Four mutually externally tangent spherical apples of radius $4$ are placed on a horizontal flat table. Then, a spherical orange of radius $3$ is placed such that it rests on all the apples. Find the distance from the center of the orange to the table.

On a sheet divided into squares, each square measuring $2cm$, two circles are drawn such that both circles are inscribed in a square as in the figure below. Determine the minimum distance between the two circles.

Two straight lines divide a square of side length $1$ into four regions. Show that at least one of the regions has a perimeter greater than or equal to $2$. [i]Proposed by Dylan Toh[/i]

Call a positive integer [i]convenient[/i] if its digits can be partitioned into two collections of contiguous digits whose element sums are $7$ and $11$. For example, $3456$ is convenient, but $4247$ is not. Compute the number of convenient positive integers less than or equal to $10^5$.

Mr. DoBa has a bag of markers. There are 2 blue, 3 red, 4 green, and 5 yellow markers. Mr. DoBa randomly takes out two markers from the bag. The probability that these two markers are different colors can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Raina Yang[/i]

Find all prime numbers $a,b,c$ and positive integers $k$ satisfying the equation \[a^2+b^2+16c^2 = 9k^2 + 1.\] Proposed by Moldova

Consider coins with positive real denominations not exceeding 1. Find the smallest $C>0$ such that the following holds: if we have any $100$ such coins with total value $50$, then we can always split them into two stacks of $50$ coins each such that the absolute difference between the total values of the two stacks is at most $C$. [i]Merlijn Staps[/i]

Three lines passing through point $ O$ form equal angles by pairs. Points $ A_1$, $ A_2$ on the first line and $ B_1$, $ B_2$ on the second line are such that the common point $ C_1$ of $ A_1B_1$ and $ A_2B_2$ lies on the third line. Let $ C_2$ be the common point of $ A_1B_2$ and $ A_2B_1$. Prove that angle $ C_1OC_2$ is right.

A midpoint plotter is an instrument which draws the exact mid point of two point previously drawn. Starting off two points $1$ unit of distance apart and using only the midpoint plotter, you have to get two point which are strictly at a distance between $\frac{1}{2017}$ and $\frac{1}{2016}$ units, drawing the minimum amount of points. ¿Which is the minimum number of times you will need to use the midpoint plotter and what strategy should you follow to achieve it?

Let $r$ be a positive integer and let $N$ be the smallest positive integer such that the numbers $\frac{N}{n+r}\binom{2n}{n}$, $n=0,1,2,\ldots $, are all integer. Show that $N=\frac{r}{2}\binom{2r}{r}$.

We are given a certain number of identical sets of weights; each set consists of four different weights expressed by natural numbers (of weight units). Using these weights we are able to weigh out every integer mass up to $1985$ (inclusive). How many ways are there to compose such a set of weight sets given that the joint mass of all weights is the least possible?

For any triple $(a, b, c)$ of positive integers, not all equal, We are given sufficiently many rectangular blocks of size $a \times b \times c$. We use these blocks to fill up a cubic box of edge $10$. (a) Assume we have used at least $100$ blocks. Show that there are two blocks, one of which is a translate of the other. (b) Find a number smaller than $100$ (the smaller, the better) for which the above statement still holds.

In a triangle $ABC$ with $\widehat{A}=80^{\circ}$ and $\widehat{B}=60^{\circ}$, the internal angle bisector of $\widehat{C}$ meets the side $AB$ at the point $D$. The parallel from $D$ to the side $AC$, meets the side $BC$ at the point $E$. Find the measure of the angle $\angle EAB$.

