Let $f(x)=\frac{P(x)}{Q(x)}$, where $P(x), Q(x)$ are two non-constant polynomials with no common zeros and $P(0)=P(1)=0$. Suppose $f(x)f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)$ for infinitely many values of $x$. a) Show that $\text{deg}(P)<\text{deg}(Q)$. b) Show that $P'(1)=2Q'(1)-\text{deg}(Q)\cdot Q(1)$. Here, $P'(x)$ denotes the derivative of $P(x)$ as usual.

Determine all functions $f:\mathbb R\rightarrow\mathbb R$ satisfying the condition \[f(y^2+2xf(y)+f(x)^2)=(y+f(x))(x+f(y))\] for all real numbers $x$ and $y$.

Find the polynomial $f$ with the following properties: $\bullet$ its leading coefficient is $1$, $\bullet$ its coefficients are nonnegative integers, $\bullet$ $72|f(x)$ if $x$ is an integer, $\bullet$ if $g$ is another polynomial with the same properties, then $g - f$ has a nonnegative leading coecient.

Prove that: there exists only one function $f:\mathbb{N^*}\to\mathbb{N^*}$ satisfying: i) $f(1)=f(2)=1$; ii)$f(n)=f(f(n-1))+f(n-f(n-1))$ for $n\ge 3$. For each integer $m\ge 2$, find the value of $f(2^m)$.

(a) Prove that $$3-\frac{2}{(n-1)!} < \frac{2^2-2}{2!}+\frac{2^2-2}{3!}+...+\frac{n^2-2}{n!}<3$$ (b) Find some positive integers $a$, $b$ and $c$ such that for any $n > 2$, $$b-\frac{c}{(n-2)!} < \frac{2^3-a}{2!}+\frac{3^3-a}{3!}+...+\frac{n^3-a}{n!}<b$$ (V Senderov, NB Vassiliev)

Prove that for any positive integer $ n$, there exists only $ n$ degree polynomial $ f(x),$ satisfying $ f(0) \equal{} 1$ and $ (x \plus{} 1)[f(x)]^2 \minus{} 1$ is an odd function.

We know that $k$ is a positive integer and the equation \[ x^3+y^3-2y(x^2-xy+y^2)=k^2(x-y) \quad (1) \] has one solution $(x_0,y_0)$ with $x_0,y_0\in \mathbb{Z}-\{0\}$ and $x_0\neq y_0$. Prove that i) the equation (1) has a finite number of solutions $(x,y)$ with $x,y\in \mathbb{Z}$ and $x\neq y$; ii) it is possible to find $11$ addition different solutions $(X,Y)$ of the equation (1) with $X,Y\in \mathbb{Z}-\{0\}$ and $X\neq Y$ where $X,Y$ are functions of $x_0,y_0$.

Find all tuples $ (x_1,x_2,x_3,x_4,x_5)$ of positive integers with $ x_1>x_2>x_3>x_4>x_5>0$ and $ {\left \lfloor \frac{x_1+x_2}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_2+x_3}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_3+x_4}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_4+x_5}{3} \right \rfloor }^2 = 38.$

Find, with proof, all polynomials $f$ such that $f$ has nonnegative integer coefficients, $f$($1$) = $8$ and $f$($2$) = $2012$.

Let $\Omega=\{z=x+iy~\in\mathbb{C}~:~|y|\leqslant 1\}$. If $f(z)=z^2+2$, then draw a sketch of $$f\Big(\Omega\Big)=\{f(z):z\in\Omega\}$$ Justify your answer.

