given a polyhedron with at least $5$ vertices, such that exactly $3$ edges meet in each of the vertices. Prove that we can assign a rational number to every vertex of the given polyhedron such that the following conditions are met:
$(i)$ At least one of the numbers assigned to the vertices is equal to $2020$.
$(ii)$ For every polygonal face, the product of the numbers assigned to the vertices of that face is equal to $1$.
Consider the following expression
$$S = \log_2 \left( \sum^{2019}_{k=1}\sum^{2020}_{j=2}\log_{2^{1/k}} (j) \log_{j^2} \left(\sin \frac{\pi k}{2020}\right) \right).$$
Find the smallest integer $n$ which is bigger than $S$ (i.e. find $\lceil S \rceil$).
What is the largest number you can write with three $3$’s and three $8$’s, using only symbols $+,-,/,\times$ and exponentiation?
Find every twice-differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfies the functional equation
$$ f(x)^2 -f(y)^2 =f(x+y)f(x-y)$$
for all $x,y \in \mathbb{R}. $
Let $n$ be a positive integer. The real numbers $a_1,a_2,\cdots,a_n$ and $r_1,r_2,\cdots,r_n$ are such that $a_1\leq a_2\leq \cdots \leq a_n$ and $0\leq r_1\leq r_2\leq \cdots \leq r_n$.
Prove that $\sum_{i=1}^n\sum_{j=1}^n a_i a_j \min (r_i,r_j)\geq 0$
Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that
\[
a b \leqslant-\frac{1}{2019}.
\]
We will call a [i]hedgehog[/i] a graph in which one vertex is connected to all the others and there are no other edges; the number of vertices of this graph will be called the size of the hedgehog. A graph $G$ is given on $n$ vertices (where $n > 1$). For each edge $e$, we denote by $s(e)$ the size of the maximum hedgehog in graph $G$, which contains this edge. Prove the inequality (summation is carried out over all edges of the graph $G$):
\[\sum_e \frac{1}{s(e)} \leqslant \frac{n}{2}.\]
[i]Proposed by D. Malec, C. Tompkins[/i]
Let $ G$ be a simple graph. Suppose that size of largest independent set in $ G$ is $ \alpha$. Prove that:
a) Vertices of $ G$ can be partitioned to at most $ \alpha$ paths.
b) Suppose that a vertex and an edge are also cycles. Prove that vertices of $ G$ can be partitioned to at most $ \alpha$ cycles.
How many positive integers a less than $100$ such that $4a^2 + 3a + 5$ is divisible by $6$.
Let $ A$ be a $ n\times n$ matrix with complex elements. Prove that $ A^{\minus{}1} \equal{} \overline{A}$ if and only if there exists an invertible matrix $ B$ with complex elements such that $ A\equal{} B^{\minus{}1} \cdot \overline{B}$.
A convex hexagon $ABCDEF$ is given whose sides $AB$ and $DE$ are parallel. Each of the diagonals $AD, BE, CF$ divides this hexagon into two quadrilaterals of equal perimeters. Show that these three diagonals intersect at one point.
A pair of words consisting only of the letters $a$ and $b$ (with repetitions) is [i]good[/i] if it is $(a,b)$ or of one of the forms $(uv, v)$, $(u, uv)$, where $(u,v)$ is a good pair. Prove that if $(\alpha, \beta)$ is a good pair, then there exists a palindrome $\gamma$ such that $\alpha\beta = a\gamma b$.
Three distinct points $A$, $B$, and $C$ are fixed on a line in this order. Let $\Gamma$ be a circle passing through $A$ and $C$ whose center does not lie on the line $AC$. Denote by $P$ the intersection of the tangents to $\Gamma$ at $A$ and $C$. Suppose $\Gamma$ meets the segment $PB$ at $Q$. Prove that the intersection of the bisector of $\angle AQC$ and the line $AC$ does not depend on the choice of $\Gamma$.
Given several numbers, one of them, $a$, is chosen and replaced by the three numbers $\frac{a}{3}, \frac{a}{3}, \frac{a}{3}$. This process is repeated with the new set of numbers, and so on. Originally, there are $1000$ ones, and we apply the process several times. A number $m$ is called [i]good[/i] if there are $m$ or more numbers that are the same after each iteration, no matter how many or what operations are performed. Find the largest possible good number.
Let $x_1,x_2,\cdots,x_n$ $(n\geq2)$ be a non-decreasing monotonous sequence of positive numbers such that $x_1,\frac{x_2}{2},\cdots,\frac{x_n}{n}$ is a non-increasing monotonous sequence .Prove that
\[ \frac{\sum_{i=1}^{n} x_i }{n\left (\prod_{i=1}^{n}x_i \right )^{\frac{1}{n}}}\le \frac{n+1}{2\sqrt[n]{n!}}\]
For which $n{}$ is it possible that a product of $n{}$ consecutive positive integers is equal to a sum of $n{}$ consecutive (not necessarily the same) positive integers?
[i]Boris Frenkin[/i]
For each natural $n \ge 4$, find the smallest natural number $k$ that satisfies following condition:
For an arbitrary arrangement of $k$ chips of two colors on $n\times n$ board, there exists a non-empty set such that all columns and rows contain even number ($0$ is also possible) of chips each color.
Let $ ABCDEF$ be a regular hexagon. Let $ G$, $ H$, $ I$, $ J$, $ K$, and $ L$ be the midpoints of sides $ AB$, $ BC$, $ CD$, $ DE$, $ EF$, and $ AF$, respectively. The segments $ AH$, $ BI$, $ CJ$, $ DK$, $ EL$, and $ FG$ bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of $ ABCDEF$ be expressed as a fraction $ \frac {m}{n}$ where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.