Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of integers numbers. Let $\Delta^1a_n=a_{n+1}-a_n$ for a non-negative integer $n$. Define $\Delta^Ma_n= \Delta^{M-1}a_{n+1}- \Delta^{M-1}a_n$. A sequence is [i]patriota[/i] if there are positive integers $k,l$ such that $a_{n+k}=\Delta^Ma_{n+l}$ for all non-negative integers $n$. Determine, with proof, whether exists a sequence that the last value of $M$ for which the sequence is [i]patriota[/i] is $2022$.

Let $\pi$ be the set of points in a plane and $f : \pi \to \pi$ be a mapping such that the image of any triangle (as its polygonal line) is a square. Show that $f(\pi)$ is a square.

The sequence $S_0,S_1,S_2,\ldots$ is defined by[list][*]$S_n=1$ for $0\le n\le 2011$, and [*]$S_{n+2012}=S_{n+2011}+S_n$ for $n\ge 0$.[/list]Prove that $S_{2011a}-S_a$ is a multiple of $2011$ for all nonnegative integers $a$.

Find a function $f(x)$ defined for all real values of $x$ such that for all $x$, \[f(x+ 2) - f(x) = x^2 + 2x + 4,\] and if $x \in [0, 2)$, then $f(x) = x^2.$

Let $a_1, a_2, \ldots, a_n$ be real numbers lying in $[-1, 1]$ such that $a_1 + a_2 + \cdots + a_n = 0$. Prove that there is a $k \in \{1, 2, \ldots, n\}$ such that $|a_1 + 2a_2 + 3a_3 + \cdots + k a_k | \le \frac{2k+1}4$ .

Let $(k_i)$ be a sequence of unique nonzero integers such that $x^2- 5x + k_i$ has rational solutions. Find the minimum possible value of $$\frac15 \sum_{i=1}^{\infty} \frac{1}{k_i}$$

A softball team played ten games, scoring $1,2,3,4,5,6,7,8,9$, and $10$ runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score? $ \textbf {(A) } 35 \qquad \textbf {(B) } 40 \qquad \textbf {(C) } 45 \qquad \textbf {(D) } 50 \qquad \textbf {(E) } 55 $

Let $\mathbb{R}[x]$ be the set of all polynomials with real coefficients. Find all functions $f: \mathbb{R}[x] \rightarrow \mathbb{R}[x]$ satisfying the following conditions: [list] [*] $f$ maps the zero polynomial to itself, [*] for any non-zero polynomial $P \in \mathbb{R}[x]$, $\text{deg} \, f(P) \le 1+ \text{deg} \, P$, and [*] for any two polynomials $P, Q \in \mathbb{R}[x]$, the polynomials $P-f(Q)$ and $Q-f(P)$ have the same set of real roots. [/list] [i]Proposed by Anant Mudgal, Sutanay Bhattacharya, Pulkit Sinha[/i]

The sum of the numerical coefficients in the expansion of the binomial $ (a\plus{}b)^8$ is: $ \textbf{(A)}\ 32 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 7$

Find all functions $f:\mathbb R \rightarrow \mathbb R$ such that $$f(f(xy-x))+f(x+y)=yf(x)+f(y)$$ for all real numbers $x$ and $y$.

Determine all $n$ for which the system with of equations can be solved in $\mathbb{R}$: \[\sum^{n}_{k=1} x_k = 27\] and \[\prod^{n}_{k=1} x_k = \left( \frac{3}{2} \right)^{24}.\]

For a natural number $n$, with $v_2(n)$ we denote the largest integer $k\geq0$ such that $2^k|n$. Let us assume that the function $f\colon\mathbb{N}\to\mathbb{N}$ meets the conditions: $(i)$ $f(x)\leq3x$ for all natural numbers $x\in\mathbb{N}$. $(ii)$ $v_2(f(x)+f(y))=v_2(x+y)$ for all natural numbers $x,y\in\mathbb{N}$. Prove that for every natural number $a$ there exists exactly one natural number $x$ such that $f(x)=3a$.

Evaluate $\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+...}}}}$.

We denote the set of nonzero integers and the set of non-negative integers with $\mathbb Z^*$ and $\mathbb N_0$, respectively. Find all functions $f:\mathbb Z^* \to \mathbb N_0$ such that: $a)$ $f(a+b)\geq min(f(a), f(b))$ for all $a,b$ in $\mathbb Z^*$ for which $a+b$ is in $\mathbb Z^*$. $b)$ $f(ab)=f(a)+f(b)$ for all $a,b$ in $\mathbb Z^*$.

Solve the system of equations $$\begin{cases} \sin^3 x+\sin^4 y=1 \\ \cos^4 x+\cos^5 y =1\end{cases}$$

Prove that the sequence $(a_n)$, where $$a_n=\sum_{k=1}^n\left\{\frac{\left\lfloor2^{k-\frac12}\right\rfloor}2\right\}2^{1-k},$$converges, and determine its limit as $n\to\infty$.

