Find all functions $ f : R \to R$ that satisfies $$xf(y) - yf(x)= f\left(\frac{y}{x}\right)$$ for all $x, y \in R$.

How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is $ax^2+bx+c,a\neq 0,$ and the roots are $r$ and $s,$ then the requirement is that $\{a,b,c\}=\{r,s\}$.) $\textbf{(A) } 3 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } 6 \qquad\textbf{(E) } \text{infinitely many}$

Determine all real solutions $(x,y)$ of the following system of equations: \begin{align*} x&=3x^2y-y^3,\\ y &= x^3-3xy^2 \end{align*}

Let $x,y$ be real numbers satisfying the condition: \[x-3\sqrt {x+1}=3\sqrt{y+2} -y\] Find the greatest value and the smallest value of: \[P=x+y\]

The largest integer not exceeding $[(n+1)a]-[na]$ where $n$ is a natural number, $a=\frac{\sqrt{2013}}{\sqrt{2014}}$ is: (A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.

Let $a_1,a_2,...,a_n$ and $b_1,b_2,...,b_n$ be real numbers between $1001$ and $2002$ inclusive. Suppose $ \sum_{i=1}^n a_i^2= \sum_{i=1}^n b_i^2$. Prove that $$\sum_{i=1}^n\frac{a_i^3}{b_i} \le \frac{17}{10} \sum_{i=1}^n a_i^2$$ Determine when equality holds.

If positive integers $p,q,r$ are such that the quadratic equation $px^2-qx+r=0$ has two distinct real roots in the open interval $(0,1)$, find the minimum value of $p$.

Suppose that n numbers $x_1, x_2, . . . , x_n$ are chosen randomly from the set $\{1, 2, 3, 4, 5\}$. Prove that the probability that $x_1^2+ x_2^2 +\cdots+ x_n^2 \equiv 0 \pmod 5$ is at least $\frac 15.$

Prove that for any integer $n$ one can find integers $a$ and $b$ such that \[n=\left[ a\sqrt{2}\right]+\left[ b\sqrt{3}\right] \]

Let $\overline{a_{n}a_{n-1}\ldots a_{1}a_{0}}$ be the decimal representation of a prime positive integer such that $n>1$ and $a_{n}>1$. Prove that the polynomial $P(x)=a_{n}x^{n}+\ldots +a_{1}x+a_{0}$ cannot be written as a product of two non-constant integer polynomials.

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] What is $2018 - 3018 + 4018$? [b]p2.[/b] What is the smallest integer greater than $100$ that is a multiple of both $6$ and $8$? [b]p3.[/b] What positive real number can be expressed as both $\frac{b}{a}$ and $a:b$ in base $10$ for nonzero digits $a$ and $b$? Express your answer as a decimal. [b]p4.[/b] A non-degenerate triangle has sides of lengths $1$, $2$, and $\sqrt{n}$, where $n$ is a positive integer. How many possible values of $n$ are there? [b]p5.[/b] When three integers are added in pairs, and the results are $20$, $18$, and $x$. If all three integers sum to $31$, what is $x$? [b]p6.[/b] A cube's volume in cubic inches is numerically equal to the sum of the lengths of all its edges, in inches. Find the surface area of the cube, in square inches. [b]p7.[/b] A $12$ hour digital clock currently displays$ 9 : 30$. Ignoring the colon, how many times in the next hour will the clock display a palindrome (a number that reads the same forwards and backwards)? [b]p8.[/b] SeaBay, an online grocery store, offers two different types of egg cartons. Small egg cartons contain $12$ eggs and cost $3$ dollars, and large egg cartons contain $18$ eggs and cost $4$ dollars. What is the maximum number of eggs that Farmer James can buy with $10$ dollars? [b]p9.[/b] What is the sum of the $3$ leftmost digits of $\underbrace{999...9}_{2018\,\,\ 9' \,\,s}\times 12$? [b]p10.[/b] Farmer James trisects the edges of a regular tetrahedron. Then, for each of the four vertices, he slices through the plane containing the three trisection points nearest to the vertex. Thus, Farmer James cuts off four smaller tetrahedra, which he throws away. How many edges does the remaining shape have? [b]p11.[/b] Farmer James is ordering takeout from Kristy's Krispy Chicken. The base cost for the dinner is $\$14.40$, the sales tax is $6.25\%$, and delivery costs $\$3.00$ (applied after tax). How much did Farmer James pay, in dollars? [b]p12.[/b] Quadrilateral $ABCD$ has $ \angle ABC = \angle BCD = \angle BDA = 90^o$. Given that $BC = 12$ and $CD = 9$, what is the area of $ABCD$? [b]p13.[/b] Farmer James has $6$ cards with the numbers $1-6$ written on them. He discards a card and makes a $5$ digit number from the rest. In how many ways can he do this so that the resulting number is divisible by $6$? [b]p14.[/b] Farmer James has a $5 \times 5$ grid of points. What is the smallest number of triangles that he may draw such that each of these $25$ points lies on the boundary of at least one triangle? [b]p15.[/b] How many ways are there to label these $15$ squares from $1$ to $15$ such that squares $1$ and $2$ are adjacent, squares $2$ and $3$ are adjacent, and so on? [img]https://cdn.artofproblemsolving.com/attachments/e/a/06dee288223a16fbc915f8b95c9e4f2e4e1c1f.png[/img] [b]p16.[/b] On Farmer James's farm, there are three henhouses located at $(4, 8)$, $(-8,-4)$, $(8,-8)$. Farmer James wants to place a feeding station within the triangle formed by these three henhouses. However, if the feeding station is too close to any one henhouse, the hens in the other henhouses will complain, so Farmer James decides the feeding station cannot be within 6 units of any of the henhouses. What is the area of the region where he could possibly place the feeding station? [b]p17.[/b] At Eggs-Eater Academy, every student attends at least one of $3$ clubs. $8$ students attend frying club, $12$ students attend scrambling club, and $20$ students attend poaching club. Additionally, $10$ students attend at least two clubs, and $3$ students attend all three clubs. How many students are there in total at Eggs-Eater Academy? [b]p18.[/b] Let $x, y, z$ be real numbers such that $8^x = 9$, $27^y = 25$, and $125^z = 128$. What is the value of $xyz$? [b]p19.[/b] Let $p$ be a prime number and $x, y$ be positive integers. Given that $9xy = p(p + 3x + 6y)$, find the maximum possible value of $p^2 + x^2 + y^2$. [b]p20.[/b] Farmer James's hens like to drop eggs. Hen Hao drops $6$ eggs uniformly at random in a unit square. Farmer James then draws the smallest possible rectangle (by area), with sides parallel to the sides of the square, that contain all $6$ eggs. What is the probability that at least one of the $6$ eggs is a vertex of this rectangle? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that the first 3 digits after the decimal point in the decimal expression of the number $\frac{0.123456789101112 . . . 495051}{0.515049 . . . 121110987654321}$ are $239$.

