Let $ABC$ be an arbitrary triangle, and let $M, N, P$ be any three points on the sides $BC, CA, AB$ such that the lines $AM, BN, CP$ concur. Let the parallel to the line $AB$ through the point $N$ meet the line $MP$ at a point $E$, and let the parallel to the line $AB$ through the point $M$ meet the line $NP$ at a point $F$. Then, the lines $CP, MN$ and $EF$ are concurrent. [hide=MOP 97 problem]Let $ABC$ be a triangle, and $M$, $N$, $P$ the points where its incircle touches the sides $BC$, $CA$, $AB$, respectively. The parallel to $AB$ through $N$ meets $MP$ at $E$, and the parallel to $AB$ through $M$ meets $NP$ at $F$. Prove that the lines $CP$, $MN$, $EF$ are concurrent. [url=https://artofproblemsolving.com/community/c6h22324p143462]also[/url][/hide]

Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its inscribed circle is equal to the arithmetic mean of the three numbers at its vertices. Prove that all points in the plane are assigned the same number.

The Purple Plant Garden Store sells grass seed in ten-pound bags and fifteen-pound bags. Yesterday half of the grass seed they had was in ten-pound bags. This morning the store received a shipment of 27 more ten-pound bags, and now they have twice as many ten-pound bags as fifteen-pound bags. Find the total weight in pounds of grass seed the store now has.

Let be $z_1$ and $z_2$ two complex numbers such that $|z_1+2z_2|=|2z_1+z_2|$.Prove that for all real numbers $a$ is true $|z_1+az_2|=|az_1+z_2|$

$I$ - incenter , $M$- midpoint of arc $BAC$ of circumcircle, $AL$ - angle bisector of triangle $ABC$. $MI$ intersect circumcircle in $K$. Circumcircle of $AKL$ intersect $BC$ at $L$ and $P$. Prove that $\angle AIP=90$

The average weight of 6 boys is 150 pounds and the average weight of 4 girls is 120 pounds. The average weight of the 10 children is $ \text{(A)}\ 135\text{ pounds}\qquad\text{(B)}\ 137\text{ pounds}\qquad\text{(C)}\ 138\text{ pounds}\qquad\text{(D)}\ 140\text{ pounds}\qquad\text{(E)}\ 141\text{ pounds} $

In a quadrilateral \(ABCD\), it is known that \(\angle ABC = \angle ADC = 90^{\circ}\). On the ray \(AB\) beyond point \(B\), a point \(K\) is chosen such that \(\angle AKD = \angle ADB\). Point \(L\) is the projection of point \(K\) onto the line \(AD\), and point \(N\) is the projection of point \(D\) onto the line \(CL\). Find the degree measure of \(\angle ANK\). [i]Proposed by Mykhailo Shtandenko[/i]

A list of positive integers is called good if the maximum element of the list appears exactly once. A sublist is a list formed by one or more consecutive elements of a list. For example, the list $10,34,34,22,30,22$ the sublist $22,30,22$ is good and $10,34,34,22$ is not. A list is very good if all its sublists are good. Find the minimum value of $k$ such that there exists a very good list of length $2019$ with $k$ different values on it.

Let $ABCD$ be a convex cyclic quadrilateral satisfying $AB\cdot CD=AD\cdot BC$. Let the inscribed circle $\omega$ of triangle $ABC$ be tangent to sides $BC$, $CA$ and $AB$ at points $A', B'$ and $C'$, respectively. Let point $K$ be the intersection of line $ID$ and the nine-point circle of triangle $A'B'C'$ that is inside line segment $ID$. Let $S$ denote the centroid of triangle $A'B'C'$. Prove that lines $SK$ and $BB'$ intersect each other on circle $\omega$. [i]Proposed by Áron Bán-Szabó, Budapest[/i]

The country has $n \ge 3$ airports, some pairs of which are connected by bidirectional flights. Every day, the government closes the airport with the strictly highest number of flights going out of it. What is the maximum number of days this can continue? [i]Proposed by Fedir Yudin[/i]

a) For integer $n \ge 3$, suppose that $0 < a_1 < a_2 < ...< a_n$ is a arithmetic sequence and $0 < b_1 < b_2 < ... < b_n$ is a geometric sequence with $a_1 = b_1, a_n = b_n$. Prove that a_k > b_k for all $k = 2,3,..., n -1$. b) Prove that for every positive integer $n \ge 3$, there exist an integer arithmetic sequence $(a_n)$ and an integer geometric sequence $(b_n)$ such that $0 < b_1 < a_1 < b_2 < a_2 < ... < b_n < a_n$.

The incircle of $ABC$ touches $BC$, $CA$, $AB$ at $K$, $L$, $M$ respectively. The line through $A$ parallel to $LK$ meets $MK$ at $P$, and the line through $A$ parallel to $MK$ meets $LK$ at $Q$. Show that the line $PQ$ bisects $AB$ and bisects $AC$.

There are four basketball players $A,B,C,D$. Initially the ball is with $A$. The ball is always passed from one person to a different person. In how many ways can the ball come back to $A$ after $\textbf{seven}$ moves? (for example $A\rightarrow C\rightarrow B\rightarrow D\rightarrow A\rightarrow B\rightarrow C\rightarrow A$, or $A\rightarrow D\rightarrow A\rightarrow D\rightarrow C\rightarrow A\rightarrow B\rightarrow A)$.

