Let $m_1< m_2 < \ldots m_{k-1}< m_k$ be $k$ distinct positive integers such that their reciprocals are in arithmetic progression. 1.Show that $k< m_1 + 2$. 2. Give an example of such a sequence of length $k$ for any positive integer $k$.

Find the maximum constant $C$ such that, whenever $\{a_n \}_{n=1}^{\infty}$ is a sequence of positive real numbers satisfying $a_{n+1}-a_n=a_n(a_n+1)(a_n+2)$, we have $$\frac{a_{2023}-a_{2020}}{a_{2022}-a_{2021}}>C.$$

[b]p1.[/b] One chilly morning, $10$ penguins ate a total of $50$ fish. No fish was shared by two or more penguins. Assuming that each penguin ate at least one fish, prove that at least two penguins ate the same number of fish. [b]p2.[/b] A triangle of area $1$ has sides of lengths $a > b > c$. Prove that $b > 2^{1/2}$. [b]p3.[/b] Imagine ducks as points in a plane. Three ducks are said to be in a row if a straight line passes through all three ducks. Three ducks, Huey, Dewey, and Louie, each waddle along a different straight line in the plane, each at his own constant speed. Although their paths may cross, the ducks never bump into each other. Prove: If at three separate times the ducks are in a row, then they are always in a row. [b]p4.[/b] Two computers and a number of humans participated in a large round-robin chess tournament (i.e., every participant played every other participant exactly once). In every game, the winner of the game received one point, the loser zero. If a game ended in a draw, each player received half a point. At the end of the tournament, the sum of the two computers' scores was $38$ points, and all of the human participants finished with the same total score. Describe (with proof) ALL POSSIBLE numbers of humans that could have participated in such a tournament. [b]p5.[/b] One thousand cows labeled $000$, $001$,$...$, $998$, $999$ are requested to enter $100$ empty barns labeled $00$, $01$,$...$,$98$, $99$. One hundred Dalmatians - one at the door of each barn - enforce the following rule: In order for a cow to enter a barn, the label of the barn must be obtainable from the label of the cow by deleting one of the digits. For example, the cow labeled $357$ would be admitted into any of the barns labeled $35$, $37$ or $57$, but would not admitted into any other barns. a) Demonstrate that there is a way for all $1000$ cows to enter the barns so that at least $50$ of the barns remain empty. b) Prove that no matter how they distribute themselves, after all $1000$ cows enter the barns, at most $50$ of the barns will remain empty. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] What is the maximum amount of a $12\%$ acid solution that can be obtained from $1$ liter of $5\%$, $10\%$ and $15\%$ solutions? [b]p2.[/b] Which number is greater: $199,719,971,997^2$ or $199,719,971,996 * 19,9719,971,998$ ? [b]p3.[/b] Is there a convex $1998$-gon whose angles are all integer degrees? [b]p4.[/b] Is there a ten-digit number divisible by $11$ that uses all the digits from$ 0$ to $9$? [b]p5.[/b] There are $20$ numbers written in a circle, each of which is equal to the sum of its two neighbors. Prove that the sum of all numbers is $0$. [b]p6.[/b] Is there a convex polygon that has neither an axis of symmetry nor a center of symmetry, but which transforms into itself when rotated around some point through some angle less than $180$ degrees? [b]p7.[/b] In a convex heptagon, draw as many diagonals as possible so that no three of them are sides of the same triangle, the vertices of which are at the vertices of the original heptagon. [b]p8.[/b] Give an example of a natural number that is divisible by $30$ and has exactly $105$ different natural factors, including $1$ and the number itself. [b]p9.[/b] In the writing of the antipodes, numbers are also written with the digits $0, ..., 9$, but each of the numbers has different meanings for them and for us. It turned out that the equalities are also true for the antipodes $5 * 8 + 7 + 1 = 48$ $2 * 2 * 6 = 24$ $5* 6 = 30$ a) How will the equality $2^3 = ...$ in the writing of the antipodes be continued? b) What does the number$ 9$ mean among the Antipodes? Clarifications: a) It asks to convert $2^3$ in antipodes language, and write with what number it is equal and find a valid equality in both numerical systems. b) What does the digit $9$ mean among the antipodes, i.e. with which digit is it equal in our number system? [b]p10.[/b] Is there a convex quadrilateral that can be cut along a straight line into two parts of the same size and shape, but neither the diagonal nor the straight line passing through the midpoints of opposite sides divides it into two equal parts? PS.1. There was typo in problem $9$, it asks for $2^3$ and not $23$. PS.2. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

