Niki and Kyle play a triangle game. Niki first draws $\triangle ABC$ with area $1$, and Kyle picks a point $X$ inside $\triangle ABC$. Niki then draws segments $\overline{DG}$, $\overline{EH}$, and $\overline{FI}$, all through $X$, such that $D$ and $E$ are on $\overline{BC}$, $F$ and $G$ are on $\overline{AC}$, and $H$ and $I$ are on $\overline{AB}$. The ten points must all be distinct. Finally, let $S$ be the sum of the areas of triangles $DEX$, $FGX$, and $HIX$. Kyle earns $S$ points, and Niki earns $1-S$ points. If both players play optimally to maximize the amount of points they get, who will win and by how much?

[b]p1.[/b] Three camps are located in the vertices of an equilateral triangle. The roads connecting camps are along the sides of the triangle. Captain America is inside the triangle and he needs to know the distances between camps. Being able to see the roads he has found that the sum of the shortest distances from his location to the roads is 50 miles. Can you help Captain America to evaluate the distances between the camps? [b]p2.[/b] $N$ regions are located in the plane, every pair of them have a non-empty overlap. Each region is a connected set, that means every two points inside the region can be connected by a curve all points of which belong to the region. Iron Man has one charge remaining to make a laser shot. Is it possible for him to make the shot that goes through all $N$ regions? [b]p3.[/b] Money in Wonderland comes in $\$5$ and $\$7$ bills. (a) What is the smallest amount of money you need to buy a slice of pizza that costs $\$1$ and get back your change in full? (The pizza man has plenty of $\$5$ and $\$7$ bills.) For example, having $\$7$ won't do since the pizza man can only give you $\$5$ back. (b) Vending machines in Wonderland accept only exact payment (do not give back change). List all positive integer numbers which CANNOT be used as prices in such vending machines. (That is, find the sums of money that cannot be paid by exact change.) [b]p4.[/b] (a) Put $5$ points on the plane so that each $3$ of them are vertices of an isosceles triangle (i.e., a triangle with two equal sides), and no three points lie on the same line. (b) Do the same with $6$ points. [b]p5.[/b] Numbers $1,2,3,…,100$ are randomly divided in two groups $50$ numbers in each. In the first group the numbers are written in increasing order and denoted $a_1,a_2, ..., a_{50}$. In the second group the numberss are written in decreasing order and denoted $b_1,b_2, ..., b_{50}$. Thus $a_1<a_2<...<a_{50}$ and $ b_1>b_2>...>b_{50}$. Evaluate $|a_1-b_1|+|a_2-b_2|+...+|a_{50}-b_{50}|$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In triangle $ABC$, $M$ is the point of intersection of the medians, $O$ is the center of the inscribed circle, $A', B', C'$ are the touchpoints with the sides $BC, CA, AB$, respectively. Prove that if $CA'= AB$, then $OM$ and $AB$ are perpendicular. PS. There is a a typo

Find all pairs $(m,n)$ of positive integers such that $m^2 + n^2 = 3(m + n)$.

A male volcano is in the shape of a hollow cone with the point side up, but with everything above a height of 6 meters removed. The resulting shape has a bottom radius of 10 meters and a top radius of 7 meters, with a height of 6 meters. He sat above his bay, watching all the couples play. His lava grew and grew until he was half full of lava. Then, he erupted, lowering the height of the lava to 2 meters. What fraction of the lava remained in the volcano? [i]Proposed by Matthew Weiss

An acute triangle $ABC$ is given. Let us denote $X$ for each of its inner points $X_a, X_b, X_c$ its images in axial symmetries sequentially along the lines $BC, CA, AB$. Prove that all $X_aX_bX_c$ triangles have a common interior point. (Josef Tkadlec)

Is there a planar polygon whose vertices have integer coordinates and whose area is $1/2$, such that this polygon is (a) a triangle with at least two sides longer than $1000$? (b) a triangle whose sides are all longer than $1000$? (c) a quadrangle?

Find all functions $f$ from positive integers to themselves such that: 1)$f(mn)=f(m)f(n)$ for all positive integers $m, n$ 2)$\{1, 2, ..., n\}=\{f(1), f(2), ... f(n)\}$ is true for infinitely many positive integers $n$.

