For each integer $C>1$ decide whether there exist pairwise distinct positive integers $a_1,a_2,a_3,...$ such that for every $k\ge 1$, $a_{k+1}^k$ divides $C^ka_1a_2...a_k$. [i]Proposed by Daniel Liu

Determine the natural numbers $n\ge 2$ for which exist $x_1,x_2,...,x_n \in R^*$, such that $$x_1+x_2+...+x_n=\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}=0$$

Let $a, b, c$ be positive real numbers with $ab + bc + ac = abc$. Prove that $$\frac{bc}{a^{a+1}} +\frac{ac}{b^{b+1 }}+\frac{ab}{c^{c+1}} \ge \frac13$$

Find all $n \in \mathbb N$ such that $n = \varphi (n) + 402$, where $\varphi$ denotes the Euler phi function.

In a triangle $ABC$, $AD$ is the altitude from $A$, and $H$ is the orthocentre. Let $K$ be the centre of the circle passing through $D$ and tangent to $BH$ at $H$. Prove that the line $DK$ bisects $AC$.

Let $ a > b > c > d $ be positive integers such that \begin{align*} a^2 + ac - c^2 = b^2 + bd - d^2 \end{align*} Prove that $ ab + cd $ is a composite number.

Let $ABCD$ be a cyclic quadrilateral with its diagonals intersecting at $E$. Let $M$ be the midpoint of $AB$. Suppose that $ME$ is perpendicular to $CD$. Show that either $AC$ is perpendicular to $BD$, or $AB$ is parallel to $CD$.

A circle of radius $ 1$ is surrounded by $ 4$ circles of radius $ r$ as shown. What is $ r$? [asy]defaultpen(linewidth(.9pt)); real r = 1 + sqrt(2); pair A = dir(45)*(r + 1); pair B = dir(135)*(r + 1); pair C = dir(-135)*(r + 1); pair D = dir(-45)*(r + 1); draw(Circle(origin,1)); draw(Circle(A,r));draw(Circle(B,r));draw(Circle(C,r));draw(Circle(D,r)); draw(A--(dir(45)*r + A)); draw(B--(dir(45)*r + B)); draw(C--(dir(45)*r + C)); draw(D--(dir(45)*r + D)); draw(origin--(dir(25))); label("$r$",midpoint(A--(dir(45)*r + A)), SE); label("$r$",midpoint(B--(dir(45)*r + B)), SE); label("$r$",midpoint(C--(dir(45)*r + C)), SE); label("$r$",midpoint(D--(dir(45)*r + D)), SE); label("$1$",origin,W);[/asy]$ \textbf{(A)}\ \sqrt {2}\qquad \textbf{(B)}\ 1 \plus{} \sqrt {2}\qquad \textbf{(C)}\ \sqrt {6}\qquad \textbf{(D)}\ 3\qquad \textbf{(E)}\ 2 \plus{} \sqrt {2}$

Let $D$ be the midpoint of the side $BC$ of a triangle $ABC$ and let $G$ be the point of the segment $AD$ such that $AG = 2GD$. Let $E$ and $F$ be points on the sides $AB$ and $AC$, respectively, such that$ G$ lies on the segment $EF$. Let $M$ and $N$ be points of the segments $AE$ and $AF$, respectively, such that $ME = EB$ and $NF = FC$. a) Prove that the area of the quadrilateral $BMNC$ is equal to four times the area of the triangle $DEF$. b) Prove that the quadrilaterals $MNFE$ and $AMDN$ have the same area.

On a cyclic quadrilateral $ABCD$, $M$ is the midpoint of $AB$ and $N$ is the midpoint of $CD$. Let $E$ be the projection of $C$ onto $AB$ and $F$ the reflection of $N$ about the midpoint of $DE$. If $F$ is inside quadrilateral $ABCD$, show that $\angle BMF = \angle CBD$.

Find the sum of all positive $30$-digit palindromes. The leading digit is not allowed to be $0$.

