Prove that for every natural number $k$ ($k \geq 2$) there exists an irrational number $r$ such that for every natural number $m$, \[[r^m] \equiv -1 \pmod k .\] [i]Remark.[/i] An easier variant: Find $r$ as a root of a polynomial of second degree with integer coefficients. [i]Proposed by Yugoslavia.[/i]

Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$. Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.

Suppose a non-identically zero function $f$ satisfies $f\left(x\right)f\left(y\right)=f\left(\sqrt{x^2+y^2}\right)$ for all $x$ and $y$. Compute $$f\left(1\right)-f\left(0\right)-f\left(-1\right).$$

Prove that if \( a \), \( b \), and \( c \) are positive real numbers, then the following inequality holds: \[ \frac{a + 3c}{a + b} + \frac{c + 3a}{b + c} + \frac{4b}{c + a} \geq 6. \]

Determine all polynomials $P(x,y)$ with real coefficients such that \[P(x+y,x-y)=2P(x,y) \qquad \forall x,y\in\mathbb{R}.\]

Five students take a test on which any integer score from $0$ to $100$ inclusive is possible. What is the largest possible difference between the median and the mean of the scores? [i](The median of a set of scores is the middlemost score when the data is arranged in increasing order. It is exactly the middle score when there are an odd number of scores and it is the average of the two middle scores when there are an even number of scores.)[/i]

We say that a positive integer $n$ is [i]memorable[/i] if it has a binary representation with strictly more $1$'s than $0$'s (for example $25$ is memorable because $25=(11001)_{2}$ has more $1$'s than $0$'s). Are there infinitely many memorable perfect squares? [i]Proposed by Nikola Velov[/i]

Misha has accepted a job in the mines and will produce one ore each day. At the market, he is able to buy or sell one ore for \$3, buy or sell bundles of three wheat for \$12 each, or $\textit{sell}$ one wheat for one ore. His ultimate goal is to build a city, which requires three ore and two wheat. How many dollars must Misha begin with in order to build a city after three days of working?

A function $f : [0,1] \to R$ has the following properties: (a) $f(x) \ge 0$ for $0 < x < 1$, (b) $f(1) = 1$, (c) $f(x+y) \ge f(x)+ f(y) $ whenever $x,y,x+y \in [0,1]$. Prove that $f(x) \le 2x$ for all $x \in [0,1]$.

Let $\lfloor \bullet \rfloor$ denote the floor function. For nonnegative integers $a$ and $b$, their [i]bitwise xor[/i], denoted $a \oplus b$, is the unique nonnegative integer such that $$ \left \lfloor \frac{a}{2^k} \right \rfloor+ \left\lfloor\frac{b}{2^k} \right\rfloor - \left\lfloor \frac{a\oplus b}{2^k}\right\rfloor$$ is even for every $k \ge 0$. Find all positive integers $a$ such that for any integers $x>y\ge 0$, we have \[ x\oplus ax \neq y \oplus ay. \] [i]Carl Schildkraut[/i]

[b]p1. [/b] In basketball, teams can score $1, 2$, or $3$ points each time. Suppose that Duke basketball have scored $8$ points so far. What is the total number of possible ways (ordered) that they have scored? For example, $(1, 2, 2, 2, 1)$,$(1, 1, 2, 2, 2)$ are two different ways. [b]p2.[/b] All the positive integers that are coprime to $2021$ are grouped in increasing order, such that the nth group contains $2n - 1$ numbers. Hence the first three groups are $\{1\}$, $\{2, 3, 4\}$, $\{5, 6, 7, 8, 9\}$. Suppose that $2022$ belongs to the $k$th group. Find $k$. [b]p3.[/b] Let $A = (0, 0)$ and $B = (3, 0)$ be points in the Cartesian plane. If $R$ is the set of all points $X$ such that $\angle AXB \ge 60^o$ (all angles are between $0^o$ and $180^o$), find the integer that is closest to the area of $R$. [b]p4.[/b] What is the smallest positive integer greater than $9$ such that when its left-most digit is erased, the resulting number is one twenty-ninth of the original number? [b]p5. [/b] Jonathan is operating a projector in the cartesian plane. He sets up $2$ infinitely long mirrors represented by the lines $y = \tan(15^o)x$ and $y = 0$, and he places the projector at $(1, 0)$ pointed perpendicularly to the $x$-axis in the positive $y$ direction. Jonathan furthermore places a screen on one of the mirrors such that light from the projector reflects off the mirrors a total of three times before hitting the screen. Suppose that the coordinates of the screen is $(a, b)$. Find $10a^2 + 5b^2$. [b]p6.[/b] Dr Kraines has a cube of size $5 \times 5 \times 5$, which is made from $5^3$ unit cubes. He then decides to choose $m$ unit cubes that have an outside face such that any two different cubes don’t share a common vertex. What is the maximum value of $m$? [b]p7.[/b] Let $a_n = \tan^{-1}(n)$ for all positive integers $n$. Suppose that $$\sum_{k=4}^{\infty}(-1)^{\lfloor \frac{k}{2} \rfloor +1} \tan(2a_k)$$ is equals to $a/b$ , where $a, b$ are relatively prime. Find $a + b$. [b]p8.[/b] Rishabh needs to settle some debts. He owes $90$ people and he must pay \$ $(101050 + n)$ to the $n$th person where $1 \le n \le 90$. Rishabh can withdraw from his account as many coins of values \$ $2021$ and \$ $x$ for some fixed positive integer $x$ as is necessary to pay these debts. Find the sum of the four least values of $x$ so that there exists a person to whom Rishabh is unable to pay the exact amount owed using coins. [b]p9.[/b] A frog starts at $(1, 1)$. Every second, if the frog is at point $(x, y)$, it moves to $(x + 1, y)$ with probability $\frac{x}{x+y}$ and moves to $(x, y + 1)$ with probability $\frac{y}{x+y}$ . The frog stops moving when its $y$ coordinate is $10$. Suppose the probability that when the frog stops its $x$-coordinate is strictly less than $16$, is given by $m/n$ where $m, n$ are positive integers that are relatively prime. Find $m + n.$ [b]p10.[/b] In the triangle $ABC$, $AB = 585$, $BC = 520$, $CA = 455$. Define $X, Y$ to be points on the segment $BC$. Let $Z \ne A$ be the intersection of $AY$ with the circumcircle of $ABC$. Suppose that $XZ$ is parallel to $AC$ and the circumcircle of $XYZ$ is tangent to the circumcircle of $ABC$ at $Z$. Find the length of $XY$ . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\mathbb{N}$ denote the set of positive integers. Fix a function $f: \mathbb{N} \rightarrow \mathbb{N}$ and for any $m,n \in \mathbb{N}$ define $$\Delta(m,n)=\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(m)\ldots))-\underbrace{f(f(\ldots f}_{f(m)\text{ times}}(n)\ldots)).$$ Suppose $\Delta(m,n) \neq 0$ for any distinct $m,n \in \mathbb{N}$. Show that $\Delta$ is unbounded, meaning that for any constant $C$ there exists $m,n \in \mathbb{N}$ with $\left|\Delta(m,n)\right| > C$.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x, y\in\mathbb{R}$, we have \[f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y).\]