$a_1,a_2,\cdots,a_n$ are positive real numbers, $a_1+a_2+\cdots,a_n=1$. Prove that $$\sum_{m=1}^n\frac{a_m}{\prod\limits_{k=1}^m(1+a_k)}\leq1-\frac{1}{2^n}.$$

Let $ c\in\mathbb C$ and $ A_c \equal{} \{p\in \mathbb C[z]|p(z^2 \plus{} c) \equal{} p(z)^2 \plus{} c\}$. a) Prove that for each $ c\in C$, $ A_c$ is infinite. b) Prove that if $ p\in A_1$, and $ p(z_0) \equal{} 0$, then $ |z_0| < 1.7$. c) Prove that each element of $ A_c$ is odd or even. Let $ f_c \equal{} z^2 \plus{} c\in \mathbb C[z]$. We see easily that $ B_c: \equal{} \{z,f_c(z),f_c(f_c(z)),\dots\}$ is a subset of $ A_c$. Prove that in the following cases $ A_c \equal{} B_c$. d) $ |c| > 2$. e) $ c\in \mathbb Q\backslash\mathbb Z$. f) $ c$ is a non-algebraic number g) $ c$ is a real number and $ c\not\in [ \minus{} 2,\frac14]$.

Find all polynomials $P$ in two variables with real coefficients satisfying the identity $$P(x,y)P(z,t)=P(xz-yt,xt+yz).$$

There is a wolf in the centre of a square field, and four dogs in the corners. The wolf can easily kill one dog, but two dogs can kill the wolf. The wolf can run all over the field, and the dogs -- along the fence (border) only. Prove that if the dog's speed is $1.5$ times more than the wolf's, than the dogs can prevent the wolf escaping.

For what natural numbers $n$ for any numbers $a, b , c$, which are values of the angles of an acute triangle, the following inequality is true: $$\sin na + \sin nb + \sin nc < 0?$$

Positive numbers $x, y, z$ satisfy $x^2+y^2+z^2+xy+yz+zy \le 1$. Prove that $\big( \frac{1}{x}-1\big) \big( \frac{1}{y}-1\big)\big( \frac{1}{z}-1\big) \ge 9 \sqrt6 -19$.

Consider $5$ distinct positive integers. Can their mean be a)Exactly $3$ times larger than their largest common divisor? b)Exactly $2$ times larger than their largest common divisor?

Find some antiderivative of the function $y = 1/x^3$, the graph of which has exactly three common points with the graph of the function $y = |x|$.

What is the largest integer not exceeding $8x^3 +6x - 1$, where $x =\frac12 \left(\sqrt[3]{2+\sqrt5} + \sqrt[3]{2-\sqrt5}\right)$ ? (A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.

How many $x$ are there such that $x,[x],\{x\}$ are in harmonic progression (i.e, the reciprocals are in arithmetic progression)? (Here $[x]$ is the largest integer less than equal to $x$ and $\{x\}=x-[ x]$ ) [list=1] [*] 0 [*] 1 [*] 2 [*] 3 [/list]

Find all the positive reals $x,y,z$ satisfying the following equations: $$y=\frac6{(2x-1)^2}$$ $$z=\frac6{(2y-1)^2}$$ $$x=\frac6{(2z-1)^2}$$

Find all functions $f \colon \mathbb{R_{+}}\to \mathbb{R_{+}}$ satisfying : \[f ( f (x)-x) = 2x\] for all $x > 0$.

Show that if \[ \frac{\cos x}{\cos y}+\frac{\sin x}{\sin y}=-1 \] then \[ \frac{\cos^3 y}{\cos x}+\frac{\sin^3 y}{\sin x}=1 \]

Compute the smallest real $t$ such that there exist constants $a$, $b$ for which the roots of $x^3-ax^2+bx - \frac{ab}{t}$ are the side lengths of a right triangle

Find all polynomials $P(x)$ with integer coefficients such that for all real numbers $s$ and $t$, if $P(s)$ and $P(t)$ are both integers, then $P(st)$ is also an integer.

Let $L$ be the number of times the letter $L$ appeared on the Speed Round, $M$ be the number of times the letter $M$ appeared on the Speed Round, and $T$ be the number of times the letter $T$ appeared on the Speed Round. Find the value of $LMT$.

Different positive $a, b, c$ are such that $a^{239} = ac- 1$ and $b^{239} = bc- 1$.Prove that $238^2 (ab)^{239} <1$.

Positive integers $a, b, c$ satisfy $\frac1a +\frac1b +\frac1c<1$. Prove that $\frac1a +\frac1b +\frac1c\le \frac{41}{42}$. Also prove that equality in fact holds in the second inequality.

