Find the area of the domain expressed by the following system inequalities. \[x\geq 0,\ y\geq 0,\ x^{\frac{1}{p}}+y^{\frac{1}{p}} \leq 1\ (p=1,2,\cdots)\]

Find all triples $(a, b, c)$ of real numbers such that $ab + bc + ca = 1$ and $$a^2b + c = b^2c + a = c^2a + b.$$

For natural number $n$ and real numbers $\alpha$ and $x$ satisfy the inequalities $\alpha^{n+1}\le x\le1$ and $0<\alpha<1$. Prove that $$\prod_{k=1}^n\left|\frac{x-\alpha^k}{x+\alpha^k}\right|\le\prod_{k=1}^n\left|\frac{1-\alpha^k}{1+\alpha^k}\right|.$$ [i]Borislav Boyanov[/i]

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying \[f(x+yf(x))=f(xf(y))-x+f(y+f(x))\]

Let $x_1,x_2,x_3,\ldots ,x_{100}$ be one hundred real numbers greater than or equal to $0$ and less than or equal to $1$. Find the maximum possible value of the sum$$S=x_1(1-x_2)+x_2(1-x_3)+x_3(1-x_4)+\cdots +x_{99}(1-x_{100})+x_ {100}(1-x_1).$$

[hide=Rules]Time Limit: 25 minutes Maximum Possible Score: 81 The following is a mathematical Sudoku puzzle which is also a crossword. Your job is to fill in as many blanks as you possibly can, including all shaded squares. You do not earn extra points for showing your work; the only points you get are for correctly filled-in squares. You get one point for each correctly filled-in square. You should read through the following rules carefully before starting. $\bullet$ Your time limit for this round is $25$ minutes, in addition to the five minutes you get for reading the rules. So make use of your time wisely. The round is based more on speed than on perfect reasoning, so use your intuition well, and be fast. $\bullet$ This is a Sudoku puzzle; all the squares should be filled in with the digits $1$ through $9$ so that every row and column contains each digit exactly once. In addition, each of the nine $3\times 3$ boxes that compose the grid also contains each digit exactly once. Furthermore, this is a super-Sudoku puzzle; in addition to satisfying all these conditions, the four $3\times 3$ boxes with red outlines also contain each of $1,..., 9$ exactly once. This last property is important to keep in mind – it may help you solve the puzzle faster. $\bullet$ Just to restate the idea, you can use the digits $1$ through $9$, but not $0$. You may not use any other symbol, such as $\pi$ or $e$ or $\epsilon$. Each square gets exactly one digit. $\bullet$ The grid is also a crossword puzzle; the usual rules apply. The shaded grey squares are the “black” squares of an ordinary crossword puzzle. The white squares as well as the shaded yellow ones count as the “white” crossword squares. All squares, white or shaded, count as ordinary Sudoku squares. $\bullet$ If you obtain the unique solution to the crossword puzzle, then this solution extends to a unique solution to the Sudoku puzzle. $\bullet$ You may use a graphing calculator to help you solve the clues. The following hints and tips may prove useful while solving the puzzle. $\bullet$ Use the super-Sudoku structure described in the first rule; use all the symmetries you have. Remember that we are not looking for proofs or methods, only for correctly filled-in squares. $\bullet$ If you find yourself stuck on a specific clue, it is nothing to worry about. You can obtain the solution to that clue later on by solving other clues and figuring out certain digits of your desired solution. Just move on to the rest of the puzzle. $\bullet$ As you progress through the puzzle, keep filling in all squares you have found on your solution sheet, including the shaded ones. Remember that for scoring, the shaded grey squares count the same as the