Given positive integer $ n > 1$ and define \[ S \equal{} \{1,2,\dots,n\}. \] Suppose \[ T \equal{} \{t \in S: \gcd(t,n) \equal{} 1\}. \] Let $ A$ be arbitrary non-empty subset of $ A$ such thar for all $ x,y \in A$, we have $ (xy\mod n) \in A$. Prove that the number of elements of $ A$ divides $ \phi(n)$. ($ \phi(n)$ is Euler-Phi function)

The figure shows a triangular pool table with sides $a$, $b$ and $c$. Located at point $S$ on $c$ a sphere - which can be assumed as a point. After kick-off, as indicated in the figure, it runs through as a result of reflections to $a, b, a, b$ and $c$ (in $S$) always the same track. The reflection occurs according to law of reflection. Characterize entilrely all triangles $ABC$, which allow such an orbit, and determine the locus of $S$. [img]https://cdn.artofproblemsolving.com/attachments/5/b/7662943e5b9ad321226e0c5f5daa3c4ac9faaa.png[/img]

Let $n$ be a positive integer. Each number $1, 2, ..., 1000$ has been colored with one of $n$ colours. Each two numbers , such that one is a divisor of second of them, are colored with different colours. Determine minimal number $n$ for which it is possible.

$a_1,a_2,\dots,a_k$ and $b_1,b_2,\dots,a_n$ are integer sequences with $a_1=b_1=0$, $a_k=26,b_n=25$, and $k+n=23$. It is known that $a_{i+1}-a_i=2\enspace\text{or}\enspace3$ and $b_{j+1}-b_j=2\enspace\text{or}\enspace3$ for all applicable $i$ and $j$, and the numbers $a_2,\dots,a_k,b_2,\dots,b_n$ are pairwise different. Determine the total number of such pairs of sequences.

Let be two matrices $ A,B\in M_2\left(\mathbb{R}\right) $ that satisfy the equality $ \left( A-B\right)^2 =O_2. $ [b]a)[/b] Show that $ \det\left( A^2-B^2\right) =\left( \det A -\det B\right)^2. $ [b]b)[/b] Demonstrate that $ \det\left( AB-BA\right) =0\iff \det A=\det B. $

Alice is once again very bored in class. On a whim, she chooses three primes $p$, $q$, $r$ independently and uniformly at random from the set of primes at most 30. She then calculates the roots of $px^2+qx+r$. What is the probability that at least one of her roots is an integer?

Isosceles, similar triangles $QPA$ and $SPB$ are constructed (outwards) on the sides of parallelogram $PQRS$ (where $PQ = AQ$ and $PS = BS$). Prove that triangles $RAB$, $QPA$ and $SPB$ are similar.

Let $n$ be a natural number, with the prime factorisation \[ n = p_1^{e_1} p_2^{e_2} \cdots p_r^{e_r} \] where $p_1, \ldots, p_r$ are distinct primes, and $e_i$ is a natural number. Define \[ rad(n) = p_1p_2 \cdots p_r \] to be the product of all distinct prime factors of $n$. Determine all polynomials $P(x)$ with rational coefficients such that there exists infinitely many naturals $n$ satisfying $P(n) = rad(n)$.

Horizontal line $m$ passes the center of circle $\odot O$. Line $l\perp m$, $l$ and $m$ intersect at $M$, and $M$ is on the right side of $O$. Three points $A,B,C$ ($B$ is in the middle) lie on line $l$, which are outside the circle, above line $m$. $AP,BQ,CR$ are tangent to $\odot O$ at $P,Q,R$. Prove: [b](a)[/b] If $l$ is tangent to $\odot O$, then $AB\cdot CR+BC\cdot AP=AC\cdot BQ$. [b](b)[/b] If $l$ and $\odot O$ intersect, then $AB\cdot CR+BC\cdot AP<AC\cdot BQ$. [b](c)[/b] If $l$ and $\odot O$ are apart, then $AB\cdot CR+BC\cdot AP>AC\cdot BQ$.

