Triangle $ABC$ has side lengths $AC = 3$, $BC = 4$, and $AB = 5$. Let $I$ be the incenter of $\triangle{ABC}$, and let $\mathcal{P}$ be the parabola with focus $I$ and directrix $\overleftrightarrow{AC}$. Suppose that $\mathcal{P}$ intersects $\overline{AB}$ and $\overline{BC}$ at points $D$ and $E$, respectively. Find $DI+EI$.

Refer to the diagram below. Let $ABCD$ be a cyclic quadrilateral with center $O$. Let the internal angle bisectors of $\angle A$ and $\angle C$ intersect at $I$ and let those of $\angle B$ and $\angle D$ intersect at $J$. Now extend $AB$ and $CD$ to intersect $IJ$ and $P$ and $R$ respectively and let $IJ$ intersect $BC$ and $DA$ at $Q$ and $S$ respectively. Let the midpoints of $PR$ and $QS$ be $M$ and $N$ respectively. Given that $O$ does not lie on the line $IJ$, show that $OM$ and $ON$ are perpendicular.

Let $ x$, $ y$ and $ z$ be non negative numbers. Prove that \[ \frac{x^2\plus{}y^2\plus{}z^2\plus{}xy\plus{}yz\plus{}zx}{6}\le \frac{x\plus{}y\plus{}z}{3}\cdot\sqrt{\frac{x^2\plus{}y^2\plus{}z^2}{3}}\]

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

Find all real numbers $x, y, z$ that satisfy the following system $$\sqrt{x^3 - y} = z - 1$$ $$\sqrt{y^3 - z} = x - 1$$ $$\sqrt{z^3 - x} = y - 1$$

Bob has $30$ identical unit cubes. He can join two cubes together by gluing a face on one cube to a face on the other cube. He must join all the cubes together into one connected solid. Over all possible solids that Bob can build, what is the largest possible surface area of the solid? [i]Proposed by Nathan Xiong[/i]

We color some unit squares in a $ 99\times 99 $ square grid with one of $ 5 $ given distinct colors, such that each color appears the same number of times. On each row and on each column there are no differently colored unit squares. Find the maximum possible number of colored unit squares.

Let $a$ and $b$ be positive integers such that $a>b$ and the difference between $a^2+b$ and $a+b^2$ is prime. Compute all possible pairs $(a,b)$.

Let $I \subset \mathbb{R}$ be a nonempty open interval and let $f: I \cap \mathbb{Q} \to \mathbb{R}$ be a function such that for all $x, y \in I \cap \mathbb{Q}$, \[ 4f\left(\frac{3x + y}{4}\right)+ 4f\left(\frac{x + 3y}{4}\right) \le f(x) + 6f\left(\frac{x + y}{2}\right)+ f(y). \] Show that $f$ can be continuously extended to $I$.

Points $A_1, B_1, C_1$ inside an acute-angled triangle $ABC$ are selected on the altitudes from $A, B, C$ respectively so that the sum of the areas of triangles $ABC_1, BCA_1$, and $CAB_1$ is equal to the area of triangle $ABC$. Prove that the circumcircle of triangle $A_1B_1C_1$ passes through the orthocenter $H$ of triangle $ABC$.

Consider a finite collection of 3-element sets $A_i$, no two of which share more than one element, whose union has cardinality 2017. Show that the elements of this union can be coloured with two colors, blue and red, so that at least 64 elements are blue and each $A_i$ has at least one red element.

In acute triangle $ABC$, let $D$, $E$, and $F$ be the feet of the altitudes from $A$, $B$, and $C$ respectively, and let $L$, $M$, and $N$ be the midpoints of $BC$, $CA$ and $AB$, respectively. Lines $DE$ and $NL$ intersect at $X$, lines $DF$ and $LM$ intersect at $Y$, and lines $XY$ and $BC$ intersect at $Z$. Find $\frac{ZB}{ZC}$ in terms of $AB$, $AC$, and $BC$.

Let $BC=6$, $BX=3$, $CX=5$, and let $F$ be the midpoint of $\overline{BC}$. Let $\overline{AX}\perp\overline{BC}$ and $AF=\sqrt{247}$. If $AC$ is of the form $\sqrt{b}$ and $AB$ is of the form $\sqrt{c}$ where $b$ and $c$ are nonnegative integers, find $2c+3b$.