white ones. Good luck! [/hide] [asy] // place label "s" in row i, column j void labelsq(int i, int j, string s) { label("$"+s+"$",(j-0.5,7.5-i),fontsize(14)); } // for example, use the command // labelsq(1,7,"2"); // to put the digit 2 in the top right box // **** rest of code **** size(250); defaultpen(linewidth(1)); pair[] labels = {(1,1),(1,4),(1,6),(1,7),(1,9),(2,1),(2,6),(3,4),(4,1),(4,8),(5,1),(6,3),(6,5),(6,6),(7,1),(7,2),(7,7),(7,9),(8,1),(8,4),(9,1),(9,6)}; pair[] blacksq = {(1,5),(2,5),(3,2),(3,3),(3,8),(5,5),(5,6),(5,7),(5,9),(6,2),(6,7),(6,9),(8,3),(9,5),(9,8)}; path peachsq = shift(1,1)*scale(3)*unitsquare; pen peach = rgb(0.98,0.92,0.71); pen darkred = red + linewidth(2); fill(peachsq,peach); fill(shift(4,0)*peachsq,peach); fill(shift(4,4)*peachsq,peach); fill(shift(0,4)*peachsq,peach); for(int i = 0; i < blacksq.length; ++i) fill(shift(blacksq[i].y-1, 9-blacksq[i].x)*unitsquare, gray(0.6)); for(int i = 0; i < 10; ++i) { pen sudokuline = linewidth(1); if(i == 3 || i == 6) sudokuline = linewidth(2); draw((0,i)--(9,i),sudokuline); draw((i,0)--(i,9),sudokuline); } draw(peachsq,darkred); draw(shift(4,0)*peachsq,darkred); draw(shift(4,4)*peachsq,darkred); draw(shift(0,4)*peachsq,darkred); for(int i = 0; i < labels.length; ++i) label(string(i+1), (labels[i].y-1, 10-labels[i].x), SE, fontsize(10)); // **** draw letters **** draw(shift(.5,.5)*((0,6)--(0,8)--(2,8)--(2,7)--(0,7)^^(3,8)--(3,6)--(5,6)--(5,8)^^(6,6)--(6,8)--(7,8)--(7,7)--(7,8)--(8,8)--(8,6)^^(0,3)--(0,5)--(2,5)--(2,3)--(2,4)--(0,4)^^(5,3)--(3,3)--(3,5)--(5,5)),linewidth(1)+rgb(0.94,0.74,0.58)); // **** end rest of code ****[/asy] [b][u][i]Across[/i][/u][/b] [b]1 Across.[/b] The following is a normal addition where each letter represents a (distinct) digit: $$GOT + TO + GO + TO = TOP$$This certainly does not have a unique solution. However, you discover suddenly that $G = 2$ and $P \notin \{4, 7\}$. Then what is the numeric value of the expression $GOT \times TO$? [b]3 Across.[/b] A strobogrammatic number which reads the same upside down, e.g. $619$. On the other hand, a triangular number is a number of the form $n(n + 1)/2$ for some $n \in N$, e.g. $15$ (therefore, the $i^{th}$ triangular number $T_i$ is the sum of $1$ through $i$). Let $a$ be the third strobogrammatic prime number. Let $b$ be the smaller number of the smallest pair of triangular numbers whose sum and difference are also triangular numbers. What is the value of $ab$? [b]6 Across.[/b] A positive integer $m$ is said to be palindromic in base $\ell$ if, when written in base $\ell$ , its digits are the same front-to-back and back-to-front. For $j, k \in N$, let $\mu (j, k)$ be the smallest base-$10$ integer that is palindromic in base $j$ as well as in base$ k$, and let $\nu (j, k) := (j + k) \cdot \mu (j, k)$. Find the value of $\nu (5, 9)$. [b]7 Across.[/b] Suppose you have the unique solution to this Sudoku puzzle. In that solution, let $X$ denote the sum of all digits in the shaded grey squares. Similarly, let $Y$ denote the sum of all numbers in the shaded yellow squares on the upper left block (i.e. the $3 \times 3$ box outlined red towards the top left). Concatenate $X$ with $Y$ in that order, and write that down. [b]8 Across.[/b] For any $n \in N$ such that $1 < n < 10$, define the sequence $X_{n,1}$,$X_{n,2}$,$ ...$ by $X_{n,1} = n$, and for $r \ge 2$, X_{n,r} is smallest number $k \in N$ larger than X_{n,r-1} such that $k$ and the sum of digits of $k$ are both powers of $n$. For instance, $X_{3,1 = 3}$, $X_{3,2} = 9$, $X_{3,3} = 27$, and so on. Concatenate $X_{2,2}$ with $X_{2,4}$, and write down the answer. [b]9 Across.