The numbers $a, b, c, d$ and $e$ satisfy $$a + b < c + d < e + a < b + c < d + e .$$ Which of the numbers is the smallest, and which is the largest?

Determine all real numbers $x$ and $y$ such that $x^2 + x = y^3 - y$, $y^2 + y = x^3 - x$

Find all possible values of $a\in \mathbb{R}$ and $n\in \mathbb{N^*}$ such that $f(x)=(x-1)^n+(x-2)^{2n+1}+(1-x^2)^{2n+1}+a$ is divisible by $\phi (x)=x^2-x+1$

Let $k,r \in \mathbb N$ and let $x\in (0,1)$ be a rational number given in decimal representation \[ x = 0.a_1a_2a_3a_4 \ldots . \] Show that if the decimals $a_k, a_{k+r}, a_{k+2r}, \ldots$ are canceled, the new number obtained is still rational. [i]Dan Schwarz[/i]

In a year that has $365$ days, what is the maximum number of "Tuesday the $13$th" there can be? Note: The months of April, June, September and November have $30$ days each, February has $28$ and all others have $31$ days.

A freight train departed from Moscow at $x$ hours and $y$ minutes and arrived at Saratov at $y$ hours and $z$ minutes. The length of its trip was $z$ hours and $x$ minutes. Find all possible values of $x$. [i]S. Tokarev[/i]

[b]p1.[/b] Find any integer solution of the puzzle: $WE+ST+RO+NG=128$ (different letters mean different digits between $1$ and $9$). [b]p2.[/b] A $5\times 6$ rectangle is drawn on the piece of graph paper (see the figure below). The side of each square on the graph paper is $1$ cm long. Cut the rectangle along the sides of the graph squares in two parts whose areas are equal but perimeters are different by $2$ cm. $\begin{tabular}{|l|l|l|l|l|l|} \hline & & & & & \\ \hline & & & & & \\ \hline & & & & & \\ \hline & & & & & \\ \hline \end{tabular}$ [b]p3.[/b] Three runners started simultaneously on a $1$ km long track. Each of them runs the whole distance at a constant speed. Runner $A$ is the fastest. When he runs $400$ meters then the total distance run by runners $B$ and $C$ together is $680$ meters. What is the total combined distance remaining for runners $B$ and $C$ when runner $A$ has $100$ meters left? [b]p4.[/b] There are three people in a room. Each person is either a knight who always tells the truth or a liar who always tells lies. The first person said «We are all liars». The second replied «Only you are a liar». Is the third person a liar or a knight? [b]p5.[/b] A $5\times 8$ rectangle is divided into forty $1\times 1$ square boxes (see the figure below). Choose 24 such boxes and one diagonal in each chosen box so that these diagonals don't have common points. $\begin{tabular}{|l|l|l|l|l|l|l|l|} \hline & & & & & & & \\ \hline & & & & & & & \\ \hline & & & & & & & \\ \hline & & & & & & & \\ \hline \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If a polynomial $f(x)$ is divided by $(x-a)^{2} (x-b)$, where $a\ne b$, derive a formula for the remainder.

Find all real numbers $x$ that satisfy the following inequality $|x^2-4x+1|>|x^2-4x+5|$

(a) Prove that $(x+y+z)^2 \le 3(x^2 +y^2 +z^2)$ for any real numbers $x,y,z$. (b) If positive numbers $a,b,c$ satisfy $a+b+c \ge abc$, prove that $a^2 +b^2 +c^2 \ge \sqrt3 abc$

Prove that there exists a surjective function $ f:\mathbb{N}\longrightarrow\mathbb{N} $ having the property that for all natural numbers $ n\ge 2, $ there exists an infinite set $ A_n $ such that $ f(x)=n, $ for all $ x\in A_n. $

For each positive integer $n$ let the set $A_n$ consist of all numbers $\pm 1 \pm 2 \pm ...\pm n$. For example, $$A_1 = \{-1,1\}, A_2 = \{ -3 ,-1 ,1 ,3 \} , A_3 = \{ -6 ,-4 ,-2 ,0 ,2 ,4 ,6 \}.$$ Find the number of elements in $A_n$ .

Suppose $a$ and $b$ are two positive real numbers such that the roots of the cubic equation $x^3 - ax + b = 0$ are all real. If $\alpha$ is a root of this cubic with minimal absolute value, prove that \[ \dfrac{b}{a} < \alpha < \dfrac{3b}{2a}. \]

Suppose $a_1$, $a_2$, $\ldots$, $a_{10}$ are nonnegative integers such that \[\sum_{k=1}^{10}a_k=15\qquad\text{and}\qquad \sum_{k=1}^{10}ka_k = 80.\] Let $M$ and $m$ denote the maximum and minimum respectively of $\sum_{k=1}^{10}k^2a_k$. Compute $M-m$.