Let $n$ be an integer greater than or equal to $2$. Consider real numbers $a_1, a_2, \dots, a_{2n}$ satisfying the condition \[ |a_k - a_{n+k}| \geqslant 1 \quad \text{for all } 1 \leqslant k \leqslant n. \] Determine the minimum possible value of \[ (a_1 - a_2)^2 + (a_2 - a_3)^2 + \dots + (a_{2n-1} - a_{2n})^2 + (a_{2n} - a_1)^2. \]

Find all solutions to $3^x-9^{x-1} = 2.$

Prove that in any set of $2000$ distinct real numbers there exist two pairs $a>b$ and $c>d$ with $a \neq c$ or $b \neq d $, such that \[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]

Compute the largest root of $x^4-x^3-5x^2+2x+6$.

Let $m,n$ be positive integers such that the equation (in respect of $x$) \[(x+m)(x+n)=x+m+n\] has at least one integer root. Prove that $\frac{1}{2}n<m<2n$.

Solve the following system of equations: $$x+\frac{1}{x^3}=2y,\quad y+\frac{1}{y^3}=2z,\quad z+\frac{1}{z^3}=2w,\quad w+\frac{1}{w^3}=2x.$$

A real number $\alpha$ is given. Find all functions $f : R^+ \to R^+$ satisfying $\alpha x^2f\left(\frac{1}{x}\right) +f(x) =\frac{x}{x+1}$ for all $x > 0$.

Let $a_1, \ldots, a_n$ be $n$ positive numbers and $0 < q < 1.$ Determine $n$ positive numbers $b_1, \ldots, b_n$ so that: [i]a.)[/i] $ a_{k} < b_{k}$ for all $k = 1, \ldots, n,$ [i]b.)[/i] $q < \frac{b_{k+1}}{b_{k}} < \frac{1}{q}$ for all $k = 1, \ldots, n-1,$ [i]c.)[/i] $\sum \limits^n_{k=1} b_k < \frac{1+q}{1-q} \cdot \sum \limits^n_{k=1} a_k.$

Let $(G,\cdot)$ be a grup with neutral element $e{}$, and let $H{}$ and $K$ be proper subgroups of $G$, satisfying $H\cap K=\{e\}$. It is known that $(G\setminus(H\cup K))\cup\{e\}$ is closed under the operation of $G$. Prove that $x^2=e$ for all the elements $x{}$ of $G{}$.

If for the real numbers $x, y,z, k$ the following conditions are valid, $x \ne y \ne z \ne x$ and $x^3 +y^3 +k(x^2 +y^2) = y^3 +z^3 +k(y^2 +z^2) = z^3 +x^3 +k(z^2 +x^2) = 2008$, find the product $xyz$.

Prove that infinitely many positive integers can be represented as $(a-1)/b + (b-1)/c + (c-1)/a$, where $a$, $b$ and $c$ are pairwise distinct positive integers greater than 1.

Find all the functions $f:(0,+\infty) \to R $ that satisfy the equation $$f(x)f(y)+f\big(\frac{\lambda}{x})f(\frac{\lambda}{y})=2f(xy)$$ for all pairs of $x,y$ real and positive numbers, where $\lambda$ is a positive real number such that $f(\lambda )=1$