Solve equation $x \lfloor{x}\rfloor+\{x\}=2018$, where $x$ is real number

[b]p1.[/b] What is the sum of the first $12$ positive integers? [b]p2.[/b] How many positive integers less than or equal to $100$ are multiples of both $2$ and $5$? [b]p3. [/b]Alex has a bag with $4$ white marbles and $4$ black marbles. She takes $2$ marbles from the bag without replacement. What is the probability that both marbles she took are black? Express your answer as a decimal or a fraction in lowest terms. [b]p4.[/b] How many $5$-digit numbers are there where each digit is either $1$ or $2$? [b]p5.[/b] An integer $a$ with $1\le a \le 10$ is randomly selected. What is the probability that $\frac{100}{a}$ is an integer? Express your answer as decimal or a fraction in lowest terms. [b]p6.[/b] Two distinct non-tangent circles are drawn so that they intersect each other. A third circle, distinct from the previous two, is drawn. Let $P$ be the number of points of intersection between any two circles. How many possible values of $P$ are there? [b]p7.[/b] Let $x, y, z$ be nonzero real numbers such that $x + y + z = xyz$. Compute $$\frac{1 + yz}{yz}+\frac{1 + xz}{xz}+\frac{1 + xy}{xy}.$$ [b]p8.[/b] How many positive integers less than $106$ are simultaneously perfect squares, cubes, and fourth powers? [b]p9.[/b] Let $C_1$ and $C_2$ be two circles centered at point $O$ of radii $1$ and $2$, respectively. Let $A$ be a point on $C_2$. We draw the two lines tangent to $C_1$ that pass through $A$, and label their other intersections with $C_2$ as $B$ and $C$. Let x be the length of minor arc $BC$, as shown. Compute $x$. [img]https://cdn.artofproblemsolving.com/attachments/7/5/915216d4b7eba0650d63b26715113e79daa176.png[/img] [b]p10.[/b] A circle of area $\pi$ is inscribed in an equilateral triangle. Find the area of the triangle. [b]p11.[/b] Julie runs a $2$ mile route every morning. She notices that if she jogs the route $2$ miles per hour faster than normal, then she will finish the route $5$ minutes faster. How fast (in miles per hour) does she normally jog? [b]p12.[/b] Let $ABCD$ be a square of side length $10$. Let $EFGH$ be a square of side length $15$ such that $E$ is the center of $ABCD$, $EF$ intersects $BC$ at $X$, and $EH$ intersects $CD$ at $Y$ (shown below). If $BX = 7$, what is the area of quadrilateral $EXCY$ ? [img]https://cdn.artofproblemsolving.com/attachments/d/b/2b2d6de789310036bc42d1e8bcf3931316c922.png[/img] [b]p13.[/b] How many solutions are there to the system of equations $$a^2 + b^2 = c^2$$ $$(a + 1)^2 + (b + 1)^2 = (c + 1)^2$$ if $a, b$, and $c$ are positive integers? [b]p14.[/b] A square of side length $ s$ is inscribed in a semicircle of radius $ r$ as shown. Compute $\frac{s}{r}$. [img]https://cdn.artofproblemsolving.com/attachments/5/f/22d7516efa240d00d6a9743a4dc204d23d190d.png[/img] [b]p15.[/b] $S$ is a collection of integers n with $1 \le n \le 50$ so that each integer in $S$ is composite and relatively prime to every other integer in $S$. What is the largest possible number of integers in $S$? [b]p16.[/b] Let $ABCD$ be a regular tetrahedron and let $W, X, Y, Z$ denote the centers of faces $ABC$, $BCD$, $CDA$, and $DAB$, respectively. What is the ratio of the volumes of tetrahedrons $WXYZ$ and $WAYZ$? Express your answer as a decimal or a fraction in lowest terms. [b]p17.[/b] Consider a random permutation $\{s_1, s_2, ... , s_8\}$ of $\{1, 1, 1, 1, -1, -1, -1, -1\}$. Let $S$ be the largest of the numbers $s_1$, $s_1 + s_2$, $s_1 + s_2 + s_3$, $...$ , $s_1 + s_2 + ... + s_8$. What is the probability that $S$ is exactly $3$? Express your answer as a decimal or a fraction in lowest terms. [b]p18.[/b] A positive integer is called [i]almost-kinda-semi-prime[/i] if it has a prime number of positive integer divisors. Given that there $are 168$ primes less than $1000$, how many almost-kinda-semi-prime numbers are there less than $1000$? [b]p19.[/b] Let $ABCD$ be a unit square and let $X, Y, Z$ be points on sides $AB$, $BC$, $CD$, respectively, such that $AX = BY = CZ$. If the area of triangle $XYZ$ is $\frac13$ , what is the maximum value of the ratio $XB/AX$? [img]https://cdn.artofproblemsolving.com/attachments/5/6/cf77e40f8e9bb03dea8e7e728b21e7fb899d3e.png[/img] [b]p20.[/b] Positive integers $a \le b \le c$ have the property that each of $a + b$, $b + c$, and $c + a$ are prime. If $a + b + c$ has exactly $4$ positive divisors, find the fourth smallest possible value of the product $c(c + b)(c + b + a)$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a \in \mathbb R$ and let $z_1, z_2, \ldots, z_n$ be complex numbers of modulus $1$ satisfying the relation \[\sum_{k=1}^n z_k^3=4(a+(a-n)i)- 3 \sum_{k=1}^n \overline{z_k}\] Prove that $a \in \{0, 1,\ldots, n \}$ and $z_k \in \{1, i \}$ for all $k.$