[b]p16[/b] Madelyn is being paid $\$50$/hour to find useful [i]Non-Functional Trios[/i], where a Non-Functional Trio is defined as an ordered triple of distinct real numbers $(a, b, c)$, and a Non- Functional Trio is [i]useful [/i] if $(a, b)$, $(b, c)$, and $(c, a)$ are collinear in the Cartesian plane. Currently, she’s working on the case $a+b+c = 2022$. Find the number of useful Non-Functional Trios $(a, b, c)$ such that $a + b + c = 2022$. [b]p17[/b] Let $p(x) = x^2 - k$, where $k$ is an integer strictly less than $250$. Find the largest possible value of k such that there exist distinct integers $a, b$ with $p(a) = b$ and $p(b) = a$. [b]p18[/b] Let $ABC$ be a triangle with orthocenter $H$ and circumcircle $\Gamma$ such that $AB = 13$, $BC = 14$, and $CA = 15$. $BH$ and $CH$ meet $\Gamma$ again at points $D$ and $E$, respectively, and $DE$ meets $AB$ and $AC$ at $F$ and $G$, respectively. The circumcircles of triangles $ABG$ and $ACF$ meet BC again at points $P$ and $Q$. If $PQ$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd (a, b) = 1$, find $a + b$.

Let $x,y$ be positive real numbers such that $x+y=1$. Prove that $\left(1+\frac {1}{x}\right)\left(1+\frac {1}{y}\right)\geq 9$.

Let $a$, $b$, $c$ be positive real numbers with $abc \leq a+b+c$. Show that \[ a^2 + b^2 + c^2 \geq \sqrt 3 abc. \] [i]Cristinel Mortici, Romania[/i]

The sequence $a_n$ is defined by the following conditions: $a_1=1$, and for any $n\in \mathbb N$, the number $a_{n+1}$ is obtained from $a_n$ by adding three if $n$ is a member of this sequence, and two if it is not. Prove that $a_n<(1+\sqrt 2)n$ for all $n$. [i]Proposed by Mikhail Ivanov[/i]

[b]8.1[/b] On the median drawn from the vertex of the triangle to the base, point $A$ is taken. The sum of the distances from $A$ to the sides of the triangle is equal to $s$. Find the distances from $A$ to the sides if the lengths of the sides are equal to $x$ and $y$. [b]8.2[/b] Fraction $0, abc...$ is composed according to the following rule: $a$ and $c$ are arbitrary digits, and each next digit is equal to the remainder of the sum of the previous two digits when divided by $10$. Prove that this fraction is purely periodic. [b]8.3[/b] Two convex polygons with $m$ and $n$ sides are drawn on the plane ($m>n$). What is the greatest possible number of parts, they can break the plane? [b]8.4 [/b]The sum of three integers that are perfect squares is divisible by $9$. Prove that among them, there are two numbers whose difference is divisible by $9$. [b]8.5 / 9.5[/b] Given $k+2$ integers. Prove that among them there are two integers such that either their sum or their difference is divisible by $2k$. [b]8.6[/b] A right angle rotates around its vertex. Find the locus of the midpoints of the segments connecting the intersection points sides of an angle and a given circle. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here[/url].

Consider the sequence $z_n = (1+i)(2+i)...(n+i)$. Prove that the sequence $Im$ $z_n$ contains infinitely many positive and infinitely many negative numbers.

Let $f : \mathbb R \to \mathbb R$ be a continuous function. Suppose that the restriction of $f$ to the set of irrational numbers is injective. What can we say about $f$? Answer the analogous question if $f$ is restricted to rationals.

