[u]Round 1[/u] [b]p1.[/b] At a fortune-telling exam, $13$ witches are sitting in a circle. To pass the exam, a witch must correctly predict, for everybody except herself and her two neighbors, whether they will pass or fail. Each witch predicts that each of the $10$ witches she is asked about will fail. How many witches could pass? [b]p2.[/b] Out of $152$ coins, $7$ are counterfeit. All counterfeit coins have the same weight, and all real coins have the same weight, but counterfeit coins are lighter than real coins. How can you find $19$ real coins if you are allowed to use a balance scale three times? [b]p3.[/b] The digits of a number $N$ increase from left to right. What could the sum of the digits of $9 \times N$ be? [b]p4.[/b] The sides and diagonals of a pentagon are colored either blue or red. You can choose three vertices and flip the colors of all three lines that join them. Can every possible coloring be turned all blue by a sequence of such moves? [img]https://cdn.artofproblemsolving.com/attachments/5/a/644aa7dd995681fc1c813b41269f904283997b.png[/img] [b]p5.[/b] You have $100$ pancakes, one with a single blueberry, one with two blueberries, one with three blueberries, and so on. The pancakes are stacked in a random order. Count the number of blueberries in the top pancake and call that number $N$. Pick up the stack of the top $N$ pancakes and flip it upside down. Prove that if you repeat this counting-and-flipping process, the pancake with one blueberry will eventually end up at the top of the stack. [u]Round 2[/u] [b]p6.[/b] A circus owner will arrange $100$ fleas on a long string of beads, each flea on her own bead. Once arranged, the fleas start jumping using the following rules. Every second, each flea chooses the closest bead occupied by one or more of the other fleas, and then all fleas jump simultaneously to their chosen beads. If there are two places where a flea could jump, she jumps to the right. At the start, the circus owner arranged the fleas so that, after some time, they all gather on just two beads. What is the shortest amount of time it could take for this to happen? [b]p7.[/b] The faraway land of Noetheria has $2016$ cities. There is a nonstop flight between every pair of cities. The price of a nonstop ticket is the same in both directions, but flights between different pairs of cities have different prices. Prove that you can plan a route of $2015$ consecutive flights so that each flight is cheaper than the previous one. It is permissible to visit the same city several times along the way. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We define $(x_1, x_2, \ldots , x_n) \Delta (y_1, y_2, \ldots , y_n) = \left( \sum_{i=1}^{n}x_iy_{2-i}, \sum_{i=1}^{n}x_iy_{3-i}, \ldots , \sum_{i=1}^{n}x_iy_{n+1-i} \right)$, where the indices are taken modulo $n$. Besides this, if $v$ is a vector, we define $v^k = v$, if $k=1$, or $v^k = v \Delta v^{k-1}$, otherwise. Prove that, if $(x_1, x_2, \ldots , x_n)^k = (0, 0, \ldots , 0)$, for some natural number $k$, then $x_1 = x_2 = \ldots = x_n = 0$.

Let $f : R \to R$ satisfy $f(x + f(y)) = 2x + 4y + 2547$ for all reals $x, y$. Compute $f(0)$.

Let $a \not = 0, b, c$ be real numbers. If $x_{1}$ is a root of the equation $ax^{2}+bx+c = 0$ and $x_{2}$ a root of $-ax^{2}+bx+c = 0$, show that there is a root $x_{3}$ of $\frac{a}{2}\cdot x^{2}+bx+c = 0$ between $x_{1}$ and $x_{2}$.

Let $p > 2$ be a fixed prime number. Find all functions $f: \mathbb Z \to \mathbb Z_p$, where the $\mathbb Z_p$ denotes the set $\{0, 1, \ldots , p-1\}$, such that $p$ divides $f(f(n))- f(n+1) + 1$ and $f(n+p) = f(n)$ for all integers $n$. [i]Proposed by Grant Yu[/i]

A polynomial $p$ with integer coefficients is such that $p(-n) < p(n) < n$ for some integer $n$. Prove that $p(-n) < -n$.

In a class, there are $n \geqslant 3$ students and a teacher with $M$ marbles. The teacher then play a [i]Marble distribution[/i] according to the following rules. At the start, the teacher distributed all her marbles to students, so that each student receives at least $1$ marbles from the teacher. Then, the teacher chooses a student , who has never been chosen before, such that the number of marbles that he owns in a multiple of $2(n-1)$. That chosen student then equally distribute half of his marbles to $n-1$ other students. The same goes on until the teacher is not able to choose anymore student. Find all integer $M$, such that for some initial numbers of marbles that the students receive, the teacher can choose all the student(according to the rule above), so that each student receiving equal amount of marbles at the end.

