Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$. (1) Prove that there exists an equilateral triangle whose vertices lie in different discs. (2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$. [i]Radu Gologan, Romania[/i] [hide="Remark"] The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url]. [/hide]

Mable and Nora play a game according to the following steps in order: 1. Mable writes down any 2015 distinct prime numbers in ascending order in a row. The product of these primes is Marble's score. 2. Nora writes down a positive integer 3. Mable draws a vertical line between two adjacent primes she has written in step 1, and compute the product of the prime(s) on the left of the vertical line 4. Nora must add the product obtained by Marble in step 3 to the number she has written in step 2, and the sum becomes Nora's score. If Marble and Nora's scores have a common factor greater than 1, Marble wins, otherwise Nora wins. Who has a winning strategy?

Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws $25$ more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting these two points. Find the expected number of regions into which these $27$ line segments divide the disk.

Find the number of integer solutions of $\left[\frac{x}{100} \left[\frac{x}{100}\right]\right]= 5$

In the triangle $ABC$, let $A'$ be the intersection of the perpendicular bisector of $AB$ and the angle bisector of $\angle BAC$ and define $B', C'$ analogously. Prove that a) The triangle $ABC$ is equilateral if and only if $A' =B'.$ b) If $A', B'$ and $C'$ are distinct, we have $\angle B' A' C' = 90^{\circ} - \frac{1}{2} \angle BAC.$

Solve the equation $3^x=2^xy+1$ in positive integers.

[b]p1.[/b] $17$ rooks are placed on an $8\times 8$ chess board. Prove that there must be at least one rook that is attacking at least $2$ other rooks. [b]p2.[/b] In New Scotland there are three kinds of coins: $1$ cent, $6$ cent, and $36$ cent coins. Josh has $99$ of the $36$-cent coins (and no other coins). He is allowed to exchange a $36$ cent coin for $6$ coins of $6$ cents, and to exchange a $6$ cent coin for $6$ coins of $1$ cent. Is it possible that after several exchanges Josh will have $500$ coins? [b]p3.[/b] Find all solutions $a, b, c, d, e, f, g, h, i$ if these letters represent distinct digits and the following multiplication is correct: $\begin{tabular}{ccccc} & & a & b & c \\ x & & & d & e \\ \hline & f & a & c & c \\ + & g & h & i & \\ \hline f & f & f & c & c \\ \end{tabular}$ [b]p4.[/b] Pinocchio rode a bicycle for $3.5$ hours. During every $1$-hour period he went exactly $5$ km. Is it true that his average speed for the trip was $5$ km/h? Explain your reasoning. [b]p5.[/b] Let $a, b, c$ be odd integers. Prove that the equation $ax^2 + bx + c = 0$ cannot have a rational solution. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle. The tangent in $A$ of the circumcircle of $ABC$ cuts the line $(BC)$ in $X$. Let $A'$ be the symetric of $A$ by $X$ and $C'$ the symetric of $C$ by the line $(AX)$ Prove that the points $A, C', A'$ and $B$ are concyclic.

Compute $$\int_{1/2}^{2} \frac{x^2 + 1}{x^2(x^{2015} + 1)} dx.$$

Henry the donkey has a very long piece of pasta. He takes a number of bites of pasta, each time eating $3$ inches of pasta from the middle of one piece. In the end, he has $10$ pieces of pasta whose total length is $17$ inches. How long, in inches, was the piece of pasta he started with? $\textbf{(A)} ~34\qquad\textbf{(B)} ~38\qquad\textbf{(C)} ~41\qquad\textbf{(D)} ~44\qquad\textbf{(E)} ~47\qquad$

Find all functions $f:\mathbb R_{\ge0}\times\mathbb R_{\ge0}\to\mathbb R_{\ge0}$ such that $1$. $f(x,0)=f(0,x)=x$ for all $x\in\mathbb R_{\ge0}$, $2$. $f(f(x,y),z)=f(x,f(y,z))$ for all $x,y,z\in\mathbb R_{\ge0}$ and $3$. there exists a real $k$ such that $f(x+y,x+z)=kx+f(y,z)$ for all $x,y,z\in\mathbb R_{\ge0}$.

