[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide] [b]D1.[/b] What is the solution to the equation $3 \cdot x \cdot 5 = 4 \cdot 5 \cdot 6$? [b]D2.[/b] Mr. Rose is making Platonic solids! If there are five different types of Platonic solids, and each Platonic solid can be one of three colors, how many different colored Platonic solids can Mr. Rose make? [b]D3.[/b] What fraction of the multiples of $5$ between $1$ and $100$ inclusive are also multiples of $20$? [b]D4.[/b] What is the maximum number of times a circle can intersect a triangle? [b]D5 / L1.[/b] At an interesting supermarket, the nth apple you purchase costs $n$ dollars, while pears are $3$ dollars each. Given that Layla has exactly enough money to purchase either $k$ apples or $2k$ pears for $k > 0$, how much money does Layla have? [b]D6 / L3.[/b] For how many positive integers $1 \le n \le 10$ does there exist a prime $p$ such that the sum of the digits of $p$ is $n$? [b]D7 / L2.[/b] Real numbers $a, b, c$ are selected uniformly and independently at random between $0$ and $1$. What is the probability that $a \ge b \le c$? [b]D8.[/b] How many ordered pairs of positive integers $(x, y)$ satisfy $lcm(x, y) = 500$? [b]D9 / L4.[/b] There are $50$ dogs in the local animal shelter. Each dog is enemies with at least $2$ other dogs. Steven wants to adopt as many dogs as possible, but he doesn’t want to adopt any pair of enemies, since they will cause a ruckus. Considering all possible enemy networks among the dogs, find the maximum number of dogs that Steven can possibly adopt. [b]D10 / L7.[/b] Unit circles $a, b, c$ satisfy $d(a, b) = 1$, $d(b, c) = 2$, and $d(c, a) = 3,$ where $d(x, y)$ is defined to be the minimum distance between any two points on circles $x$ and $y$. Find the radius of the smallest circle entirely containing $a$, $b$, and $c$. [b]D11 / L8.[/b] The numbers $1$ through $5$ are written on a chalkboard. Every second, Sara erases two numbers $a$ and $b$ such that $a \ge b$ and writes $\sqrt{a^2 - b^2}$ on the board. Let M and m be the maximum and minimum possible values on the board when there is only one number left, respectively. Find the ordered pair $(M, m)$. [b]D12 / L9.[/b] $N$ people stand in a line. Bella says, “There exists an assignment of nonnegative numbers to the $N$ people so that the sum of all the numbers is $1$ and the sum of any three consecutive people’s numbers does not exceed $1/2019$.” If Bella is right, find the minimum value of $N$ possible. [b]D13 / L10.[/b] In triangle $\vartriangle ABC$, $D$ is on $AC$ such that $BD$ is an altitude, and $E$ is on $AB$ such that $CE$ is an altitude. Let F be the intersection of $BD$ and $CE$. If $EF = 2FC$, $BF = 8DF$, and $DC = 3$, then find the area of $\vartriangle CDF$. [b]D14 / L11.[/b] Consider nonnegative real numbers $a_1, ..., a_6$ such that $a_1 +... + a_6 = 20$. Find the minimum possible value of $$\sqrt{a^2_1 + 1^2} +\sqrt{a^2_2 + 2^2} +\sqrt{a^2_3 + 3^2} +\sqrt{a^2_4 + 4^2} +\sqrt{a^2_5 + 5^2} +\sqrt{a^2_6 + 6^2}.$$ [b]D15 / L13.[/b] Find an $a < 1000000$ so that both $a$ and $101a$ are triangular numbers. (A triangular number is a number that can be written as $1 + 2 +... + n$ for some $n \ge 1$.) Note: There are multiple possible answers to this problem. You only need to find one. [b]L6.[/b] How many ordered pairs of positive integers $(x, y)$, where $x$ is a perfect square and $y$ is a perfect cube, satisfy $lcm(x, y) = 81000000$? [b]L12.[/b] Given two points $A$ and $B$ in the plane with $AB = 1$, define $f(C)$ to be the incenter of triangle $ABC$, if it exists. Find the area of the region of points $f(f(X))$ where $X$ is arbitrary. [b]L14.[/b] Leptina and Zandar play a game. At the four corners of a square, the numbers $1, 2, 3$, and $4$ are written in clockwise order. On Leptina’s turn, she must swap a pair of adjacent numbers. On Zandar’s turn, he must choose two adjacent numbers $a$ and $b$ with $a \ge b$ and replace $a$ with $ a - b$. Zandar wants to reduce the sum of the numbers at the four corners of the square to $2$ in as few turns as possible, and Leptina wants to delay this as long as possible. If Leptina goes first and both players play optimally, find the minimum number of turns Zandar can take after which Zandar is guaranteed to have reduced the sum of the numbers to $2$. [b]L15.[/b] There exist polynomials $P, Q$ and real numbers $c_0, c_1, c_2, ... , c_{10}$ so that the three polynomials $P, Q$, and $$c_0P^{10} + c_1P^9Q + c_2P^8Q^2 + ... + c_{10}Q^{10}$$ are all polynomials of degree 2019. Suppose that $c_0 = 1$, $c_1 = -7$, $c_2 = 22$. Find all possible values of $c_{10}$. Note: The answer(s) are rational numbers. It suffices to give the prime factorization(s) of the numerator(s) and denominator(s). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The product of three integers is $60$. What is the least possible positive sum of the three integers? $\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 13$