[/b] Find positive integers $x, y,z$ satisfying the following properties: $y$ is obtained by subtracting $93$ from $x$, and $z$ is obtained by subtracting $183$ from $y$, furthermore, $x, y$ and $z$ in their base-$10$ representations contain precisely all the digits from $1$ through $9$ once (i.e. concatenating $x, y$ and $z$ yields a valid $9$-digit Sudoku answer). Obviously, write down the concatenation of $x, y$ and $z$ in that order. [b]11 Across.[/b] Find the largest pair of two-digit consecutive prime numbers $a$ and $b$ (with $a < b$) such that the sum of the digits of a plus the sum of the digits of b is also a prime number. Write the concatenation of $a$ and $b$. [b]12 Across.[/b] Suppose you have a strip of $2n + 1$ squares, with n frogs on the $n$ squares on the left, and $n$ toads on the $n$ squares on the right. A move consists either of a toad or a frog sliding to an adjacent square if it is vacant, or of a toad or a frog jumping one square over another one and landing on the next square if it is vacant. For instance, the starting position [img]https://cdn.artofproblemsolving.com/attachments/a/a/6c97f15304449284dc282ff86014f526322e4a.png[/img] has the position [img]https://cdn.artofproblemsolving.com/attachments/e/6/e2c9520731bd94dc0aa37f540c2b9d1bce6432.png[/img] or the position [img]https://cdn.artofproblemsolving.com/attachments/3/f/06868eca80d649c4f80425dc9dc5c596cb2a4e.png[/img] as results of valid first moves. What is the minimum number of moves needed to swap the toads with the frogs if $n = 5$? How about $n = 6$? Concatenate your answers. [b]15 Across.[/b] Let $w$ be the largest number such that $w$, $2w$ and $3w$ together contain every digit from $1$ through $9$ exactly once. Let $x$ be the smallest integer with the property that its first $5$ multiples contain the digit $9$. A Leyland number is an integer of the form $m^n + n^m$ for integers $m, n > 1$. Let $y$ be the fourth Leyland number. A Pillai prime is a prime number $p$ for which there is an integer $n > 0$ such that $n! \equiv - 1 (mod \,\, p)$, but $p \not\equiv 1 (mod \,\, n)$. Let $z$ be the fourth Pillai prime. Concatenate $w$, $x, y$ and $z$ in that order to obtain a permutation of $1,..., 9$. Write down this permutation. [b]19 Across.[/b] A hoax number $k \in N$ is one for which the sum of its digits (in base $10$) equals the sum of the digits of its distinct prime factors (in base $10$). For instance, the distinct prime factors of $22$ are $2$ and $11$, and we have $2+2 = 2+(1+1)$. In fact, $22$ is the first hoax number. What is the second? [b]20 Across.[/b] Let $a, b$ and $c$ be distinct $2$-digit numbers satisfying the following properties: – $a$ is the largest integer expressible as $a = x^y = y^x$, for distinct integers $x$ and $y$. – $b$ is the smallest integer which has three partitions into three parts, which all give the same product (which turns out to be $1200$) when multiplied. – $c$ is the largest number that is the sum of the digits of its cube. Concatenate $a, b$ and $c$, and write down the resulting 6-digit prime number. [b]21 Across.[/b] Suppose $N = \underline{a}\, \underline{b} \, \underline{c} \, \underline{d}$ is a $4$-digit number with digits $a, b, c$ and $d$, such that $N = a \cdot b \cdot c \cdot d^7$. Find $N$. [b]22 Across.[/b] What is the smallest number expressible as the sum of $2, 3, 4$, or $5$ distinct primes? [b][u][i]Down [/i][/u][/b] [b]1 Down.[/b] For some $a, b, c \in N$, let the polynomial $$p(x) = x^5 - 252x^4 + ax^3 - bx^2 + cx - 62604360$$ have five distinct roots that are positive integers. Four of these are 2-digit numbers, while the last one is single-digit. Concatenate all five roots in decreasing order, and write down the result. [b]2 Down.