The function $f$ from the set $\mathbb{N}$ of positive integers into itself is defined by the equality \[f(n)=\sum_{k=1}^{n} \gcd(k,n),\qquad n\in \mathbb{N}.\] a) Prove that $f(mn)=f(m)f(n)$ for every two relatively prime ${m,n\in\mathbb{N}}$. b) Prove that for each $a\in\mathbb{N}$ the equation $f(x)=ax$ has a solution. c) Find all ${a\in\mathbb{N}}$ such that the equation $f(x)=ax$ has a unique solution.

If $(a_n)$ is a sequence of real numbers such that for $n \geq 1$ $$(2-a_n )a_{n+1} =1,$$ prove that $\lim_{n\to \infty} a_n =1.$

If the incircle of a right triangle with area $a$ is the circumcircle of a right triangle with area $b$, what is the minimum value of $\frac{a}{b}$? $ \textbf{(A)}\ 3 + 2\sqrt2 \qquad\textbf{(B)}\ 1+\sqrt2 \qquad\textbf{(C)}\ 2\sqrt2 \qquad\textbf{(D)}\ 2+\sqrt3 \qquad\textbf{(E)}\ 2\sqrt3$

Let $m, n$ be positive integers $(m, n>=2)$. Given an $n$-element set $A$ of integers $(A=\{a_1,a_2,\cdots ,a_n\})$, for each pair of elements $a_i, a_j(j>i)$, we make a difference by $a_j-a_i$. All these $C^2_n$ differences form an ascending sequence called “derived sequence” of set $A$. Let $\bar{A}$ denote the derived sequence of set $A$. Let $\bar{A}(m)$ denote the number of terms divisible by $m$ in $\bar{A}$ . Prove that $\bar{A}(m)\ge \bar{B}(m)$ where $A=\{a_1,a_2,\cdots ,a_n\}$ and $B=\{1,2,\cdots ,n\}$.

Two thieves stole an open chain with $2k$ white beads and $2m$ black beads. They want to share the loot equally, by cutting the chain to pieces in such a way that each one gets $k$ white beads and $m$ black beads. What is the minimal number of cuts that is always sufficient?

Let $A$ be a set consist of finite real numbers,$A_1,A_2,\cdots,A_n$ be nonempty sets of $A$, such that [b](a)[/b] The sum of the elements of $A$ is $0,$ [b](b)[/b] For all $x_i \in A_i(i=1,2,\cdots,n)$,we have $x_1+x_2+\cdots+x_n>0$. Prove that there exist $1\le k\le n,$ and $1\le i_1<i_2<\cdots<i_k\le n$, such that \[|A_{i_1}\bigcup A_{i_2} \bigcup \cdots \bigcup A_{i_k}|<\frac{k}{n}|A|.\] Where $|X|$ denote the numbers of the elements in set $X$.

The measure of the angles of a triangle are in arithmetic progression and the lengths of its altitudes are as well. Show that such a triangle is equilateral.

Find $\displaystyle \sum_{k=0}^{49}(-1)^k\binom{99}{2k}$, where $\binom{n}{j}=\frac{n!}{j!(n-j)!}$. $ \textbf{(A)}\ -2^{50} \qquad\textbf{(B)}\ -2^{49} \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 2^{49} \qquad\textbf{(E)}\ 2^{50} $

A [i]binary string[/i] is a sequence, each of whose terms is $0$ or $1$. A set $\mathcal{B}$ of binary strings is defined inductively according to the following rules. [list] [*]The binary string $1$ is in $\mathcal{B}$.[/*] [*]If $s_1,s_2,\dotsc ,s_n$ is in $\mathcal{B}$ with $n$ odd, then both $s_1,s_2,\dotsc ,s_n,0$ and $0,s_1,s_2,\dotsc ,s_n$ are in $\mathcal{B}$.[/*] [*]If $s_1,s_2,\dotsc ,s_n$ is in $\mathcal{B}$ with $n$ even, then both $s_1,s_2,\dotsc ,s_n,1$ and $1,s_1,s_2,\dotsc ,s_n$ are in $\mathcal{B}$.[/*] [*]No other binary strings are in $\mathcal{B}$.[/*] [/list] For each positive integer $n$, let $b_n$ be the number of binary strings in $\mathcal{B}$ of length $n$. [list=a] [*]Prove that there exist constants $c_1,c_2>0$ and $1.6<\lambda_1,\lambda_2<1.9$ such that $c_1\lambda_1^n<b_n<c_2\lambda_2^n$ for all positive integer $n$.[/*] [*]Determine $\liminf_{n\to \infty} {\sqrt[n]{b_n}}$ and $\limsup_{n\to \infty} {\sqrt[n]{b_n}}$[/*] [/list] [i]Note: The problem is open in the sense that no solution is currently known to part (b).[/i]

A square with sides of length $1$ cm is given. There are many different ways to cut the square into four rectangles. Let $S$ be the sum of the four rectangles’ perimeters. Describe all possible values of $S$ with justification.

In a circle $O$, there are six points, $ A$, $ B$, $C$, $D$, $E$, $F$ in a counterclockwise order such that $BD \perp CF$ , and $CF$, $BE$, $AD$ are concurrent. Let the perpendicular from $B$ to $AC$ be $M$, and the perpendicular from $D$ to $CE$ be $N$. Prove that $AE \parallel MN$.

Yesterday, Alex, Beth, and Carl raked their lawn. First, Alex and Beth raked half of the lawn together in $30$ minutes. While they took a break, Carl raked a third of the remaining lawn in $60$ minutes. Finally, Beth joined Carl and together they finished raking the lawn in $24$ minutes. If they each rake at a constant rate, how many hours would it have taken Alex to rake the entire lawn by himself?