Let $n \geq 2$ be a positive integer. On a spaceship, there are $n$ crewmates. At most one accusation of being an imposter can occur from one crewmate to another crewmate. Multiple accusations are thrown, with the following properties: • Each crewmate made a different number of accusations. • Each crewmate received a different number of accusations. • A crewmate does not accuse themself. Prove that no two crewmates made accusations at each other.

A good approximation of $\pi$ is $3.14.$ Find the least positive integer $d$ such that if the area of a circle with diameter $d$ is calculated using the approximation $3.14,$ the error will exceed $1.$

For any permutation $p$ of the set $\{1,2,...,n\}$, let us denote $d(p) = |p(1)-1|+|p(2)-2|+...+|p(n)-n|$. Let $i(p)$ be the number of inversions of $p$, i.e. the number of pairs $1 \le i < j \le n$ with $p(i) > p(j)$. Prove that $d(p)\le 2i(p)$$.

Let $M$ be a positive real number. Determine the least positive real number $k$ with the following property: for each integer $n>M$, the interval $(n,kn]$ contains a power of $2$.

Let $\{a_n\}$ be the sequence such that $a_0=2019$ and $$a_n=-\frac{2020}{n}\sum_{k=0}^{n-1}a_k.$$ Compute the last three digits of $\sum_{n=1}^{2020}2020^na_nn$.

The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?

(A.Myakishev, 9--10) Given a triangle $ ABC$. Point $ A_1$ is chosen on the ray $ BA$ so that segments $ BA_1$ and $ BC$ are equal. Point $ A_2$ is chosen on the ray $ CA$ so that segments $ CA_2$ and $ BC$ are equal. Points $ B_1$, $ B_2$ and $ C_1$, $ C_2$ are chosen similarly. Prove that lines $ A_1A_2$, $ B_1B_2$, $ C_1C_2$ are parallel.

$x_1$ is a natural constant. Prove that there does not exist any natural number $m> 2500$ such that the recursive sequence $\{x_i\} _{i=1} ^ \infty $ defined by $x_{n+1} = x_n^{s(n)} + 1$ becomes eventually periodic modulo $m$. (That is there does not exist natural numbers $N$ and $T$ such that for each $n\geq N$, $m\mid x_n - x_{n+T}$). ($s(n)$ is the sum of digits of $n$.)

Let $ X,Y,Z$ be the midpoints of the small arcs $ BC,CA,AB$ respectively (arcs of the circumcircle of $ ABC$). $ M$ is an arbitrary point on $ BC$, and the parallels through $ M$ to the internal bisectors of $ \angle B,\angle C$ cut the external bisectors of $ \angle C,\angle B$ in $ N,P$ respectively. Show that $ XM,YN,ZP$ concur.