[b]p1.[/b] A robot is at position $0$ on a number line. Each second, it randomly moves either one unit in the positive direction or one unit in the negative direction, with probability $\frac12$ of doing each. Find the probability that after $4$ seconds, the robot has returned to position $0$. [b]p2.[/b] How many positive integers $n \le 20$ are such that the greatest common divisor of $n$ and $20$ is a prime number? [b]p3.[/b] A sequence of points $A_1$, $A_2$, $A_3$, $...$, $A_7$ is shown in the diagram below, with $A_1A_2$ parallel to $A_6A_7$. We have $\angle A_2A_3A_4 = 113^o$, $\angle A_3A_4A_5 = 100^o$, and $\angle A_4A_5A_6 = 122^o$. Find the degree measure of $\angle A_1A_2A_3 + \angle A_5A_6A_7$. [center][img]https://cdn.artofproblemsolving.com/attachments/d/a/75b06a6663b2f4258e35ef0f68fcfbfaa903f7.png[/img][/center] [b]p4.[/b] Compute $$\log_3 \left( \frac{\log_3 3^{3^{3^3}}}{\log_{3^3} 3^{3^3}} \right)$$ [b]p5.[/b] In an $8\times 8$ chessboard, a pawn has been placed on the third column and fourth row, and all the other squares are empty. It is possible to place nine rooks on this board such that no two rooks attack each other. How many ways can this be done? (Recall that a rook can attack any square in its row or column provided all the squares in between are empty.) [b]p6.[/b] Suppose that $a, b$ are positive real numbers with $a > b$ and $ab = 8$. Find the minimum value of $\frac{a^2+b^2}{a-b} $. [b]p7.[/b] A cone of radius $4$ and height $7$ has $A$ as its apex and $B$ as the center of its base. A second cone of radius $3$ and height $7$ has $B$ as its apex and $A$ as the center of its base. What is the volume of the region contained in both cones? [b]p8.[/b] Let $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$ be a permutation of the numbers $1$, $2$, $3$, $4$, $5$, $6$. We say $a_i$ is visible if $a_i$ is greater than any number that comes before it; that is, $a_j < a_i$ for all $j < i$. For example, the permutation $2$, $4$, $1$, $3$, $6$, $5$ has three visible elements: $2$, $4$, $6$. How many such permutations have exactly two visible elements? [b]p9.[/b] Let $f(x) = x+2x^2 +3x^3 +4x^4 +5x^5 +6x^6$, and let $S = [f(6)]^5 +[f(10)]^3 +[f(15)]^2$. Compute the remainder when $S$ is divided by $30$. [b]p10.[/b] In triangle $ABC$, the angle bisector from $A$ and the perpendicular bisector of $BC$ meet at point $D$, the angle bisector from $B$ and the perpendicular bisector of $AC$ meet at point $E$, and the perpendicular bisectors of $BC$ and $AC$ meet at point $F$. Given that $\angle ADF = 5^o$, $\angle BEF = 10^o$, and $AC = 3$, find the length of $DF$. [img]https://cdn.artofproblemsolving.com/attachments/6/d/6bb8409678a4c44135d393b9b942f8defb198e.png[/img] [b]p11.[/b] Let $F_0 = 0$, $F_1 = 1$, and $F_n = F_{n-1} + F_{n-2}$. How many subsets $S$ of $\{1, 2,..., 2011\}$ are there such that $$F_{2012} - 1 =\sum_{i \in S}F_i?$$ [b]p12.[/b] Let $a_k$ be the number of perfect squares $m$ such that $k^3 \le m < (k + 1)^3$. For example, $a_2 = 3$ since three squares $m$ satisfy $2^3 \le m < 3^3$, namely $9$, $16$, and $25$. Compute$$ \sum^{99}_{k=0} \lfloor \sqrt{k}\rfloor a_k, $$ where $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$. [b]p13.[/b] Suppose that $a, b, c, d, e, f$ are real numbers such that $$a + b + c + d + e + f = 0,$$ $$a + 2b + 3c + 4d + 2e + 2f = 0,$$ $$a + 3b + 6c + 9d + 4e + 6f = 0,$$ $$a + 4b + 10c + 16d + 8e + 24f = 0,$$ $$a + 5b + 15c + 25d + 16e + 120f = 42.$$ Compute $a + 6b + 21c + 36d + 32e + 720f.$ [b]p14.[/b] In Cartesian space, three spheres centered at $(-2, 5, 4)$, $(2, 1, 4)$, and $(4, 7, 5)$ are all tangent to the $xy$-plane. The $xy$-plane is one of two planes tangent to all three spheres; the second plane can be written as the equation $ax + by + cz = d$ for some real numbers $a$, $b$, $c$, $d$. Find $\frac{c}{a}$ . [b]p15.[/b] Find the number of pairs of positive integers $a$, $b$, with $a \le 125$ and $b \le 100$, such that $a^b - 1$ is divisible by $125$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For a given value $t$, we consider number sequences $a_1, a_2, a_3,...$ such that $a_{n+1} =\frac{a_n + t}{a_n + 1}$ for all $n \ge 1$. (a) Suppose that $t = 2$. Determine all starting values $a_1 > 0$ such that $\frac43 \le a_n \le \frac32$ holds for all $n \ge 2$. (b) Suppose that $t = -3$. Investigate whether $a_{2020} = a_1$ for all starting values $a_1$ different from $-1$ and $1$.

Let $a\le 1$ be a real number. Sequence $\{x_n\}$ satisfies $x_0=0, x_{n+1}= 1-a\cdot e^{x_n}$, for all $n\ge 1$, where $e$ is the natural logarithm. Prove that for any natural $n$, $x_n\ge 0$.

If $x^5 - x ^3 + x = a,$ prove that $x^6 \geq 2a - 1$.