Determine if there are triplets ($x,y,z)$ of real numbers such that $$\begin{cases} x+y+z=7 \\ xy+yz+zx=11\end{cases}$$ If the answer is affirmative, find the minimum and maximum values ​​of $z$ in such a triplet.

Let real numbers $x_1, x_2, \cdots , x_n$ satisfy $0 < x_1 < x_2 < \cdots< x_n < 1$ and set $x_0 = 0, x_{n+1} = 1$. Suppose that these numbers satisfy the following system of equations: \[\sum_{j=0, j \neq i}^{n+1} \frac{1}{x_i-x_j}=0 \quad \text{where } i = 1, 2, . . ., n.\] Prove that $x_{n+1-i} = 1- x_i$ for $i = 1, 2, . . . , n.$

Compute the sum of the real solutions to $\lfloor x \rfloor \{x\} = 2020x$. Here, $\lfloor x \rfloor$ is defined as the greatest integer less than or equal to $x$, and$ \{x\} = x -\lfloor x \rfloor$.

Prove that among the elements of the sequence $\left( \left\lfloor n \sqrt 2 \right\rfloor + \left\lfloor n \sqrt 3 \right\rfloor \right)_{n \geq 0}$ are an infinity of even numbers and an infinity of odd numbers.

$\definecolor{A}{RGB}{190,0,60}\color{A}\fbox{A1.}$ Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $$\definecolor{A}{RGB}{80,0,200}\color{A} x^4+y^4+z^4\ge f(xy)+f(yz)+f(zx)\ge xyz(x+y+z)$$holds for all $a,b,c\in\mathbb{R}$. [i]Proposed by [/i][b][color=#FFFF00]usjl[/color][/b]. [color=#B6D7A8]#1733[/color]

Let $G$ be a finite group and $f: G \to G$ a map, such that $f (xy) = f (x) f (y)$ for all $x, y \in G$ and $f (x) = x^{-1}$ for more than $\frac34$ of all $x \in G$ is fulfilled. Show that $f (x) =x^{-1}$ even holds for all $x \in G$ holds.

The following figure shows a [i]walk[/i] of length 6: [asy] unitsize(20); for (int x = -5; x <= 5; ++x) for (int y = 0; y <= 5; ++y) dot((x, y)); label("$O$", (0, 0), S); draw((0, 0) -- (1, 0) -- (1, 1) -- (0, 1) -- (-1, 1) -- (-1, 2) -- (-1, 3)); [/asy] This walk has three interesting properties: [list] [*] It starts at the origin, labelled $O$. [*] Each step is 1 unit north, east, or west. There are no south steps. [*] The walk never comes back to a point it has been to.[/list] Let's call a walk with these three properties a [i]northern walk[/i]. There are 3 northern walks of length 1 and 7 northern walks of length 2. How many northern walks of length 6 are there?

The polynomial $x^8 +x^7$ is written on a blackboard. In a move, Peter can erase the polynomial $P(x)$ and write down $(x+1)P(x)$ or its derivative $P'(x).$ After a while, the linear polynomial $ax+b$ with $a\ne 0$ is written on the board. Prove that $a-b$ is divisible by $49.$

Find all pairs of integers $(m, n)$ which satisfy the equation \[(2n^2+5m-5n-mn)^2=m^3n\]

Let $x, y, z$ be positive real numbers such that $\frac{1}{1 + x}+\frac{1}{1 + y}+\frac{1}{1 + z}= 2$. Prove that $8xyz \le 1$.

Let $ f(x)$ be a polynomial and $ C$ be a real number. Find the $ f(x)$ and $ C$ such that $ \int_0^x f(y)dy\plus{}\int_0^1 (x\plus{}y)^2f(y)dy\equal{}x^2\plus{}C$.

Solve the system of equations: \[ x+y+z=a \] \[ x^2+y^2+z^2=b^2 \] \[ xy=z^2 \] where $a$ and $b$ are constants. Give the conditions that $a$ and $b$ must satisfy so that $x,y,z$ are distinct positive numbers.

In the coordinate plane, define $M = \{(a, b),a,b \in Z\}$. A transformation $S$, which is defined on $M$, sends $(a,b)$ to $(a + b, b)$. Transformation $T$, also defined on $M$, sends $(a, b)$ to $(-b, a)$. Prove that for all $(a, b) \in M$, we can use $S,T$ denitely to map it to $(g,0)$.