A cunning dragon guards a princess. To overcome the dragon and to win the princess you must solve the following task: The dragon has in some of the fields $i$ the columned hall (see figure) with the numbers $1-8$. Even in the rest of the fields you can place numbers $9-36$. The numbers $1-36$ must be arranged so that any turn that starts with one enters from either the south or the west, and ends up going out towards either the north or east, goes through at least one number from the $5$ table. (On the figure are north, south, east and west indicated by N, S, E and W). Georg wants to win the princess. Is it possible to be done? [img]https://cdn.artofproblemsolving.com/attachments/0/7/2ad1b7f944847ee6d3c614ea6c2656865808e7.png[/img]

Steve plants ten trees every three minutes. If he continues planting at the same rate, how long will it take him to plant 2500 trees? $\textbf{(A)}\ 1~1/4 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12~1/2$

A number $N$ has three digits when expressed in base $7$. When $N$ is expressed in base $9$ the digits are reversed. Then the middle digit is: $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5$

In an election for the Peer Pressure High School student council president, there are 2019 voters and two candidates Alice and Celia (who are voters themselves). At the beginning, Alice and Celia both vote for themselves, and Alice's boyfriend Bob votes for Alice as well. Then one by one, each of the remaining 2016 voters votes for a candidate randomly, with probabilities proportional to the current number of the respective candidate's votes. For example, the first undecided voter David has a $\tfrac{2}{3}$ probability of voting for Alice and a $\tfrac{1}{3}$ probability of voting for Celia. What is the probability that Alice wins the election (by having more votes than Celia)?

Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $AC=15$. Let $M$ be the midpoint of $BC$ and let $\Gamma$ be the circle passing through $A$ and tangent to line $BC$ at $M$. Let $\Gamma$ intersect lines $AB$ and $AC$ at points $D$ and $E$, respectively, and let $N$ be the midpoint of $DE$. Suppose line $MN$ intersects lines $AB$ and $AC$ at points $P$ and $O$, respectively. If the ratio $MN:NO:OP$ can be written in the form $a:b:c$ with $a,b,c$ positive integers satisfying $\gcd(a,b,c)=1$, find $a+b+c$. [i]James Tao[/i]

Samia set off on her bicycle to visit her friend, traveling at an average speed of 17 kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at 5 kilometers per hour. In all it took her 44 minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk? $\textbf{(A)}\ 2.0 \qquad \textbf{(B)}\ 2.2\qquad \textbf{(C)}\ 2.8 \qquad \textbf{(D)}\ 3.4 \qquad \textbf{(E)}\ 4.4$

Find all positive integers n with the following property: There exists a positive integer $k$ and mutually distinct integers $x_1,x_2,\ldots,x_n$ such that the set $\{x_i+x_j\mid1\le i<j\le n\}$ is a set of distinct powers of $k$.

Two mirror walls are placed to form an angle of measure $\alpha$. There is a candle inside the angle. How many reflections of the candle can an observer see?

Two players play the following game on a square of $N \times N$ squares. They color one square in turn so that no two colored squares are on the same diagonal. A player who cannot make a move loses. For what values of $N$ does the first player have a winning strategy?

Let $f(x)=ax^2+bx+c$ be a quadratic function, the graph of which doesn't intersect the x-axis. Prove that $$a(2a+3b+6c)>0.$$

Let $a,b$, be two real numbers. If for any $x>0$ holds that $|a-b|<x$, then prove that $a=b$.

a) Prove that a convex $n$-gon cannot have more than $3$ interior angles acute. b) Prove that a convex $n$-gon that has $3$ interior angles equal to $60^0,$ is equilateral.

A bug is standing at each of the vertices of a regular hexagon $ABCDEF$. At the same time each bug picks one of the vertices of the hexagon, which it is not currently in, and immediately starts moving towards that vertex. Each bug travels in a straight line from the vertex it was in originally to the vertex it picked. All bugs travel at the same speed and are of negligible size. Once a bug arrives at a vertex it picked, it stays there. In how many ways can the bugs move to the vertices so that no two bugs are ever in the same spot at the same time?

Assuming $a,b,c$ are the side-lengths of a triangle, show that \begin{align*} \frac{a^2+b^2-c^2}{ab} + \frac{b^2+c^2-a^2}{bc} + \frac{c^2+a^2-b^2}{ca} > 2. \end{align*} Also show that the inequality does not necessarily hold if you replace $2$ (on the right-hand side) by a bigger by a bigger number.