For $2k$ real numbers $a_1, a_2, ..., a_k$, $b_1, b_2, ..., b_k$ define a sequence of numbers $X_n$ by \[ X_n = \sum_{i=1}^k [a_in + b_i] \quad (n=1,2,...). \] If the sequence $X_N$ forms an arithmetic progression, show that $\textstyle\sum_{i=1}^k a_i$ must be an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.

Find all fuctions $f,g:\mathbb{R}\rightarrow \mathbb{R}$ such that: $f(x-3f(y))=xf(y)-yf(x)+g(x) \forall x,y\in\mathbb{R}$ and $g(1)=-8$

If $x=\sqrt2+\sqrt3+\sqrt6$ is a root of $x^4+ax^3+bx^2+cx+d=0$ where $a,b,c,d$ are integers, what is the value of $|a+b+c+d|$?

Find all real positive solutions (if any) to \begin{align*} x^3+y^3+z^3 &= x+y+z, \mbox{ and} \\ x^2+y^2+z^2 &= xyz. \end{align*}

Prove that any real solution of $x^3+px+q=0$, where $p,q$ are real numbers, satisfies the inequality $4qx\le p^2$.

Let $p$ be a certain prime number. Find all non-negative integers $n$ for which polynomial $P(x)=x^4-2(n+p)x^2+(n-p)^2$ may be rewritten as product of two quadratic polynomials $P_1, \ P_2 \in \mathbb{Z}[X]$.

The sequence $\{a_{n}\}_{n\geq 0}$ is defined by $a_{0}=20,a_{1}=100,a_{n+2}=4a_{n+1}+5a_{n}+20(n=0,1,2,...)$. Find the smallest positive integer $h$ satisfying $1998|a_{n+h}-a_{n}\forall n=0,1,2,...$

Let $a_1,a_2,\dots$ be a sequence of positive numbers satisfying, for any positive integers $k,l,m,n$ such that $k+n=m+l$, $$\frac{a_k+a_n}{1+a_ka_n}=\frac{a_m+a_l}{1+a_ma_l}.$$Show that there exist positive numbers $b,c$ so that $b\le a_n\le c$ for any positive integer $n$.

Let be a bipartition of the set formed by the first $ 13 $ nonnegative numbers. Prove that at least one of these two subsets that form this partition contains an arithmetic progression.

If $ a$ and $ b$ are two of the roots of $ x^4\plus{}x^3\minus{}1\equal{}0$, prove that $ ab$ is a root of $ x^6\plus{}x^4\plus{}x^3\minus{}x^2\minus{}1\equal{}0$.

Given $a,x\in\mathbb{R}$ and $x\geq 0$,$a\geq 0$ . Also $\sin(\sqrt{x+a})=\sin(\sqrt{x})$ . What can you say about $a$??? Justify your answer.

Find all natural $n$ such that for some different natural $a, b, c$ and $d$ among numbers $$\frac{(a-c)(b-d)}{(b-c)(a-d)} , \frac{(b-c)(a-d)}{(a-c)(b-d)} , \frac{(a-b)(d-c)}{(a-d)(b-c)} , \frac{(a-c)(b-d)}{(a-b)(c-d)} ,$$ there are at least two numbers equal to $n$.

$(USS 3)$ $(a)$ Prove that if $0 \le a_0 \le a_1 \le a_2,$ then $(a_0 + a_1x - a_2x^2)^2 \le (a_0 + a_1 + a_2)^2\left(1 +\frac{1}{2}x+\frac{1}{3}x^2+\frac{1}{2}x^3+x^4\right)$ $(b)$ Formulate and prove the analogous result for polynomials of third degree.

[b]7.1 / 6.1[/b] There are $8$ rooks on the chessboard such that no two of them they don't hit each other. Prove that the black squares contain an even number of rooks. [b]7.2[/b] The sides of triangle $ABC$ are extended as shown in the figure. At this $AA' = 3 AB$,, $BB' = 5BC$ , $CC'= 8 CA$. How many times is the area of the triangle $ABC$ less than the area of the triangle $A'B'C' $? [img]https://cdn.artofproblemsolving.com/attachments/9/f/06795292291cd234bf2469e8311f55897552f6.png[/img] [url=https://artofproblemsolving.com/community/c893771h1860178p12579333]7.3[/url] Prove the equality $$\frac{2}{x^2-1}+\frac{4}{x^2-4} +\frac{6}{x^2-9}+...+\frac{20}{x^2-100} =\frac{11}{(x-1)(x+10)}+\frac{11}{(x-2)(x+9)}+...+\frac{11}{(x-10)(x+1)}$$ [url=https://artofproblemsolving.com/community/c893771h1861966p12597273]7.4* / 8.4 *[/url] (asterisk problems in separate posts) [b]7.5 [/b]. The collective farm consists of $4$ villages located in the peaks of square with side $10$ km. It has the means to conctruct 28 kilometers of roads . Can a collective farm build such a road system so that was it possible to get from any village to any other? [b]7.6 / 6.6[/b] Two brilliant mathematicians were told in natural terms number and were told that these numbers differ by one. After that they take turns asking each other the same question: “Do you know my number?" Prove that sooner or later one of them will answer positively. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988085_1969_leningrad_math_olympiad]here[/url].