[/b] Gene, Ashwath and Cosmin together have $2511$ math books. Gene now buys as many math books as he already has, and Cosmin sells off half his math books. This leaves them with $2919$ books in total. After this, Ashwath suddenly sells off all his books to buy a private jet, leaving Gene and Cosmin with a total of $2184$ books. How many books did Gene, Ashwath and Cosmin have to begin with? Concatenate the three answers (in the order Gene, Ashwath, Cosmin) and write down the result. [b]3 Down.[/b] A regular octahedron is a convex polyhedron composed of eight congruent faces, each of which is an equilateral triangle; four of them meet at each vertex. For instance, the following diagram depicts a regular octahedron: [img]https://cdn.artofproblemsolving.com/attachments/c/1/6a92f12d5e9f56b0699531ae8369a0ab8ab813.png[/img] Let $T$ be a regular octahedron of edge length $28$. What is the total surface area of $T$ , rounded to the nearest integer? [b]4 Down.[/b] Evaluate the value of the expression $$\sum^{T_{25}}_{k=T_{24}+1}k, $$ where $T_i$ denotes the $i^{th}$ triangular number (the sum of the integers from $1$ through $i$). [b]5 Down.[/b] Suppose $r$ and $s$ are consecutive multiples of$ 9$ satisfying the following properties: – $r$ is the smallest positive integer that can be written as the sum of $3$ positive squares in $3$ different ways. – $s$ is the smallest $2$-digit number that is a Woodall number as well as a base-$10$ Harshad number. A Woodall number is any number of the form $n \cdot 2^n - 1$ for some $n \in N$. A base-$10$ Harshad number is divisible by the sum of its digits in base $10$. Concatenate $r$ and $s$ and write down the result. [b]10 Down.[/b] For any $k \in N$, let $\phi_p(k)$ denote the sum of the distinct prime factors of $k$. Suppose $N$ is the largest integer less than $50000$ satisfying $\phi_p(N) =\phi_p(N + 1)$, where the common value turns out to be a meager $55$. What is$ N$? [b]13 Down.[/b] The $n^{th}$ $s$-gonal number $P(s, n)$ is defined as $$P(s, n) = (s - 3)T_{n-1} + T_n$$ where $T_i$ is the $i^{th}$ triangular number (recall that the $i^{th}$ triangular number is the sum of the numbers $1$ through $i$). Find the least $N$ such that $N$ is both a $34$-gonal number, and a $163$-gonal number. [b]14 Down.[/b] A biprime is a positive integer that is the product of precisely two (not necessarily distinct) primes. A cluster of biprimes is an ordered triple $(m,m + 1,m + 2)$ of consecutive integers that are biprimes. There are precisely three clusters of biprimes below 100. Denote these by, say, $$\{(p, p + 1, p + 2), (q, q + 1,q + 2), (r, r + 1, r + 2)\}$$ and add the condition that $p + 2 < q < r - 2$ to fix the three clusters. Interestingly, $p + 1$ and $q$ are both multiples of $17$. Concatenate $q$ with $p + 1$ in that order, and write down the result. [b]16 Down.[/b] Find the least positive integer $m$ (written in base $10$ as $m = \underline{a} \, \underline{b} \, \underline{c} $, with digits $a, b,c$), such that $m = (b + c)^a$. [b]17 Down.[/b] Let $X$ be a set containing $32$ elements, and let $Y\subseteq X$ be a subset containing $29$ elements. How many $2$-element subsets of $X$ are there which have nonempty intersection with $Y$? [b]18 Down.[/b] Find a positive integer $K < 196$, which is a strange twin of the number $196$, in the sense that $K^2$ shares the same digits as $196^2$, and $K^3$ shares the same digits as $196^3$. PS. You should use hide for answers.