An unequal acute-angled triangle $ABC$ with an orthocenter $H$ is given, $M$ is the midpoint of side $BC$. Points $K$ and $L$ lie on a line passing through $H$ and perpendicular to $AM$ such a $KB$ and $LC$ perpendicular to $BC$. Point $N$ lies on the line $HM$, and the lines $AN$ and $AH$ are symmetric with respect to the line $AM$. Prove that a circle with a diameter $AN$ touches two circles: centered at $K$ and with a radius $KB$ and with a center $L$ and radius $LC$.

[u]Round 1[/u] [b]p1.[/b] How many distinct ways are there to scramble the letters in $EXETER$? [b]p2.[/b] Given that $\frac{x - y}{x - z}= 3$, find $\frac{x - z}{y - z}$. [b]p3.[/b] When written in base $10$, $9^9 =\overline{ABC420DEF}.$ Find the remainder when $A + B + C + D + E + F$ is divided by $9$. [u]Round 2[/u] [b]p4.[/b] How many positive integers, when expressed in base $7$, have exactly $3$ digits, but don't contain the digit $3$? [b]p5.[/b] Pentagon $JAMES$ is such that its internal angles satisfy $\angle J = \angle A = \angle M = 90^o$ and $\angle E = \angle S$. If $JA = AM = 4$ and $ME = 2$, what is the area of $JAMES$? [b]p6.[/b] Let $x$ be a real number such that $x = \frac{1+\sqrt{x}}{2}$ . What is the sum of all possible values of $x$? [u]Round 3[/u] [b]p7.[/b] Farmer James sends his favorite chickens, Hen Hao and PEAcock, to compete at the Fermi Estimation All Star Tournament (FEAST). The first problem at the FEAST requires the chickens to estimate the number of boarding students at Eggs-Eater Academy given the number of dorms $D$ and the average number of students per dorm $A$. Hen Hao rounds both $D$ and $A$ down to the nearest multiple of $10$ and multiplies them, getting an estimate of $1200$ students. PEAcock rounds both $D$ and $A$ up to the nearest multiple of $10$ and multiplies them, getting an estimate of $N$ students. What is the maximum possible value of $N$? [b]p8.[/b] Farmer James has decided to prepare a large bowl of egg drop soup for the Festival of Eggs-Eater Annual Soup Tasting (FEAST). To flavor the soup, Hen Hao drops eggs into it. Hen Hao drops $1$ egg into the soup in the first hour, $2$ eggs into the soup in the second hour, and so on, dropping $k$ eggs into the soup in the $k$th hour. Find the smallest positive integer $n$ so that after exactly n hours, Farmer James finds that the number of eggs dropped in his egg drop soup is a multiple of $200$. [b]p9.[/b] Farmer James decides to FEAST on Hen Hao. First, he cuts Hen Hao into $2018$ pieces. Then, he eats $1346$ pieces every day, and then splits each of the remaining pieces into three smaller pieces. How many days will it take Farmer James to eat Hen Hao? (If there are fewer than $1346$ pieces remaining, then Farmer James will just eat all of the pieces.) [u]Round 4[/u] [b]p10.[/b] Farmer James has three baskets, and each basket has one magical egg. Every minute, each magical egg disappears from its basket, and reappears with probability $\frac12$ in each of the other two baskets. Find the probability that after three minutes, Farmer James has all his eggs in one basket. [b]p11.[/b] Find the value of $\frac{4 \cdot 7}{\sqrt{4 +\sqrt7} +\sqrt{4 -\sqrt7}}$. [b]p12.[/b] Two circles, with radius $6$ and radius $8$, are externally tangent to each other. Two more circles, of radius $7$, are placed on either side of this configuration, so that they are both externally tangent to both of the original two circles. Out of these $4$ circles, what is the maximum distance between any two centers? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2949222p26406222]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let be three nonzero rational numbers $ a,b,c $ under the relation $ (a+b)(b+c)(c+a)=a^2b^2c^2. $ Show that the expression $ \sqrt[3]{3+1/a^3+1/b^3+1/c^3} $ is rational. [i]Ion Bursuc[/i]