Let $n \geq 3$ be a positive integer. Find the smallest positive real $k$, satisfying the following condition: if $G$ is a connected graph with $n$ vertices and $m$ edges, then it is always possible to delete at most $k(m-\lfloor \frac{n} {2} \rfloor)$ edges, so that the resulting graph has a proper vertex coloring with two colors.

In triangle $ABC$, the points $M$ and $N$ are the midpoints of the sides $AB$ and $AC$, respectively. The bisectors of interior angles $\angle ABC$ and $\angle BCA$ intersect the line $MN$ at points $P$ and $Q$, respectively. Let $p$ be the tangent to the circumscribed circle of the triangle $AMP$ passing through point $P$, and $q$ be the tangent to the circumscribed circle of the triangle $ANQ$ passing through point $Q$. Prove that the lines $p$ and $q$ intersect on line $BC$.

A parliament has $n$ senators. The senators form 10 parties and 10 committees, such that any senator belongs to exactly one party and one committee. Find the least possible $n$ for which it is possible to label the parties and the committees with numbers from 1 to 10, such that there are at least 11 senators for which the numbers of the corresponding party and committee are equal.

Determine all positive integers that cannot be written as $\dfrac{a}{b}+\dfrac{a+1}{b+1}$, where $a$ and $b$ are positive integers.

Suppose for some positive integer $n$, the numbers $2^n$ and $5^n$ have equal first digit. What are the possible values of this first digit? Note: solved [url=https://artofproblemsolving.com/community/c6h312638p1685546]here[/url]

Julián writes five positive integers, not necessarily different, such that their product is equal to their sum. What could be the numbers that Julian writes?

An ordered pair of integers $(m,n)$ with $1<m<n$ is said to be a [i]Benelux couple[/i] if the following two conditions hold: $m$ has the same prime divisors as $n$, and $m+1$ has the same prime divisors as $n+1$. (a) Find three Benelux couples $(m,n)$ with $m\leqslant 14$. (b) Prove that there are infinitely many Benelux couples

Given that $ \minus{} 4\le x\le \minus{} 2$ and $ 2\le y\le 4$, what is the largest possible value of $ (x \plus{} y)/x$? $ \textbf{(A)}\ \minus{}\!1\qquad \textbf{(B)}\ \minus{}\!\frac {1}{2}\qquad \textbf{(C)}\ 0\qquad \textbf{(D)}\ \frac {1}{2}\qquad \textbf{(E)}\ 1$

Let $N \geqslant 3$ be an integer. In the country of Sibyl, there are $N^2$ towns arranged as the vertices of an $N \times N$ grid, with each pair of towns corresponding to an adjacent pair of vertices on the grid connected by a road. Several automated drones are given the instruction to traverse a rectangular path starting and ending at the same town, following the roads of the country. It turned out that each road was traversed at least once by some drone. Determine the minimum number of drones that must be operating. [i]Proposed by Sutanay Bhattacharya and Anant Mudgal[/i]

Boy A and $2016$ flags are on a circumference whose length is $1$ of a circle. He wants to get all flags by moving on the circumference. He can get all flags by moving distance $l$ regardless of the positions of boy A and flags. Find the possible minimum value as $l$ like this. Note that boy A doesn’t have to return to the starting point to leave gotten flags.

Given is an acute triangle $ABC$. Its heights $BB_1$ and $CC_1$ are extended past points $B_1$ and $C_1$. On these extensions, points $P$ and $Q$ are chosen, such that angle $PAQ$ is right. Let $AF$ be a height of triangle $APQ$. Prove that angle $BFC$ is a right angle.

Let $ AD$ is centroid of $ ABC$ triangle. Let $ (d)$ is the perpendicular line with $ AD$. Let $ M$ is a point on $ (d)$. Let $ E, F$ are midpoints of $ MB, MC$ respectively. The line through point $ E$ and perpendicular with $ (d)$ meet $ AB$ at $ P$. The line through point $ F$ and perpendicular with $ (d)$ meet $ AC$ at $ Q$. Let $ (d')$ is a line through point $ M$ and perpendicular with $ PQ$. Prove $ (d')$ always pass a fixed point.

In $\Delta ABC$, $AD \perp BC$ at $D$. $E,F$ lie on line $AB$, such that $BD=BE=BF$. Let $I,J$ be the incenter and $A$-excenter. Prove that there exist two points $P,Q$ on the circumcircle of $\Delta ABC$ , such that $PB=QC$, and $\Delta PEI \sim \Delta QFJ$ .