Let $a\in(0,1)$ ,$f(x)=ax^3+(1-4a)x^2+(5a-1)x-5a+3 $ , $g(x)=(1-a)x^3-x^2+(2-a)x-3a-1 $. Prove that:For any real number $x$ ,at least one of $|f(x)|$ and $|g(x)|$ not less than $a+1$.

If $a$ and $b$ are positive real numbers with sum $3$, and $x, y, z$ positive real numbers with product $1$, prove that : $(ax+b)(ay+b)(az+b)\geq 27$

The triangle table is constructed according to the rule: You put the natural number $a>1$ in the upper row, and then you write under the number $k$ from the left side $k^2$, and from the right side -- $(k+1)$. For example, if $a = 2$, you get the table on the picture. Prove that all the numbers on each particular line are different. 2 / \ / \ 4 3 / \ / \ 16 5 9 4 / \ / \ /\ / \

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Today is Saturday, April $25$, $2020$. What is the value of $6 + 4 + 25 + 2020$? [b]p2.[/b] The figure below consists of a $2$ by $3$ grid of squares. How many squares of any size are in the grid? $\begin{tabular}{|l|l|l|} \hline & & \\ \hline & & \\ \hline \end{tabular}$ [b]p3.[/b] James is playing a game. He first rolls a six-sided dice which contains a different number on each side, then randomly picks one of twelve dierent colors, and finally ips a quarter. How many different possible combinations of a number, a color and a flip are there in this game? [b]p4.[/b] What is the sum of the number of diagonals and sides in a regular hexagon? [b]p5.[/b] Mickey Mouse and Minnie Mouse are best friends but they often fight. Each of their fights take up exactly one hour, and they always fight on prime days. For example, they fight on January $2$nd, $3$rd, but not the $4$th. Knowing this, how many total times do Mickey and Minnie fight in the months of April, May and June? [b]p6.[/b] Apple always loved eating watermelons. Normal watermelons have around $13$ black seeds and $25$ brown seeds, whereas strange watermelons had $45$ black seeds and $2$ brown seeds. If Apple bought $14$ normal watermelons and $7$ strange watermelons, then let $a$ be the total number of black seeds and $b$ be the total number of brown seeds. What is $a - b$? [b]p7.[/b] Jerry and Justin both roll a die once. The probability that Jerry's roll is greater than Justin's can be expressed as a fraction in the form $\frac{m}{n}$ in simplified terms. What is $m + n$? [b]p8.[/b] Taylor wants to color the sides of an octagon. What is the minimum number of colors Taylor will need so that no adjacent sides of the octagon will be filled in with the same color? [b]p9.[/b] The point $\frac23$ of the way from ($-6, 8$) to ($-3, 5$) can be expressed as an ordered pair $(a, b)$. What is $|a - b|$? [b]p10.[/b] Mary Price Maddox laughs $7$ times per class. If she teaches $4$ classes a day for the $5$ weekdays every week but doesn't laugh on Wednesdays, then how many times does she laugh after $5$ weeks of teaching? [b]p11.[/b] Let $ABCD$ be a unit square. If $E$ is the midpoint of $AB$ and $F$ lies inside $ABCD$ such that $CFD$ is an equilateral triangle, the positive difference between the area of $CED$ and $CFD$ can be expressed in the form $\frac{a-\sqrt{b}}{c}$ , where $a$, $b$, $c$ are in lowest simplified terms. What is $a + b + c$? [b]p12.[/b] Eddie has musician's syndrome. Whenever a song is a $C$, $A$, or $F$ minor, he begins to cry and his body becomes very stiff. On the other hand, if the song is in $G$ minor, $A$ at major, or $E$ at major, his eyes open wide and he feels like the happiest human being ever alive. There are a total of $24$ keys. How many different possibilities are there in which he cries while playing one song with two distinct keys? [b]p13.