[b]p1.[/b] An organization decides to raise funds by holding a $\$60$ a plate dinner. They get prices from two caterers. The first caterer charges $\$50$ a plate. The second caterer charges according to the following schedule: $\$500$ set-up fee plus $\$40$ a plate for up to and including $61$ plates, and $\$2500$ $\log_{10}\left(\frac{p}{4}\right)$ for $p > 61$ plates. a) For what number of plates $N$ does it become at least as cheap to use the second caterer as the first? b) Let $N$ be the number you found in a). For what number of plates $X$ is the second caterer's price exactly double the price for $N$ plates? c) Let $X$ be the number you found in b). When X people appear for the dinner, how much profit does the organization raise for itself by using the second caterer? [b]p2.[/b] Let $N$ be a positive integer. Prove the following: a) If $N$ is divisible by $4$, then $N$ can be expressed as the sum of two or more consecutive odd integers. b) If $N$ is a prime number, then $N$ cannot be expressed as the sum of two or more consecutive odd integers. c) If $N$ is twice some odd integer, then $N$ cannot be expressed as the sum of two or more consecutive odd integers. [b]p3.[/b] Let $S =\frac{1}{1^2} +\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...$ a) Find, in terms of $S$, the value of $S =\frac{1}{2^2} +\frac{1}{4^2}+\frac{1}{6^2}+\frac{1}{8^2}+...$ b) Find, in terms of $S$, the value of$S =\frac{1}{1^2} +\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+...$ c) Find, in terms of $S$, the value of$S =\frac{1}{1^2} -\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+...$ [b]p4.[/b] Let $\{P_1, P_2, P_3, ...\}$ be an infinite set of points on the $x$-axis having positive integer coordinates, and let $Q$ be an arbitrary point in the plane not on the $x$-axis. Prove that infinitely many of the distances $|P_iQ|$ are not integers. a) Draw a relevant picture. b) Provide a proof. [b]p5.[/b] Point $P$ is an arbitrary point inside triangle $ABC$. Points $X$, $Y$ , and $Z$ are constructed to make segments $PX$, $PY$ , and $PZ$ perpendicular to $AB$, $BC$, and $CA$, respectively. Let $x$, $y$, and $z$ denote the lengths of the segments $PX$, $PY$ , and $PZ$, respectively. a) If triangle $ABC$ is an equilateral triangle, prove that $x + y + z$ does not change regardless of the location of $P$ inside triangle ABC. b) If triangle $ABC$ is an isosceles triangle with $|BC| = |CA|$, prove that $x + y + z$ does not change when $P$ moves along a line parallel to $AB$. c) Now suppose that triangle $ABC$ is scalene (i.e., $|AB|$, $|BC|$, and $|CA|$ are all different). Prove that there exists a line for which $x+y+z$ does not change when $P$ moves along this line. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The teacher drew a coordinate plane on the board and marked some points on this plane. Unfortunately, Vasya's second-grader, who was on duty, erased almost the entire drawing, except for two points $A (1, 2)$ and $B (3,1)$. Will the excellent Andriyko be able to follow these two points to construct the beginning of the coordinate system point $O (0, 0)$? Point A on the board located above and to the left of point $B$.

Find the remainder when $$\binom{\binom{3}{2}}{2} + \binom{\binom{4}{2}}{2} + \dots + \binom{\binom{40}{2}}{2}$$ is divided by $1000$.

In each square of an $11\times 11$ board, we are to write one of the numbers $-1$, $0$, or $1$ in such a way that the sum of the numbers in each column is nonnegative and the sum of the numbers in each row is nonpositive. What is the smallest number of zeros that can be written on the board? Justify your answer.

Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$. [i]South Africa [/i]

The points $ (6,12)$ and $ (0, \minus{} 6)$ are connected by a straight line. Another point on this line is: $ \textbf{(A)}\ (3,3) \qquad\textbf{(B)}\ (2,1) \qquad\textbf{(C)}\ (7,16) \qquad\textbf{(D)}\ ( \minus{} 1, \minus{} 4) \qquad\textbf{(E)}\ ( \minus{} 3, \minus{} 8)$

For all integers $n\geq 2$, let $f(n)$ denote the largest positive integer $m$ such that $\sqrt[m]{n}$ is an integer. Evaluate \[f(2)+f(3)+\cdots+f(100).\]

Let $j$ and $k$ be integers. Prove that the inequality $$\lfloor(j+k)\alpha\rfloor+\lfloor(j+k)\beta\rfloor\ge\lfloor j\alpha\rfloor+\lfloor j\beta\rfloor+\lfloor k(\alpha+\beta)\rfloor$$holds for all real numbers $\alpha,\beta$ if and only if $j=k$.

Evaluate the infinite sum $ \sum_{n\equal{}1}^{\infty}\frac{n}{n^4\plus{}4}$.

In the set of integers solve the equation $\frac{1}{x}+\frac{1}{y}=\frac{1}{p}$, where $p$ is a prime number.

For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying \[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\] [i]Marko Radovanović, Serbia[/i]

Consider the sequence of integers $0, 1, 2, 4, 6, 9, 12,...$ obtained by starting with zero, adding $1$, then adding $1$ again, then adding $2$, and adding $2$ again, then adding $3$, and adding $3$ again, and so on. If we call the subsequent terms of this sequence $a_0, a_1, a_2, ...$, then we have $a_0 = 0$, and $a_{2n-1} = a_{2n-2} + n$ , $a_{2n} = a_{2n-1} + n$ for all integers $n \ge 1$. Find all integers $k \ge 0$ for which $a_k$ is the square of an integer.