The following figure is made up of $12$ segments and $8$ circles. As you can see, at the beginning all the circles are empty. In each operation an empty circle is chosen, it is painted red and inside it the number of red neighboring circles that the chosen circle has is written (in the first operation the chosen circle is painted red and the number $0$ is written). After $8$ operations all the circles are painted red and each one has a number written on it. Prove that, no matter how the operations are done, the sum of all the numbers at the end is the same. [img]https://cdn.artofproblemsolving.com/attachments/3/a/8cd74a0fdc7bb9bc5d1bc764e80ffb58159c0c.png[/img]

Let $n \in \mathbb{N}$ and $z \in \mathbb{C}^{*}$. Prove that $\left | n\textrm{e}^{z} - \sum_{j=1}^{n}\left (1+\frac{z}{j^2}\right )^{j^2}\right | < \frac{1}{3}\textrm{e}^{|z|}\left (\frac{\pi|z|}{2}\right)^2$.

Starting with the sequence $F_1 = (1,2,3,4, \ldots)$ of the natural numbers further sequences are generated as follows: $F_{n+1}$ is created from $F_n$ by the following rule: the order of elements remains unchanged, the elements from $F_n$ which are divisible by $n$ are increased by 1 and the other elements from $F_n$ remain unchanged. Example: $F_2 = (2,3,4,5 \ldots)$ and $F_3 = (3,3,5,5, \ldots)$. Determine all natural numbers $n$ such that exactly the first $n-1$ elements of $F_n$ take the value $n.$

Let $m$ be a positive integers. A square room with corners at $(0,0), (2m,0), (0,2m),$ $(2m,2m)$ has mirrors as walls. At each integer lattice point $(i,j)$ with $0 < i, j < 2m$ a single small double sided mirror is oriented parallel to either the $x$ or $y$ axis. A beam of light is shone from a corner making a $45^\circ$ angle with each of the walls. Prove that the opposite corner is not lit.

[u]Round 1[/u] [b]p1.[/b] Define the operation $\clubsuit$ so that $a \,\clubsuit \, b = a^b + b^a$. Then, if $2 \,\clubsuit \,b = 32$, what is $b$? [b]p2. [/b] A square is changed into a rectangle by increasing two of its sides by $p\%$ and decreasing the two other sides by $p\%$. The area is then reduced by $1\%$. What is the value of $p$? [b]p3.[/b] What is the sum, in degrees, of the internal angles of a heptagon? [b]p4.[/b] How many integers in between $\sqrt{47}$ and $\sqrt{8283}$ are divisible by $7$? [u]Round 2[/u] [b]p5.[/b] Some mutant green turkeys and pink elephants are grazing in a field. Mutant green turkeys have six legs and three heads. Pink elephants have $4$ legs and $1$ head. There are $100$ legs and $37$ heads in the field. How many animals are grazing? [b]p6.[/b] Let $A = (0, 0)$, $B = (6, 8)$, $C = (20, 8)$, $D = (14, 0)$, $E = (21, -10)$, and $F = (7, -10)$. Find the area of the hexagon $ABCDEF$. [b]p7.[/b] In Moscow, three men, Oleg, Igor, and Dima, are questioned on suspicion of stealing Vladimir Putin’s blankie. It is known that each man either always tells the truth or always lies. They make the following statements: (a) Oleg: I am innocent! (b) Igor: Dima stole the blankie! (c) Dima: I am innocent! (d) Igor: I am guilty! (e) Oleg: Yes, Igor is indeed guilty! If exactly one of Oleg, Igor, and Dima is guilty of the theft, who is the thief?? [b]p8.[/b] How many $11$-letter sequences of $E$’s and $M$’s have at least as many $E$’s as $M$’s? [u]Round 3[/u] [b]p9.[/b] John is entering the following summation $31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39$ in his calculator. However, he accidently leaves out a plus sign and the answer becomes $3582$. What is the number that comes before the missing plus sign? [b]p10.[/b] Two circles of radius $6$ intersect such that they share a common chord of length $6$. The total area covered may be expressed as $a\pi + \sqrt{b}$, where $a$ and $b$ are integers. What is $a + b$? [b]p11.[/b] Alice has a rectangular room with $6$ outlets lined up on one wall and $6$ lamps lined up on the opposite wall. She has $6$ distinct power cords (red, blue, green, purple, black, yellow). If the red and green power cords cannot cross, how many ways can she plug in all six lamps? [b]p12.[/b] Tracy wants to jump through a line of $12$ tiles on the floor by either jumping onto the next block, or jumping onto the block two steps ahead. An example of a path through the $12$ tiles may be: $1$ step, $2$ steps, $2$ steps, $2$ steps, $1$ step, $2$ steps, $2$ steps. In how many ways can Tracy jump through these $12$ tiles? PS. You should use hide for answers. Last rounds have been posted [url=https://artofproblemsolving.com/community/c4h2784268p24464984]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ \alpha$ be a rational number with $ 0 < \alpha < 1$ and $ \cos (3 \pi \alpha) \plus{} 2\cos(2 \pi \alpha) \equal{} 0$. Prove that $ \alpha \equal{} \frac {2}{3}$.