[b]p1.[/b] Given any four digit number $X = \underline{ABCD}$, consider the quantity $Y(X) = 2 \cdot \underline{AB}+\underline{CD}$. For example, if $X = 1234$, then $Y(X) = 2 \cdot 12+34 = 58$. Find the sum of all natural numbers $n \le 10000$ such that over all four digit numbers $X$, the number $n$ divides $X$ if and only if it also divides $Y(X)$. [b]p2.[/b] A sink has a red faucet, a blue faucet, and a drain. The two faucets release water into the sink at constant but different rates when turned on, and the drain removes water from the sink at a constant rate when opened. It takes $5$ minutes to fill the sink (from empty to full) when the drain is open and only the red faucet is on, it takes $10$ minutes to fill the sink when the drain is open and only the blue faucet is on, and it takes $15$ seconds to fill the sink when both faucets are on and the drain is closed. Suppose that the sink is currently one-thirds full of water, and the drain is opened. Rounded to the nearest integer, how many seconds will elapse before the sink is emptied (keeping the two faucets closed)? [b]p3.[/b] One of the bases of a right triangular prism is a triangle $XYZ$ with side lengths $XY = 13$, $YZ = 14$, $ZX = 15$. Suppose that a sphere may be positioned to touch each of the five faces of the prism at exactly one point. A plane parallel to the rectangular face of the prism containing $\overline{YZ}$ cuts the prism and the sphere, giving rise to a cross-section of area $A$ for the prism and area $15\pi$ for the sphere. Find the sum of all possible values of $A$. [b]p4.[/b] Albert, Brian, and Christine are hanging out by a magical tree. This tree gives each of them a stick, each of which have a non-negative real length. Say that Albert gets a branch of length $x$, Brian a branch of length $y$, and Christine a branch of length $z$, and the lengths follow the condition that $x+y+z = 2$. Let $m$ and $n$ be the minimum and maximum possible values of $xy+yz+xz-xyz$, respectively. What is $m+n$? [b]p5.[/b] Let $S := MATHEMATICSMATHEMATICSMATHE...$ be the sequence where $7$ copies of the word $MATHEMATICS$ are concatenated together. How many ways are there to delete all but five letters of $S$ such that the resulting subsequence is $CHMMC$? [b]p6.[/b] Consider two sequences of integers $a_n$ and $b_n$ such that $a_1 = a_2 = 1$, $b_1 = b_2 = 1$ and that the following recursive relations are satisfied for integers $n > 2$: $$a_n = a_{n-1}a_{n-2}-b_{n-1}b_{n-2},$$ $$b_n = b_{n-1}a_{n-2}+a_{n-1}b_{n-2}.$$ Determine the value of $$\sum_{1\le n\le2023,b_n \ne 0} \frac{a_n}{b_n}.$$ [b]p7.[/b] Suppose $ABC$ is a triangle with circumcenter $O$. Let $A'$ be the reflection of $A$ across $\overline{BC}$. If $BC =12$, $\angle BAC = 60^o$, and the perimeter of $ABC$ is $30$, then find $A'O$. [b]p8.[/b] A class of $10$ students wants to determine the class president by drawing slips of paper from a box. One of the students, Bob, puts a slip of paper with his name into the box. Each other student has a $\frac12$ probability of putting a slip of paper with their own name into the box and a $\frac12$ probability of not doing so. Later, one slip is randomly selected from the box. Given that Bob’s slip is selected, find the expected number of slips of paper in the box before the slip is selected. [b]p9.[/b] Let $a$ and $b$ be positive integers, $a > b$, such that $6! \cdot 11$ divides $x^a -x^b$ for all positive integers $x$. What is the minimum possible value of $a+b$? [b]p10.[/b] Find the number of pairs of positive integers $(m,n)$ such that $n < m \le 100$ and the polynomial $x^m+x^n+1$ has a root on the unit circle. [b]p11.[/b] Let $ABC$ be a triangle and let $\omega$ be the circle passing through $A$, $B$, $C$ with center $O$. Lines $\ell_A$, $\ell_B$, $\ell_C$ are drawn tangent to $\omega$ at $A$, $B$, $C$ respectively. The intersections of these lines form a triangle $XYZ$ where $X$ is the intersection of $\ell_B$ and $\ell_C$, $Y$ is the intersection of $\ell_C$ and $\ell_A$, and $Z$ is the intersection of $\ell_A$ and $\ell_B$. Let $P$ be the intersection of lines $\overline{OX}$ and $\overline{YZ}$. Given $\angle ACB = \frac32 \angle ABC$ and $\frac{AC}{AB} = \frac{15}{16}$ , find $\frac{ZP}{YP}$. [b]p12.[/b] Compute the remainder when $$\sum_{1\le a,k\le 2021} a^k$$ is divided by $2022$ (in the above summation $a,k$ are integers). [b]p13.[/b] Consider a $7\times 2$ grid of squares, each of which is equally likely to be colored either red or blue. Madeline would like to visit every square on the grid exactly once, starting on one of the top two squares and ending on one of the bottom two squares. She can move between two squares if they are adjacent or diagonally adjacent. What is the probability that Madeline may visit the squares of the grid in this way such that the sequence of colors she visits is alternating (i.e., red, blue, red,... or blue, red, blue,... )? [b]p14.[/b] Let $ABC$ be a triangle with $AB = 8$, $BC = 10$, and $CA = 12$. Denote by $\Omega_A$ the $A$-excircle of $ABC$, and suppose that $\Omega_A$ is tangent to $\overline{AB}$ and $\overline{AC}$ at $F$ and $E$, respectively. Line $\ell \ne \overline{BC}$ is tangent to $\Omega_A$ and passes through the midpoint of $\overline{BC}$. Let $T$ be the intersection of $\overline{EF}$ and $\ell$. Compute the area of triangle $ATB$. [b]p15.[/b] For any positive integer $n$, let $D_n$ be the set of ordered pairs of positive integers $(m,d)$ such that $d$ divides $n$ and gcd$(m,n) = 1$, $1 \le m \le n$. For any positive integers $a$, $b$, let $r(a,b)$ be the non-negative remainder when $a$ is divided by $b$. Denote by $S_n$ the sum $$S_n = \sum_{(m,d)\in D_n} r(m,d).$$ Determine the value of $S_{396}$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If $ n$ runs through all the positive integers, then $ f(n) \equal{} \left[n \plus{} \sqrt {\frac {n}{3}} \plus{} \frac {1}{2} \right]$ runs through all positive integers skipping the terms of the sequence $ a_n \equal{} 3 \cdot n^2 \minus{} 2 \cdot n.$