Initially the equation $$\star \frac{1}{x-1} \star \frac{1}{x-2} \star \frac{1}{x-4} ... \star \frac{1}{x-2^{2023}}=0$$ is written on the board. In each turn Aslı and Zehra deletes one of the stars in the equation and writes $+$ or $-$ instead. The first move is performed by Aslı and continues in order. What is the maximum number of real solutions Aslı can guarantee after all the stars have been replaced by signs?

[u]Set 1[/u] [b]p1.[/b] Farmer John has $4000$ gallons of milk in a bucket. On the first day, he withdraws $10\%$ of the milk in the bucket for his cows. On each following day, he withdraws a percentage of the remaining milk that is $10\%$ more than the percentage he withdrew on the previous day. For example, he withdraws $20\%$ of the remaining milk on the second day. How much milk, in gallons, is left after the tenth day? [b]p2.[/b] Will multiplies the first four positive composite numbers to get an answer of $w$. Jeremy multiplies the first four positive prime numbers to get an answer of $j$. What is the positive difference between $w$ and $j$? [b]p3.[/b] In Nathan’s math class of $60$ students, $75\%$ of the students like dogs and $60\%$ of the students like cats. What is the positive difference between the maximum possible and minimum possible number of students who like both dogs and cats? [u]Set 2[/u] [b]p4.[/b] For how many integers $x$ is $x^4 - 1$ prime? [b]p5.[/b] Right triangle $\vartriangle ABC$ satisfies $\angle BAC = 90^o$. Let $D$ be the foot of the altitude from $A$ to $BC$. If $AD = 60$ and $AB = 65$, find the area of $\vartriangle ABC$. [b]p6.[/b] Define $n! = n \times (n - 1) \times ... \times 1$. Given that $3! + 4! + 5! = a^2 + b^2 + c^2$ for distinct positive integers $a, b, c$, find $a + b + c$. [u]Set 3[/u] [b]p7.[/b] Max nails a unit square to the plane. Let M be the number of ways to place a regular hexagon (of any size) in the same plane such that the square and hexagon share at least $2$ vertices. Vincent, on the other hand, nails a regular unit hexagon to the plane. Let $V$ be the number of ways to place a square (of any size) in the same plane such that the square and hexagon share at least $2$ vertices. Find the nonnegative difference between $M$ and $V$ . [b]p8.[/b] Let a be the answer to this question, and suppose $a > 0$. Find $\sqrt{a +\sqrt{a +\sqrt{a +...}}}$ . [b]p9.[/b] How many ordered pairs of integers $(x, y)$ are there such that $x^2 - y^2 = 2019$? [u]Set 4[/u] [b]p10.[/b] Compute $\frac{p^3 + q^3 + r^3 - 3pqr}{p + q + r}$ where $p = 17$, $q = 7$, and $r = 8$. [b]p11.[/b] The unit squares of a $3 \times 3$ grid are colored black and white. Call a coloring good if in each of the four $2 \times 2$ squares in the $3 \times 3$ grid, there is either exactly one black square or exactly one white square. How many good colorings are there? Consider rotations and reflections of the same pattern distinct colorings. [b]p12.[/b] Define a $k$-[i]respecting [/i]string as a sequence of $k$ consecutive positive integers $a_1$, $a_2$, $...$ , $a_k$ such that $a_i$ is divisible by $i$ for each $1 \le i \le k$. For example, $7$, $8$, $9$ is a $3$-respecting string because $7$ is divisible by $1$, $8$ is divisible by $2$, and $9$ is divisible by $3$. Let $S_7$ be the set of the first terms of all $7$-respecting strings. Find the sum of the three smallest elements in $S_7$. [u]Set 5[/u] [b]p13.[/b] A triangle and a quadrilateral are situated in the plane such that they have a finite number of intersection points $I$. Find the sum of all possible values of $I$. [b]p14.[/b] Mr. DoBa continuously chooses a positive integer at random such that he picks the positive integer $N$ with probability $2^{-N}$ , and he wins when he picks a multiple of 10. What is the expected number of times Mr. DoBa will pick a number in this game until he wins? [b]p15.[/b] If $a, b, c, d$ are all positive integers less than $5$, not necessarily distinct, find the number of ordered quadruples $(a, b, c, d)$ such that $a^b - c^d$ is divisible by $5$. PS. You had better use hide for answers. Last 4 sets have been posted [url=https://artofproblemsolving.com/community/c4h2777362p24370554]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that if $a,b,c>0,$ then \[ \frac{ab}{3a+4b+5c}+\frac{bc}{3b+4c+5a}+\frac{ca}{3c+4a+5b}\le \frac{a+b+c}{12}. \] [i] Nikolai Nikolov[/i]