If $-3 \le x < \frac{3}{2}$ and $x \ne 1$, define $C(x) = \frac{x^3}{1 - x}$. The real root of the cubic $2x^3 + 3x - 7$ is of the form $p C^{-1}(q)$, where $p$ and $q$ are rational numbers. What is the ordered pair $(p, q)$?

Suppose $ABC$ is a triangle with inradius $r$. The incircle touches the sides $BC, CA,$ and $AB$ at $D,E$ and $F$ respectively. If $BD=x, CE=y$ and $AF=z$, then show that \[r^2=\frac{xyz}{x+y+z}\]

The positive integers $p$, $q$ are such that for each real number $x$ $(x+1)^p (x-3)^q=x^n+a_1 x^{n-1}+a_2 x^{n-2}+\dots +a_{n-1} x+a_n$ where $n=p+q$ and $a_1,\dots ,a_n$ are real numbers. Prove that there exists infinitely many pairs $(p,q)$ for which $a_1=a_2$.

A pair of positive integers $(x, y)$ is [i] good [/i] if they satisfy $\text{rad}(x) = \text{rad}(y)$ and they do not divide each-other. Given coprime positive integers $a$ and $b$, show that there exist infinitely many $n$ for which there exists a positive integer $m$ such that $(a^n + bm, b^n + am)$ is [i] good[/i]. (Here, $\text{rad}(x)$ denotes the product of $x$'s prime divisors, as usual.)

Let $ ABC$ be an acute-angled triangle. We consider the equilateral triangle $ A'UV$, where $ A' \in (BC)$, $ U\in (AC)$ and $ V\in(AB)$ such that $ UV \parallel BC$. We define the points $ B',C'$ in the same way. Prove that $ AA'$, $ BB'$ and $ CC'$ are concurrent.

Let $n$ be an integer. Dayang are given $n$ sticks of lengths $1,2, 3,..., n$. She may connect the sticks at their ends to form longer sticks, but cannot cut them. She wants to use all these sticks to form a square. For example, for $n = 8$, she can make a square of side length $9$ using these connected sticks: $1 + 8$, $2 + 7$, $3 + 6$, and $4 + 5$. How many values of $n$, with $1 \le n \le 2018$, that allow her to do this?

Determine the smallest integer $n$ for which there exist integers $x_1,\ldots,x_n$ and positive integers $a_1,\ldots,a_n$ so that \begin{align*} x_1+\cdots+x_n &=0,\\ a_1x_1+\cdots+a_nx_n&>0, \text{ and }\\ a_1^2x_1+\cdots+a_n^2x_n &<0. \end{align*}

$ x \plus{} 19y \equiv 0 \pmod {23}$ and $ x \plus{} y < 69$. How many pairs of $ (x,y)$ are there in positive integers? $\textbf{(A)}\ 100 \qquad\textbf{(B)}\ 102 \qquad\textbf{(C)}\ 105 \qquad\textbf{(D)}\ 109 \qquad\textbf{(E)}\ \text{None}$