Let $ f:[0,1]\longrightarrow\mathbb{R} $ be a countinuous function such that $$ \int_0^1 f(x)dx=\int_0^1 xf(x)dx. $$ Show that there is a $ c\in (0,1) $ such that $ f(c)=\int_0^c f(x)dx. $

In multiplying two positive integers $a$ and $b$, Ron reversed the digits of the two-digit number $a$. His errorneous product was $161$. What is the correct value of the product of $a$ and $b$? $ \textbf{(A)}\ 116 \qquad \textbf{(B)}\ 161 \qquad \textbf{(C)}\ 204 \qquad \textbf{(D)}\ 214 \qquad \textbf{(E)}\ 224 $

Simplify \[ \sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}} \]$ \textbf{(A)}\ \sqrt{x} \qquad \textbf{(B)}\ \sqrt[3]{x^2} \qquad \textbf{(C)}\ \sqrt[27]{x^2} \qquad \textbf{(D)}\ \sqrt[54]{x} \qquad \textbf{(E)}\ \sqrt[81]{x^{80}}$

Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$ [i]Michael Ren and Ankan Bhattacharya, USA[/i]

[i]25 problems for 30 minutes.[/i] [b]p1.[/b] Chad, Ravi, Kevin, and Meena are four of the $551$ residents of Chadwick, Illinois. Expressing your answer to the nearest percent, how much of the population do they represent? [b]p2.[/b] Points $A$, $B$, and $C$ are on a line for which $AB = 625$ and $BC = 256$. What is the sum of all possible values of the length $AC$? [b]p3.[/b] An increasing arithmetic sequence has first term $2014$ and common difference $1337$. What is the least odd term of this sequence? [b]p4.[/b] How many non-congruent scalene triangles with integer side lengths have two sides with lengths $3$ and $4$? [b]p5.[/b] Let $a$ and $b$ be real numbers for which the function $f(x) = ax^2+bx+3$ satisfies $f(0)+2^0 = f(1)+2^1 = f(2) + 2^2$. What is $f(0)$? [b]p6.[/b] A pentomino is a set of five planar unit squares that are joined edge to edge. Two pentominoes are considered the same if and only if one can be rotated and translated to be identical to the other. We say that a pentomino is compact if it can fit within a $2$ by $3$ rectangle. How many distinct compact pentominoes exist? [b]p7.[/b] Consider a hexagon with interior angle measurements of $91$, $101$, $107$, $116$, $152$, and $153$ degrees. What is the average of the interior angles of this hexagon, in degrees? [b]p8.[/b] What is the smallest positive number that is either one larger than a perfect cube and one less than a perfect square, or vice versa? [b]p9.[/b] What is the first time after $4:56$ (a.m.) when the $24$-hour expression for the time has three consecutive digits that form an increasing arithmetic sequence with difference $1$? (For example, $23:41$ is one of those moments, while $23:12$ is not.) [b]p10.[/b] Chad has trouble counting. He wants to count from $1$ to $100$, but cannot pronounce the word "three," so he skips every number containing the digit three. If he tries to count up to $100$ anyway, how many numbers will he count? [b]p11.[/b] In square $ABCD$, point $E$ lies on side $BC$ and point $F$ lies on side $CD$ so that triangle $AEF$ is equilateral and inside the square. Point $M$ is the midpoint of segment $EF$, and $P$ is the point other than $E$ on $AE$ for which $PM = FM$. The extension of segment $PM$ meets segment $CD$ at $Q$. What is the measure of $\angle CQP$, in degrees? [b]p12.[/b] One apple is five cents cheaper than two bananas, and one banana is seven cents cheaper than three peaches. How much cheaper is one apple than six peaches, in cents? [b]p13.[/b] How many ordered pairs of integers $(a, b)$ exist for which |a| and |b| are at most $3$, and $a^3-a = b^3-b$? [b]p14.[/b] Five distinct boys and four distinct girls are going to have lunch together around a table. They decide to sit down one by one under the following conditions: no boy will sit down when more boys than girls are already seated, and no girl will sit down when more girls than boys are already seated. How many possible sequences of taking seats exist? [b]p15.[/b] Jordan is swimming laps in a pool. For each lap after the first, the time it takes her to complete is five seconds more than that of the previous lap. Given that she spends 10 minutes on the first six laps, how long does she spend on the next six laps, in minutes? [b]p16.