Two numbers $1+\sqrt[3]{2}+\sqrt[3]{4}$ and $1+2\sqrt[3]{2}+3\sqrt[3]{4}$ are given. In one move you can do one of the following operations: 1. Replace one of the numbers $a$ with either $a-\sqrt[3]{2}$ or $-2a$ 2. Replace both numbers $a$ and $b$ with $a-b$ and $a+b$ (you can choose the order of $a$ and $b$ yourself) Prove that the obtained numbers are always non-zero

Let $f_1(x)$ be a polynomial of degree $2$ with the leading coefficient positive and $f_{n+1}(x) =f_1(f_n(x))$ for $n\ge 1.$ Prove that if the equation $f_2(x)=0$ has four different non-positive real roots, then for arbitrary $n$ then $f_n(x)$ has $2^n$ different real roots.

Let $a_0 < a_1 < a_2 < \dots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that \[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\] [i]Proposed by Gerhard Wöginger, Austria.[/i]

Let $P(x)$ be a polynomial with complex coefficients such that $P(0)\neq 0$. Prove that there exists a multiple of $P(x)$ with real positive coefficients if and only if $P(x)$ has no real positive root.

Show that for any $8$ distinct positive real numbers, one can choose a quadraple of them $(a,b,c,d)$ , all distinct such that $$(ac+bd)^2 \ge \frac{2+\sqrt3}{4}\left(a^2+b^2 \right)\left(c^2+d^2 \right)$$ [i]Proposed by Evan Chang (squareman), USA[/i]

After the schoolday is over, Juku must attend an extra math class. The teacher writes a quadratic equation $ x^2\plus{} p_1x\plus{}q_1 \equal{} 0$ with integer coefficients on the blackboard and Juku has to find its solutions. If they are not both integers, Jukumay go home. If the solutions are integers, then the teacher writes a new equation $ x^2 \plus{} p_2x \plus{} q_2 \equal{} 0,$ where $ p_2$ and $ q_2$ are the solutions of the previous equation taken in some order, and everything starts all over. Find all possible values for $ p_1$ and $ q_1$ such that the teacher can hold Juku at school forever.

The sequence $x_0, x_1, x_2, ...$ is defined by $x_0 = 0$, $x_{k+1} = [(n - \sum_0^k x_i)/2]$. Show that $x_k = 0$ for all sufficiently large $k$ and that the sum of the non-zero terms $x_k$ is $n-1$.