[/b] What positive integer must be added to both the numerator and denominator of $\frac{12}{40}$ to make a fraction that is equivalent to $\frac{4}{11}$ ? [b]p14.[/b] The number $0$ is written on the board. Each minute, Gene the genie either multiplies the number on the board by $3$ or $9$, each with equal probability, and then adds either $1$,$2$, or $3$, each with equal probability. Find the expected value of the number after $3$ minutes. [b]p15.[/b] $x$ satisfies $\dfrac{1}{x+ \dfrac{1}{1+\frac{1}{2}}}=\dfrac{1}{2+ \dfrac{1}{1- \dfrac{1}{2+\frac{1}{2}}}}$ Find $x$. [b]p16.[/b] How many different points in a coordinate plane can a bug end up on if the bug starts at the origin and moves one unit to the right, left, up or down every minute for $8$ minutes? [b]p17.[/b] The triplets Addie, Allie, and Annie, are racing against the triplets Bobby, Billy, and Bonnie in a relay race on a track that is $100$ feet long. The first person of each team must run around the entire track twice and tag the second person for the second person to start running. Then, the second person must run once around the entire track and tag the third person, and finally, the third person would only have to run around half the track. Addie and Bob run first, Allie and Billy second, Annie and Bonnie third. Addie, Allie, and Annie run at $50$ feet per minute (ft/m), $25$ ft/m, and $20$ ft/m, respectively. If Bob, Billy, and Bonnie run half as fast as Addie, Allie, and Annie, respectively, then how many minutes will it take Bob, Billy, and Bonnie to finish the race. Assume that everyone runs at a constant rate. [b]p18.[/b] James likes to play with Jane and Jason. If the probability that Jason and Jane play together is $\frac13$, while the probability that James and Jason is $\frac14$ and the probability that James and Jane play together is $\frac15$, then the probability that they all play together is $\frac{\sqrt{p}}{q}$ for positive integers $p$, $q$ where $p$ is not divisible by the square of any prime. Find $p + q$. [b]p19.[/b] Call an integer a near-prime if it is one more than a prime number. Find the sum of all near-primes less than$ 1000$ that are perfect powers. (Note: a perfect power is an integer of the form $n^k$ where $n, k \ge 2$ are integers.) [b]p20.[/b] What is the integer solution to $\sqrt{\frac{2x-6}{x-11}} = \frac{3x-7}{x+6}$ ? [b]p21.[/b] Consider rectangle $ABCD$ with $AB = 12$ and $BC = 4$ with $F$,$G$ trisecting $DC$ so that $F$ is closer to $D$. Then $E$ is on $AB$. We call the intersection of $EF$ and $DB$ $X$, and the intersection of $EG$ and $DB$ is $Y$. If the area of $\vartriangle XY E$ is \frac{8}{15} , then what is the length of $EB$? [b]p22.[/b] The sum $$\sum^{\infty}_{n=2} \frac{1}{4n^2-1}$$ can be expressed as a common fraction $\frac{a}{b}$ in lowest terms. Find $a + b$. [b]p23.[/b] In square $ABCD$, $M$, $N$, $O$, $P$ are points on sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$ and $\overline{DA}$, respectively. If $AB = 4$, $AM = BM$ and $DP = 3AP$, the least possible value of $MN + NO + OP$ can be expressed as $\sqrt{x}$ forsome integer x. Find x: [b]p24.[/b] Grand-Ovich the ant is at a vertex of a regular hexagon and he moves to one of the adjacent vertices every minute with equal probability. Let the probability that after $8$ minutes he will have returned to the starting vertex at least once be the common fraction $\frac{a}{b}$ in lowest terms. What is $a + b$? [b]p25.[/b] Find the last two non-zero digits at the end of $2020!$ written as a two digit number. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $P$ be a fourth degree polynomial, with derivative $P^\prime$, such that $P(1)=P(3)=P(5)=P^\prime (7)=0$. Find the real number $x\neq 1,3,5$ such that $P(x)=0$.