[b]p1.[/b] A matrix is a rectangular array of numbers. For example, $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $\begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}$ are $2 \times 2$ matrices. A [i]saddle [/i] point in a matrix is an entry which is simultaneously the smallest number in its row and the largest number in its column. a. Write down a $2 \times 2$ matrix which has a saddle point, and indicate which entry is the saddle point. b. Write down a $2 \times 2$ matrix which has no saddle point c. Prove that a matrix of any size, all of whose entries are distinct, can have at most one saddle point. [b]p2.[/b] a. Find four different pairs of positive integers satisfying the equation $\frac{7}{m}+\frac{11}{n}=1$. b. Prove that the solutions you have found in part (a) are all possible pairs of positive integers satisfying the equation $\frac{7}{m}+\frac{11}{n}=1$. [b]p3.[/b] Let $ABCD$ be a quadrilateral, and let points $M, N, O, P$ be the respective midpoints of sides $AB$, $BC$, $CD$, $DA$. a. Show, by example, that it is possible that $ABCD$ is not a parallelogram, but $MNOP$ is a square. Be sure to prove that your construction satisfies all given conditions. b. Suppose that $MO$ is perpendicular to $NP$. Prove that $AC = BD$. [b]p4.[/b] A [i]Pythagorean triple[/i] is an ordered collection of three positive integers $(a, b, c)$ satisfying the relation $a^2 + b^2 = c^2$. We say that $(a, b, c)$ is a [i]primitive [/i] Pythagorean triple if $1$ is the only common factor of $a, b$, and $c$. a. Decide, with proof, if there are infinitely many Pythagorean triples. b. Decide, with proof, if there are infinitely many primitive Pythagorean triples of the form $(a, b, c)$ where $c = b + 2$. c. Decide, with proof, if there are infinitely many primitive Pythagorean triples of the form $(a, b, c)$ where $c = b + 3$. [b]p5.[/b] Let $x$ and $y$ be positive real numbers and let $s$ be the smallest among the numbers $\frac{3x}{2}$,$\frac{y}{x}+\frac{1}{x}$ and $\frac{3}{y}$. a. Find an example giving $s > 1$. b. Prove that for any positive $x$ and $y,s <2$. c. Find, with proof, the largest possible value of $s$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a rectangular coordinate system we call a horizontal line parallel to the $x$ -axis triangular if it intersects the curve with equation \[y = x^4 + px^3 + qx^2 + rx + s\] in the points $A,B,C$ and $D$ (from left to right) such that the segments $AB, AC$ and $AD$ are the sides of a triangle. Prove that the lines parallel to the $x$ - axis intersecting the curve in four distinct points are all triangular or none of them is triangular.

The function $f(x) = ax^2 + bx + c$ satisfies the following conditions: $f(\sqrt2)=3$ and $ |f(x)| \le 1$ for all $x \in [-1, 1]$. Evaluate the value of $f(\sqrt{2013})$

Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=Q(x)P^{\prime\prime}(x)$, where $Q(x)$ is a quadratic polynomial and $P^{\prime\prime}(x)$ is the second derivative of $P(x)$. Show that if $P(x)$ has at least two distinct roots then it must have $n$ distinct roots.

Paul starts at $1$ and counts by threes: $1, 4, 7, 10, ... $. At the same time and at the same speed, Penny counts backwards from $2017$ by fives: $2017, 2012, 2007, 2002,...$ . Find the one number that both Paul and Penny count at the same time.

Prove that for all even positive integers $n$ the following inequality holds a) $\{n\sqrt6\} > \frac{1}{n}$ b)$ \{n\sqrt6\}> \frac{1}{n-1/(5n)} $ (I. Voronovich)

A positive integer $d$ and a permutation of positive integers $a_1,a_2,a_3,\dots$ is given such that for all indices $i\geq 10^{100}$, $|a_{i+1}-a_{i}|\leq 2d$ holds. Prove that there exists infinity many indices $j$ such that $|a_j -j|< d$.

We call the three variable polynomial $P$ cyclic if $P(x,y,z)=P(y,z,x)$. Prove that cyclic three variable polynomials $P_1,P_2,P_3$ and $P_4$ exist such that for each cyclic three variable polynomial $P$, there exists a four variable polynomial $Q$ such that $P(x,y,z)=Q(P_1(x,y,z),P_2(x,y,z),P_3(x,y,z),P_4(x,y,z))$. [i]Solution by Mostafa Eynollahzade and Erfan Salavati[/i]

Given are a positive integer $n$ and an infinite sequence of proper fractions $x_0 = \frac{a_0}{n}$, $\ldots$, $x_i=\frac{a_i}{n+i}$, with $a_i < n+i$. Prove that there exist a positive integer $k$ and integers $c_1$, $\ldots$, $c_k$ such that \[ c_1 x_1 + \ldots + c_k x_k = 1. \] [i]Proposed by M. Dubashinsky[/i]