Let $E$ be the set of $n^2+1$ closed intervals on the real axis. Prove that there exists a subset of $n+1$ intervals that are monotonically increasing with respect to inclusion, or a subset of $n+1$ intervals none of which contains any other interval from the subset.

A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$. (a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced. (b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$. [i]Proposed by Jorge Tipe, Peru[/i]

Let $ ABC $ be a triangle, and let $M$ be midpoint of $BC$. Let $ I_b $ and $ I_c $ be incenters of $ AMB $ and $ AMC $. Prove that the second intersection of circumcircles of $ ABI_b $ and $ ACI_c $ distinct from $A$ lies on line $AM$.

Show that the equation $\{x^3\}+\{y^3\}=\{z^3\}$ has infinitely many rational non-integer solutions.

Some language has only three letters - $A, B$ and $C$. A sequence of letters is called a word iff it contains exactly 100 letters such that exactly 40 of them are consonants and other 60 letters are all $A$. What is the maximum numbers of words one can pick such that any two picked words have at least one position where they both have consonants, but different consonants?

The kingdom of Pinguinia has various cities and streets, the latter being all one-way streets always run between exactly two cities, so there are no intermediate stops. Every city has exactly two streets that lead out of it and exactly two that lead into it. Prove that the streets can be divided into black and white streets so that exactly one exit of each city is black and one is white and exactly one white and one black entrance in each city.

James writes down fifteen 1's in a row and randomly writes $+$ or $-$ between each pair of consecutive 1's. One such example is \[1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.\] What is the probability that the value of the expression James wrote down is $7$? $\begin{array}{@{\hspace{-1em}}l@{\hspace{14em}}l@{\hspace{14em}}l} \textbf{(A) }0&\textbf{(B) }\dfrac{6435}{2^{14}}&\textbf{(C) }\dfrac{6435}{2^{13}}\\\\ \textbf{(D) }\dfrac{429}{2^{12}}&\textbf{(E) }\dfrac{429}{2^{11}}&\textbf{(F) }\dfrac{429}{2^{10}}\\\\ \textbf{(G) }\dfrac1{15}&\textbf{(H) } \dfrac1{31}&\textbf{(I) }\dfrac1{30}\\\\ \textbf{(J) }\dfrac1{29}&\textbf{(K) }\dfrac{1001}{2^{15}}&\textbf{(L) }\dfrac{1001}{2^{14}}\\\\ \textbf{(M) }\dfrac{1001}{2^{13}}&\textbf{(N) }\dfrac1{2^7}&\textbf{(O) }\dfrac1{2^{14}}\\\\ \textbf{(P) }\dfrac1{2^{15}}&\textbf{(Q) }\dfrac{2007}{2^{14}}&\textbf{(R) }\dfrac{2007}{2^{15}}\\\\ \textbf{(S) }\dfrac{2007}{2^{2007}}&\textbf{(T) }\dfrac1{2007}&\textbf{(U) }\dfrac{-2007}{2^{14}}\end{array}$

Let $L$ be the number of times the letter $L$ appeared on the Speed Round, $M$ be the number of times the letter $M$ appeared on the Speed Round, and $T$ be the number of times the letter $T$ appeared on the Speed Round. Find the value of $LMT$.

Let be a point $ P $ on the diagonal $ BD $ (excluding its endpoints) of a quadrilateral $ ABCD, $ and $ Q $ be a point in the interior of $ ABD. $ The projections of $ P $ on $ AB,AD $ are $ P_1,P_2, $ respectively, and the projections of $ Q $ on $ AB,AD $ are $ Q_1,Q_2, $ respectively, and verify the equations $ AQ_1=\frac{1}{4}AB $ and $ AQ_2=\frac{1}{4}AD. $ Show that the point $ Q $ is not in the interior of $ AP_1P_2. $

Let $E$ and $F$ be the intersections of opposite sides of a convex quadrilateral $ABCD$. The two diagonals meet at $P$. Let $O$ be the foot of the perpendicular from $P$ to $EF$. Show that $\angle BOC=\angle AOD$.

Determine all integers $k$ such that there exists infinitely many positive integers $n$ [b]not[/b] satisfying \[n+k |\binom{2n}{n}\]

In a party among any four persons there are three people who are mutual acquaintances or mutual strangers. Prove that all the people can be separated into two groups $A$ and $B$ such that in $A$ everybody knows everybody else and in $B$ nobody knows anybody else.