Given that $x$ and $y$ are positive real numbers such that $\frac{5}{x}=\frac{y}{13}=\frac{x}{y}$, find the value of $x^3 + y^3$. Proposed by Ephram Chun

The value of $x^2-6x+13$ can never be less than: $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 4.5 \qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 13 $

Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

Show that there exists a polynomial $P(x)$ with integer coefficients, such that the equation $P(x) = 0$ has no integer solutions, but for each positive integer $n$ there is an $x \in \Bbb{Z}$ such that $n \mid P(x).$

What is the value of $ { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=odd} $ $ - { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=even} $

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying relation : $$f(xf(y)-f(x))=2f(x)+xy$$ $\forall x,y \in \mathbb{R}$

Find all rational numbers $k$ such that $0 \le k \le \frac{1}{2}$ and $\cos k \pi$ is rational.

Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$. [i]Proposed by Dominik Burek and Tomasz Ciesla, Poland[/i]

Let $x_1, x_2,... x_n$ be positive numbers such that $S = x_1+x_2+...+x_n =\frac{1}{x_1}+...+\frac{1}{x_n}$ Prove that $$\sum_{i=1}^{n} \frac{1}{n - 1 + x_i} \ge \sum_{i=1}^{n} \frac{1}{1+S - x_i}$$

Calculate the following indefinite integrals. [1] $\int \frac{6x+1}{\sqrt{3x^2+x+4}}dx$ [2] $\int \frac{e^x}{e^x+e^{a-x}}dx$ [3] $\int \frac{(\sqrt{x}+1)^3}{\sqrt{x}}dx$ [4] $\int x\ln (x^2-1)dx$ [5] $\int \frac{2(x+2)}{x^2+4x+1}dx$

Let $f:\mathbb R^+ \to \mathbb R^+$ be a function such that: \[ x,y > 0 \qquad f(x+f(y)) = yf(xy+1). \] a) Show that $(y-1)*(f(y)-1) \le 0$ for $y>0$. b) Find all such functions that require the given condition.

Let $f: N^+ \to N^+$ be a function that satisfies that $$kf(n) \le f (kn) \le kf(n)+ k- 1, \,\, \forall k,n \in N^+$$ Prove that $$f(a) + f(b) \le f (a + b) \le f(a) + f(b) + 1, \,\, \forall a, b \in N^+$$

We call a number $n$ [i]interesting [/i]if for each permutation $\sigma$ of $1,2,\ldots,n$ there exist polynomials $P_1,P_2,\ldots ,P_n$ and $\epsilon > 0$ such that: $i)$ $P_1(0)=P_2(0)=\ldots =P_n(0)$ $ii)$ $P_1(x)>P_2(x)>\ldots >P_n(x)$ for $-\epsilon<x<0$ $iii)$ $P_{\sigma (1)} (x)>P_{\sigma (2)}(x)> \ldots >P_{\sigma (n)} (x) $ for $0<x<\epsilon$ Find all [i]interesting [/i]$n$. [i]Proposed by Mojtaba Zare Bidaki[/i]

Three speed skaters have a friendly "race" on a skating oval. They all start from the same point and skate in the same direction, but with different speeds that they maintain throughout the race. The slowest skater does $1$ lap per minute, the fastest one does $3.14$ laps per minute, and the middle one does $L$ laps a minute for some $1 < L < 3.14$. The race ends at the moment when all three skaters again come together to the same point on the oval (which may differ from the starting point.) Determine the number of different choices for $L$ such that exactly $117$ passings occur before the end of the race. Note: A passing is defined as when one skater passes another one. The beginning and the end of the race when all three skaters are together are not counted as passings.