There are $2^{2n+1}$ towns with $2n+1$ companies and each two towns are connected with airlines from one of the companies. What’s the greatest number $k$ with the following property: We can close $k$ of the companies and their airlines in such way that we can still reach each town from any other (connected graph).

A set of ten two-digit numbers is given. Prove that one can always choose two disjoint subsets of this set such that the sum of their elements is the same. Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )

Prove that if the numbers $ x $, $ y $, $ z $ satisfy the equationw $$x + y + z = a,$$ $$ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{a},$$ then at least one of them is equal to $ a $.

How many ways are there to color the $8$ regions of a three-set Venn Diagram with $3$ colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and/or reflection.

$ (C)$ and $(C')$ are two circles which intersect in $A$ and $B$. $(D)$ is a line that moves and passes through $A$, intersecting $(C)$ in P and $(C')$ in P'. Prove that the bisector of $[PP']$ passes through a non-moving point.

Let $A$ be a $n\times n$ matrix with complex elements and let $A^\star$ be the classical adjoint of $A$. Prove that if there exists a positive integer $m$ such that $(A^\star)^m = 0_n$ then $(A^\star)^2 = 0_n$. [i]Marian Ionescu, Pitesti[/i]

Given a circle $\Gamma$ and the positive numbers $h$ and $m$, construct with straight edge and compass a trapezoid inscribed in $\Gamma$, such that it has altitude $h$ and the sum of its parallel sides is $m$.

Let $ \Delta ABC$ be a triangle with $M$ the middle of the side $[BC]$. On the line $BC$, to the left and to the right of the point $M,$ at the same distance from $M,$ let us consider $d_1$ and $d_2,$ which are perpendicular to the line BC. The perpendicular line from $M$ to $AB$ intersects $d_1$ in $P,$ and the perpendicular line from $M$ to $AC$ intersects $d_2$ in $Q.$ Prove that \[AM\perp PQ.\] [b]Author: Marin Bancoș Regional Mathematical Contest GRIGORE MOISIL, Romania, Baia Mare, 2012, 9th grade[/b]

A man has $\$$2.73 in pennies, nickels, dimes, quarters and half dollars. If he has an equal number of coins of each kind, then the total number of coins he has is \[ \text{(A)}\ 3 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 15 \]

Determine all the positive real numbers $x_1, x_2, x_3, \dots, x_{2021}$ such that $x_{i+1}=\frac{x_i^3+2}{3x_i^2}$ for every $i=1, 2, 3, \dots, 2020$ and $x_{2021}=x_1$

There are $n$ students sitting on a round table. You collect all of $ n $ name tags and give them back arbitrarily. Each student gets one of $n$ name tags. Now $n$ students repeat following operation: The students who have their own name tags exit the table. The other students give their name tags to the student who is sitting right to him. Find the number of ways giving name tags such that there exist a student who don't exit the table after 4 operations.

Let $C=\{4,6,8,9,10,\ldots\}$ be the set of composite positive integers. For each $n\in C$ let $a_n$ be the smallest positive integer $k$ such that $k!$ is divisible by $n$. Determine whether the following series converges: $$\sum_{n\in C}\left(\frac{a_n}{n}\right)^n.$$ [i]Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan[/i]

Set positive integer $m=2^k\cdot t$, where $k$ is a non-negative integer, $t$ is an odd number, and let $f(m)=t^{1-k}$. Prove that for any positive integer $n$ and for any positive odd number $a\le n$, $\prod_{m=1}^n f(m)$ is a multiple of $a$.

Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$

Let $S_{n}=\{1,2,...,n\}$. How many functions $f:S_{n} \rightarrow S_{n}$ satisfy $f(k) \leq f(k+1)$ and $f(k)=f(f(k+1))$ for $k <n?$

Find all pairs of relatively prime integers $(x, y)$ that satisfy equality $2 (x^3 - x) = 5 (y^3 - y)$.

Do there exist real numbers $b$ and $c$ such that each of the equations $x^2+bx+c = 0$ and $2x^2+(b+1)x+c+1 = 0$ have two integer roots? [i]N. Agakhanov[/i]

In an acute triangle $ABC$ with $\overline{AB} < \overline{AC}$, angle bisector of $A$ and perpendicular bisector of $\overline{BC}$ intersect at $D$. Let $P$ be an interior point of triangle $ABC$. Line $CP$ meets the circumcircle of triangle $ABP$ again at $K$. Prove that $B, D, K$ are collinear if and only if $AD$ and $BC$ meet on the circumcircle of triangle $APC$.

Given are positive integers $m, n$ and a sequence $a_1, a_2, \ldots, $ such that $a_i=a_{i-n}$ for all $i>n$. For all $1 \leq j \leq n$, let $l_j$ be the smallest positive integer such that $m \mid a_j+a_{j+1}+\ldots+a_{j+l_j-1}$. Prove that $l_1+l_2+\ldots+l_n \leq mn$.