[/b] Chad decides to go to trade school to ascertain his potential in carpentry. Chad is assigned to cut away all the vertices of a wooden regular tetrahedron with sides measuring four inches. Each vertex is cut away by a plane which passes through the three midpoints of the edges adjacent to that vertex. What is the surface area of the resultant solid, in square inches? Note: A tetrahedron is a solid with four triangular faces. In a regular tetrahedron, these faces are all equilateral triangles. [b]p17.[/b] Chad and Jordan independently choose two-digit positive integers. The two numbers are then multiplied together. What is the probability that the result has a units digit of zero? [b]p18.[/b] For art class, Jordan needs to cut a circle out of the coordinate grid. She would like to find a circle passing through at least $16$ lattice points so that her cut is accurate. What is the smallest possible radius of her circle? Note: A lattice point is defined as one whose coordinates are both integers. For example, $(5, 8)$ is a lattice point whereas $(3.5, 5)$ is not. [b]p19.[/b] Chad's ant Arctica is on one of the eight corners of Chad's toolbox, which measures two decimeters in width, three decimeters in length, and four decimeters in height. One day, Arctica wanted to go to the opposite corner of this box. Assuming she can only crawl on the surface of the toolbox, what is the shortest distance she has to crawl to accomplish this task, in decimeters? (You may assume that the toolbox is oating in the Exeter Space Station, so that Arctica can crawl on all six faces.) [b]p20.[/b] Jordan is counting numbers for fun. She starts with the number $1$, and then counts onward, skipping any number that is a divisor of the product of all previous numbers she has said. For example, she starts by counting $1$, $2$, $3$, $4$, $5$, but skips 6, a divisor of $1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 = 120$. What is the $20^{th}$ number she counts? [b]p21.[/b] Chad and Jordan are having a race in the lake shown below. The lake has a diameter of four kilometers and there is a circular island in the middle of the lake with a diameter of two kilometers. They start at one point on the edge of the lake and finish at the diametrically opposite point. Jordan makes the trip only by swimming in the water, while Chad swims to the island, runs across it, and then continues swimming. They both take the fastest possible route and, amazingly, they tie! Chad swims at two kilometers an hour and runs at five kilometers an hour. At what speed does Jordan swim? [img]https://cdn.artofproblemsolving.com/attachments/f/6/22b3b0bba97d25ab7aabc67d30821d0b12efc0.png[/img] [b]p22.[/b] Cameron has stolen Chad's barrel of oil and is driving it around on a truck on the coordinate grid on his truck. Cameron is a bad truck driver, so he can only move the truck forward one kilometer at a $4$ $EMC^2$ $2014$ Problems time along one of the gridlines. In fact, Cameron is so bad at driving the truck that between every two one-kilometer movements, he has to turn exactly $90$ degrees. After $50$ one-kilometer movements, given that Cameron's first one-kilometer movement was westward, how many points he could be on? [b]p23.[/b] Let $a$, $b$, and $c$ be distinct nonzero base ten digits. Assume there exist integers $x$ and $y$ for which $\overline{abc} \cdot \overline{cb} = 100x^2 + 1$ and $\overline{acb} \cdot \overline{bc} = 100y^2 + 1$. What is the minimum value of the number $\overline{abbc}$? Note: The notation $\overline{pqr}$ designates the number whose hundreds digit is $p$, tens digit is $q$, and units digit is $r$, not the product $p \cdot q \cdot r$. [b]p24.[/b] Let $r_1, r_2, r_3, r_4$ and $r_5$ be the five roots of the equation $x^5-4x^4+3x^2-2x+1 = 0$. What is the product of $(r_1 +r_2 +r_3 +r_4)$, $(r_1 +r_2 +r_3 +r_5)$, $(r_1 +r_2 +r_4 +r_5)$, $(r_1 +r_3 +r_4 +r_5)$, and $(r_2 +r_3 +r_4 +r_5)$? [b]p25.[/b] Chad needs seven apples to make an apple strudel for Jordan. He is currently at 0 on the metric number line. Every minute, he randomly moves one meter in either the positive or the negative direction with equal probability. Arctica's parents are located at $+4$ and $-4$ on the number line. They will bite Chad for kidnapping Arctica if he walks onto those numbers. Also, there is one apple located at each integer between $-3$ and $3$, inclusive. Whenever Chad lands on an integer with an unpicked apple, he picks it. What is the probability that Chad picks all the apples without getting bitten by Arctica's parents? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].