We say that a nonconstant polynomial $p(x)$ with real coefficients is split into two squares if it is represented as $a(x) +b(x)$ where $a(x)$ and $b(x)$ are squares of polynomials with real coefficients. Is there such a polynomial $p(x)$ that it may be split into two squares: a) in exactly one way; b) in exactly two ways? Note: two splittings that differ only in the order of summands are considered to be the same. Sergey Markelov

Any five points are taken inside or on a square with side length $1$. Let $a$ be the [i]smallest[/i] possible number with the property that it is always possible to select one pair of points from these five such that the distance between them is equal to or less than $a$. Then $a$ is: ${{ \textbf{(A)}\ \sqrt{3}/3 \qquad\textbf{(B)}\ \sqrt{2}/2 \qquad\textbf{(C)}\ 2\sqrt{2}/3 \qquad\textbf{(D)}\ 1 }\qquad\textbf{(E)}\ \sqrt{2} } $

Prove that if the product of four consecutive natural numbers is added one unit, the result is a perfect square.

Let $a, b, c, d$ be four positive real numbers. Prove that $$\frac{(a + b + c)^2}{a^2+b^2+c^2}+\frac{(b + c + d)^3}{b^3+c^3+d^3}+\frac{(c+d+a)^4}{c^4+d^4+a^4}+\frac{(d+a+b)^5}{d^5+a^5+b^5}\le 120$$

Let $n\ge2$ be a positive integer. The numbers $1,2,..., n^2$ are consecutively placed into squares of an $n\times n$, so the first row contains $1,2,...,n$ from left to right, the second row contains $n+1,n+2,...,2n$ from left to right, and so on. The [i]magic square value[/i] of a grid is defined to be the number of rows, columns, and main diagonals whose elements have an average value of $\frac{n^2 + 1}{2}$. Show that the magic-square value of the grid stays constant under the following two operations: (1) a permutation of the rows; and (2) a permutation of the columns. (The operations can be used multiple times, and in any order.) [i]Proposed by Ray Li[/i]

Find all functions $f:\mathbb{R}\backslash\{0\}\rightarrow\mathbb{R}$ for which $xf(xy) + f(-y) = xf(x)$ for all non-zero real numbers $x, y$.

$ S$ be a set of $ n$ points in the plane. No three points of $ S$ are collinear. Prove that there exists a set $ P$ containing $ 2n \minus{} 5$ points satisfying the following condition: In the interior of every triangle whose three vertices are elements of $ S$ lies a point that is an element of $ P.$

Let $S = \{1, 2, \ldots, 2016\}$, and let $f$ be a randomly chosen bijection from $S$ to itself. Let $n$ be the smallest positive integer such that $f^{(n)}(1) = 1$, where $f^{(i)}(x) = f(f^{(i-1)}(x))$. What is the expected value of $n$?

Three rectangles, each of area $6$ square inches, are placed inside a $4$ in. by $4$ in. square. Prove that, no matter how the three rectangles are shaped and arranged, (for example, like in the picture below), one can find two of them which have a common area of at least $2/3$ square inches.

Let $ABC$ be an acute angled triangle$.$ Let $D$ be the foot of the internal angle bisector of $\angle BAC$ and let $M$ be the midpoint of $AD.$ Let $X$ be a point on segment $BM$ such that $\angle MXA=\angle DAC.$ Prove that $AX$ is perpendicular to $XC.$

Show that there is a positive number $p$ such that $\int^\pi_0x^p\sin xdx=\sqrt[10]{2000}$.

$n\ge2$ is a given positive integer. $i\leq a_i \leq n$ satisfies for all $1\leq i\leq n$, and $S_i$ is defined as $a_1+a_2+...+a_i(S_0=0)$. Show that there exists such $1\leq k\leq n$ that satisfies $a_k^2+S_{n-k}<2S_n-\frac{n(n+1)}{2}$.

Pieck the Frog hops on Pascal's Triangle, where she starts at the number $1$ at the top. In a hop, Pieck can hop to one of the two numbers directly below the number she is currently on with equal probability. Given that the expected value of the number she is on after $7$ hops is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Proposed by Steven Yu[/i]

How many ways are there to choose (not necessarily distinct) integers $a,b,c$ from the set $\{1,2,3,4\}$ such that $a^{(b^c)}$ is divisible by $4$? [i]Ray Li[/i]

Let $M$ be the set of all positive integers that do not contain the digit $9$ (base $10$). If $x_1, \ldots , x_n$ are arbitrary but distinct elements in $M$, prove that \[\sum_{j=1}^n \frac{1}{x_j} < 80 .\]

The positive integer $m$ and non-negative integers $x_0, x_1,..., x_{1001}$ satisfy the following equation: $$m^{x_0} =\sum_{i=1}^{1001}m^{x_i}.$$ How many possibilities are there for the value of $m$?

Find all terms of the arithmetic progression $-1, 18, 37, 56, ...$ whose only digit is $5$.