Found problems: 85335
A bipartite graph on the sets $\{ x_1,\ldots, x_n \}$ and $\{ y_1,\ldots, y_n\}$ of vertices (that is the edges are of the form $x_iy_j$) is called tame if it has no $x_iy_jx_ky_l$ path ($i,j,k,l\in\{ 1,\ldots, n\}$) where $j<l$ and $i+j>k+l$. Calculate the infimum of those real numbers $\alpha$ for which there exists a constant $c=c(\alpha)>0$ such that for all tame graphs $e\le cn^{\alpha}$, where $e$ is the number of edges and $n$ is half of the number of vertices.
(translated by Miklós Maróti)
A group is call locally cyclic if any finitely generated subgroup is cyclic. Prove that a locally cyclic group is isomorphic to one of its proper subgroups if and only if it's isomorphic to a proper subgroup of the rational numbers with the adition.
Let $AB CD$ be a rectangle with $AB=1$. If $m ( \angle BDC) = 82^o30'$, compute the length of$ BD$ and the cosine of $82^o30'$.
How many non-negative integer solutions are there for $x^4-2y^2=1$?
For what integer values of $n$ is $1+n+\frac{n^2}{2}+\cdots +\frac{n^n}{n!}$ an integer?
Let $K$ be a cube with edge $n$, where $n>2$ is an even integer. Cube $K$ is divided into $n^3$ unit cubes. We call any set of $n^2$ unit cubes lying on the same horizontal or vertical level a layer. We dispose of $\frac{n^3}4$ colors, in each of which we paint exactly $4$ unit cubes. Prove that we can always select $n$ unit cubes of distinct colors, no two of which lie on the same layer.
In a circle $\odot O$, radius $OA$ is perpendicular to radius $OB$. Chord $AC$ intersects $OB$ at $E$ so that the length of arc $AC$ is one-third the circumference of $\odot O$. Point $D$ is chosen on $OB$ so that $CD \perp AB$. Suppose that segment $AC$ is $2$ units longer than segment $OD$. What is the length of segment $AC$?
$A_0B_0C_0$ and $A_1B_1C_1$ are acute-angled triangles. Describe, and prove, how to construct the triangle $ABC$ with the largest possible area which is circumscribed about $A_0B_0C_0$ (so $BC$ contains $B_0, CA$ contains $B_0$, and $AB$ contains $C_0$) and similar to $A_1B_1C_1.$
Using $600$ cards, $200$ of them having written the number $5$, $200$ having a $2$, and the other $200$ having a $1$, a student wants to create groups of cards such that the sum of the card numbers in each group is $9$. What is the maximum amount of groups that the student may create?
Let $(K,+,\cdot)$ be a field with the property $-x=x^{-1},\forall x\in K,x\ne0$. Prove that:
$$(K,+,\cdot)\simeq(\mathbb Z_2,+,\cdot)$$
[i]Proposed by Ovidiu Pop[/i]
Let $f (x)$ be a function mapping real numbers to real numbers. Given that $f (f (x)) =\frac{1}{3x}$, and $f (2) =\frac19$, find $ f\left(\frac{1}{6}\right)$.
[i]Proposed by Zachary Perry[/i]
Find all function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that
$f(x+y)-2f(x-y)+f(x)-2f(y)=y-2,\forall x,y\in \mathbb{R}$.
How many lines in a three dimensional rectangular coordiante system pass through four distinct points of the form $(i,j,k)$ where $i,j,$ and $k$ are positive integers not exceeding four?
$\text{(A)} \ 60 \qquad \text{(B)} \ 64 \qquad \text{(C)} \ 72 \qquad \text{(D)} \ 76 \qquad \text{(E)} \ 100$
The triplet of positive integers $(a,b,c)$ with $a<b<c$ is called a [i]fatal[/i] triplet if there exist three nonzero integers $p,q,r$ which satisfy the equation $a^p b^q c^r = 1$.
As an example, $(2,3,12)$ is a fatal triplet since $2^2 \cdot 3^1 \cdot (12)^{-1} = 1$.
The positive integer $N$ is called [i]fatal[/i] if there exists a fatal triplet $(a,b,c)$ satisfying $N=a+b+c$.
(a) Prove that 16 is not [i]fatal[/i].
(b) Prove that all integers bigger than 16 which are [b]not[/b] an integer multiple of 6 are fatal.
Let $a, b, c, d$ be positive integers such that $ab = cd$. Prove that $w = a^{2006} + b^{2006} + c^{2006} + d^{2006}$ is composite.
We consider three vectors drawn from the same initial point $O,$ of lengths $a,b$ and $c$, respectively. Let $E$ be the parallelepiped with vertex $O$ of which the given vectors are the edges and $H$ the parallelepiped with vertex $O$ of which the given vectors are the altitudes. Show that the product of the volumes of $E$ and $H$ equals $(abc)^{2}$ and generalize this result to $n$ dimensions.
$P_0 = (1,0), P_1 = (1,1), P_2 = (0,1), P_3 = (0,0)$.
$P_{n+4}$ is the midpoint of $P_nP_{n+1}$.
$Q_n$ is the quadrilateral $P_{n}P_{n+1}P_{n+2}P_{n+3}$.
$A_n$ is the interior of $Q_n$.
Find $\cap_{n \geq 0}A_n$.
Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$.
[i]Proposed by David Stoner[/i]
Show that $1994$ divides $10^{900}-2^{1000}$.
Determine all finite nonempty sets $S$ of positive integers satisfying
\[ {i+j\over (i,j)}\qquad\mbox{is an element of S for all i,j in S}, \]
where $(i,j)$ is the greatest common divisor of $i$ and $j$.
Let ${n \choose k}$ be the number of ways that $k$ objects can be chosen (regardless of order) from a set of $n$ objects. Prove that if positive integers k and l are greater than $1$ and less than $n$, then integers ${n \choose k}$ and ${n \choose l}$ have a common divisor greater than $1$.
$n$ is a positive integer. The number of solutions of $x^2+2016y^2=2017^n$ is $k$. Write $k$ with $n$.
Let $S$ be a set consisting of all positive integers less than or equal to $100$. Let $P$ be a subset of $S$ such that there do not exist two elements $x,y\in P$ such that $x=2y$. Find the maximum possible number of elements of $P$.
[i]Proposed by Nathan Ramesh
From the positive integers, $m,m+1,\dots,m+n$, only the sum of digits of $m$ and the sum of digits of $m+n$ are divisible by $8$. Find the maximum value of $n$.
The maximum value of
\[ 2\sum_{n=1}^{\infty} \frac{\sin(n\theta)}{44^n} \]
over all real numbers $\theta$ can be expressed as a common fraction $\tfrac{p}{q}$. Compute $p + q$.