Let $ f(x) \equal{} c_m x^m \plus{} c_{m\minus{}1} x^{m\minus{}1} \plus{}...\plus{} c_1 x \plus{} c_0$, where each $ c_i$ is a non-zero integer. Define a sequence $ \{ a_n \}$ by $ a_1 \equal{} 0$ and $ a_{n\plus{}1} \equal{} f(a_n)$ for all positive integers $ n$. (a) Let $ i$ and $ j$ be positive integers with $ i<j$. Show that $ a_{j\plus{}1} \minus{} a_j$ is a multiple of $ a_{i\plus{}1} \minus{} a_i$. (b) Show that $ a_{2008} \neq 0$

On the board were written the numbers from $1$ to $k$ (where $k$ is an unknown positive integer). One of the numbers was erased. The average of the remaining numbers is $25.25$. Which number was erased?

Prove that among the elements of the sequence $\left\{ \left\lfloor n\sqrt{2003} \right\rfloor \right\}_{n\geq 1}$ one can find a geometric progression having any number of terms, and having the ratio bigger than $k$, where $k$ can be any positive integer. [i]Radu Gologan[/i]

If the equation $ ax^2+(c-b)x+(e-d)=0$ has real roots greater than $1$, prove that the equation $ax^4+bx^3+cx^2+dx+e=0$ has at least one real root.

For a natural number $n$, let $P(x)$ be the polynomial of $2n$−th degree such that: $P(0) = 1$ and $P(k) = 2^{k-1}$ for $k = 1, 2, . . . , 2n$. Prove that $2P(2n + 1) - P(2n + 2) = 1$. P.S. I tried to prove it by firstly expressing this polynomial using Lagrange interpolation but get bored of computations - it seems like it can be done this way, but I'd like to see more 'clever' solution. :)

Given $n > 1$ monic square trinomials $x^2 - a_1x + b_1$,$...$, $x^2-a_nx + b_n$, and all $2n$ numbers are $a_1$,$...$, $a_n$, $b_1$,$...$, $b_n$ are different. Can it happen that each of the numbers $a_1$,$...$, $a_n$, $b_1$,$...$, $b_n is the root of one of these trinomials?

Suppose $abc\neq 0$. Express in terms of $a,b,$ and $c$ the solutions $x,y,z,u,v,w$ of the equations $$x+y=a,\quad z+u=b,\quad v+w=c,\quad ay=bz,\quad ub=cv,\quad wc=ax.\quad$$

A sequence $a_1, a_2, \ldots $ consisting of $1$'s and $0$'s satisfies for all $k>2016$ that \[ a_k=0 \quad \Longleftrightarrow \quad a_{k-1}+a_{k-2}+\cdots+a_{k-2016}>23. \] Prove that there exist positive integers $N$ and $T$ such that $a_k=a_{k+T}$ for all $k>N$.

Find all positive real solutions to the equation $x+\left\lfloor\frac x3\right\rfloor=\left\lfloor\frac{2x}3\right\rfloor+\left\lfloor\frac{3x}5\right\rfloor$