We have twenty-seven $1$ by $1$ cubes. Each face of every cube is marked with a natural number so that two opposite faces (top and bottom, front and back, left and right) are always marked with an even number and an odd number where the even number is twice that of the odd number. The twenty-seven cubes are put together to form one $3$ by $3$ cube as shown. When two cubes are placed face-to-face, adjoining faces are always marked with an odd number and an even number where the even number is one greater than the odd number. Find the sum of all of the numbers on all of the faces of all the $1$ by $1$ cubes. [asy] import graph; size(7cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((-1,7)--(-1,4)); draw((-1,9.15)--(-3.42,8.21)); draw((-1,9.15)--(1.42,8.21)); draw((-1,7)--(1.42,8.21)); draw((1.42,7.21)--(-1,6)); draw((1.42,6.21)--(-1,5)); draw((1.42,5.21)--(-1,4)); draw((1.42,8.21)--(1.42,5.21)); draw((-3.42,8.21)--(-3.42,5.21)); draw((-3.42,7.21)--(-1,6)); draw((-3.42,8.21)--(-1,7)); draw((-1,4)--(-3.42,5.21)); draw((-3.42,6.21)--(-1,5)); draw((-2.61,7.8)--(-2.61,4.8)); draw((-1.8,4.4)--(-1.8,7.4)); draw((-0.2,7.4)--(-0.2,4.4)); draw((0.61,4.8)--(0.61,7.8)); label("2",(-1.07,9.01),SE*labelscalefactor); label("9",(-1.88,8.65),SE*labelscalefactor); label("1",(-2.68,8.33),SE*labelscalefactor); label("3",(-0.38,8.72),SE*labelscalefactor); draw((-1.8,7.4)--(0.63,8.52)); draw((-0.27,8.87)--(-2.61,7.8)); draw((-2.65,8.51)--(-0.2,7.4)); draw((-1.77,8.85)--(0.61,7.8)); label("7",(-1.12,8.33),SE*labelscalefactor); label("5",(-1.9,7.91),SE*labelscalefactor); label("1",(0.58,8.33),SE*labelscalefactor); label("18",(-0.36,7.89),SE*labelscalefactor); label("1",(-1.07,7.55),SE*labelscalefactor); label("1",(-0.66,6.89),SE*labelscalefactor); label("5",(-0.68,5.8),SE*labelscalefactor); label("1",(-0.68,4.83),SE*labelscalefactor); label("2",(0.09,7.27),SE*labelscalefactor); label("1",(0.15,6.24),SE*labelscalefactor); label("2",(0.11,5.26),SE*labelscalefactor); label("1",(0.89,7.61),SE*labelscalefactor); label("3",(0.89,6.63),SE*labelscalefactor); label("9",(0.92,5.62),SE*labelscalefactor); label("18",(-3.18,7.63),SE*labelscalefactor); label("2",(-3.07,6.61),SE*labelscalefactor); label("2",(-3.09,5.62),SE*labelscalefactor); label("1",(-2.29,7.25),SE*labelscalefactor); label("3",(-2.27,6.22),SE*labelscalefactor); label("5",(-2.29,5.2),SE*labelscalefactor); label("7",(-1.49,6.89),SE*labelscalefactor); label("34",(-1.52,5.81),SE*labelscalefactor); label("1",(-1.41,4.86),SE*labelscalefactor); [/asy]

Edward's formula for the stock market predicts correctly that the price of HMMT is directly proportional to a secret quantity $ x$ and inversely proportional to $ y$, the number of hours he slept the night before. If the price of HMMT is $ \$12$ when $ x\equal{}8$ and $ y\equal{}4$, how many dollars does it cost when $ x\equal{}4$ and $ y\equal{}8$?

Let be a subset of the interval $ (0,1) $ that contains $ 1/2 $ and has the property that if a number is in this subset, then, both its half and its successor's inverse are in the same subset. Prove that this subset contains all the rational numbers of the interval $ (0,1). $

Let $S$ be the set of positive integers. For any $a$ and $b$ in the set we have $GCD(a, b)>1$. For any $a$, $b$ and $c$ in the set we have $GCD(a, b, c)=1$. Is it possible that $S$ has $2012$ elements? [i]Proposed by Ognjen Stipetić.[/i]

Find the number of subsets of $\{1, 2,... , 2100\}$ such that each has sum of the elements giving a remainder of $3$ when divided by $7$.

On the board were written the numbers from $1$ to $k$ (where $k$ is an unknown positive integer). One of the numbers was erased. The average of the remaining numbers is $25.25$. Which number was erased?