[b]p1.[/b] Alex and Sam have a friend Pat, who is younger than they are. Alex, Sam and Pat all share a birthday. When Pat was born, Alex’s age times Sam’s age was $42$. Now Pat’s age is $33$ and Alex’s age is a prime number. How old is Sam now? Show your work and justify your answer. (All ages are whole numbers.) [b]p2.[/b] Let $ABCD$ be a square with side length $2$. The four sides of $ABCD$ are diameters of four semicircles, each of which lies inside the square. The set of all points which lie on or inside two of these semicircles is a four petaled flower. Find (with proof) the area of this flower. [img]https://cdn.artofproblemsolving.com/attachments/5/5/bc724b9f74c3470434c322020997a533986d33.png[/img] [b]p3.[/b] A prime number is called [i]strongly prime[/i] if every integer obtained by permuting its digits is also prime. For example $113$ is strongly prime, since $113$, $131$, and $311$ are all prime numbers. Prove that there is no strongly prime number such that each of the digits $1, 3, 7$, and $9$ appears at least once in its decimal representation. [b]p4.[/b] Suppose $n$ is a positive integer. Let an be the number of permutations of $1, 2, . . . , n$, where $i$ is not in the $i$-th position, for all $i$ with $1 \le i \le n$. For example $a_3 = 2$, where the two permutations that are counted are $231$, and $312$. Let bn be the number of permutations of $1, 2, . . . , n$, where no $i$ is followed by $i + 1$, for all $i$ with $1 \le i \le n - 1$. For example $b_3 = 3$, where the three permutations that are counted are $132$, $213$, and $321$. For every $n \ge 1$, find (with proof) a simple formula for $\frac{a_{n+1}}{b_n}$. Your formula should not involve summations. Use your formula to evaluate $\frac{a_{2020}}{b_{2019}}$. [b]p5.[/b] Let $n \ge 2$ be an integer and $a_1, a_2, ... , a_n$ be positive real numbers such that $a_1 + a_2 +... + a_n = 1$. Prove that $$\sum^n_{k=1}\frac{a_k}{1 + a_{k+1} - a_{k-1}}\ge 1.$$ (Here $a_0 = a_n$ and $a_{n+1} = a_1$.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] In a $3$-person race, how many different results are possible if ties are allowed? [b]p2.[/b] An isosceles trapezoid has lengths $5$, $5$, $5$, and $8$. What is the sum of the lengths of its diagonals? [b]p3.[/b] Let $f(x) = (1 + x + x^2)(1 + x^3 + x^6)(1 + x^9 + x^{18})...$. Compute $f(4/5)$. [b]p4.[/b] Compute the largest prime factor of $3^{12} - 1$. [b]p5.[/b] Taren wants to throw a frisbee to David, who starts running perpendicular to the initial line between them at rate $1$ m/s. Taren throws the frisbee at rate $2$ m/s at the same instant David begins to run. At what angle should Taren throw the frisbee? [b]p6.[/b] The polynomial $p(x)$ leaves remainder $6$ when divided by $x-5$, and $5$ when divided by $x-6$. What is the remainder when $p(x)$ is divided by $(x - 5)(x - 6)$? [b]p7.[/b] Find the sum of the cubes of the roots of $x^{10} + x^9 + ... + x + 1 = 0$. [b]p8.[/b] A circle of radius $1$ is inscribed in a the parabola $y = x^2$. What is the $x$-coordinate of the intersection in the first quadrant? [b]p9.[/b] You are stuck in a cave with $3$ tunnels. The first tunnel leads you back to your starting point in $5$ hours, and the second tunnel leads you back there in $7$ hours. The third tunnel leads you out of the cave in $4$ hours. What is the expected number of hours for you to exit the cave, assuming you choose a tunnel randomly each time you come across your point of origin? [b]p10.[/b] What is the minimum distance between the line $y = 4x/7 + 1/5$ and any lattice point in the plane? (lattice points are points with integer coordinates) PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_n, b_n$ be sequences of positive reals such that,$$a_{n+1}= a_n + \frac{1}{2b_n}$$ $$b_{n+1}= b_n + \frac{1}{2a_n}$$ for all $n\in\mathbb N$. Prove that, $\text{max}\left(a_{2018}, b_{2018}\right) >44$.

Find all continuous functions $f: R \to R$ such that for all reals $x$ and $y$, $f(x+f(y)) = y+f(x+1)$.

Let $z_1$ and $z_2$ be the zeros of the polynomial $f(x) = x^2 + 6x + 11$. Compute $(1 + z^2_1z_2)(1 + z_1z_2^2)$.

Let $\star$ be an operation defined in the set of nonnegative integers with the following properties: for any nonnegative integers $x$ and $y$, (i) $(x + 1)\star 0 = (0\star x) + 1$ (ii) $0\star (y + 1) = (y\star 0) + 1$ (iii) $(x + 1)\star (y + 1) = (x\star y) + 1$. If $123\star 456 = 789$, find $246\star 135$.

Let $n$ be an integer. Prove that the polynomial $f(x)$ has at most one zero, where \[ f(x) = x^4 - 1994 x^3 + (1993+n)x^2 - 11x + n . \] [i]Greece[/i]

Find the smallest real number $M$, for which $\{a\}+\{b\}+\{c\}\leq M$ for any real positive numbers $a, b, c$ with $abc = 2024$. Here $\{a\}$ denotes the fractional part of number $a$. [i]Proposed by Fedir Yudin, Anton Trygub[/i]

It is well-known that if $\frac{p}{q}=\frac{r}{s}$, both of the expressions are also equal to $\frac{p-r}{q-s}$. Now we write the equality $$\frac{3x-b}{3x-5b}=\frac{3a-4b}{3a-8b}.$$ The previous property shows that both fractions should be equal to $$\frac{3x-b-3a+4b}{3x-5b-3a+8b}=\frac{3x-3a+3b}{3x-3a+3b}=1.$$ However, the initial fractions given may not be equal to $1$. Explain what is going on.

Let $\mathbb{Q}^+$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}^+\to \mathbb{Q}^+$ satisfying$$f(x^2f(y)^2)=f(x^2)f(y)$$for all $x,y\in\mathbb{Q}^+$

Let $k\ge 2$ be an integer and $a_1,a_2,\cdots,a_k$ be $k$ non-zero reals. Prove that there are finitely many pairs of pairwise distinct positive integers $(n_1,n_2,\cdots,n_k)$ such that $$a_1\cdot n_1!+a_2\cdot n_2!+\cdots+a_k\cdot n_k!=0.$$