p1. If $x + y + z = 2$, show that $\frac{1}{xy+z-1}+\frac{1}{yz+x-1}+\frac{1}{xz+y-1}=\frac{-1}{(x-1)(y-1)(z-1)}$. p2. Determine the simplest form of $\frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!}+\frac{5}{3!+4!+5!}+...+\frac{100}{98!+99!+100!}$ p3. It is known that $ABCD$ and $DEFG$ are two parallelograms. Point $E$ lies on $AB$ and point $C$ lie on $FG$. The area of $​​ABCD$ is $20$ units. $H$ is the point on $DG$ so that $EH$ is perpendicular to $DG$. If the length of $DG$ is $5$ units, determine the length of $EH$. [img]https://cdn.artofproblemsolving.com/attachments/b/e/42453bf6768129ed84fbdc81ab7235e780b0e1.png[/img] p4. Each room in the following picture will be painted so that every two rooms which is directly connected to the door is given a different color. If $10$ different colors are provided and $4$ of them can not be used close together for two rooms that are directly connected with a door, determine how many different ways to color the $4$ rooms. [img]https://cdn.artofproblemsolving.com/attachments/4/a/e80a464a282b3fe3cdadde832b2fd38b51a41a.png[/img] 5. The floor of a hall is rectangular $ABCD$ with $AB = 30$ meters and $BC = 15$ meters. A cat is in position $A$. Seeing the cat, the mouse in the midpoint of $AB$ ran and tried to escape from cat. The mouse runs from its place to point $C$ at a speed of $3$ meters/second. The trajectory is a straight line. Watching the mice run away, at the same time from point $A$ the cat is chasing with a speed of $5$ meters/second. If the cat's path is also a straight line and the mouse caught before in $C$, determine an equation that can be used for determine the position and time the mouse was caught by the cat.

Let \( n > 2 \) be an integer, \( k > 1 \) a real number, and \( x_1, x_2, \ldots, x_n \) be positive real numbers such that \( x_1 \cdot x_2 \cdots x_n = 1 \). Prove that: \[ \frac{1 + x_1^k}{1 + x_2} + \frac{1 + x_2^k}{1 + x_3} + \cdots + \frac{1 + x_n^k}{1 + x_1} \geq n. \] When does equality hold?

Given $2025$ pairwise distinct positive integer numbers \(a_1, a_2, \ldots, a_{2025}\), find the maximum possible number of equal numbers among the fractions of the form \[ \frac{a_i^2 + a_j^2}{a_i + a_j} \] [i]Proposed by Mykhailo Shtandenko[/i]

Find all positive real solutions to: \begin{eqnarray*} (x_1^2-x_3x_5)(x_2^2-x_3x_5) &\le& 0 \\ (x_2^2-x_4x_1)(x_3^2-x_4x_1) &\le& 0 \\ (x_3^2-x_5x_2)(x_4^2-x_5x_2) &\le& 0 \\ (x_4^2-x_1x_3)(x_5^2-x_1x_3) &\le & 0 \\ (x_5^2-x_2x_4)(x_1^2-x_2x_4) &\le& 0 \\ \end{eqnarray*}

Suppose $A\subseteq \mathbb R^m$ is closed and non-empty. Let $f:A\to A$ is a lipchitz function with constant less than 1. (ie there exist $c<1$ that $|f(x)-f(y)|<c|x-y|,\ \forall x,y \in A)$. Prove that there exists a unique point $x\in A$ such that $f(x)=x$.

Hi guys, I was just reading over old posts that I made last year ( :P ) and saw how much the level of Getting Started became harder. To encourage more people from posting, I decided to start a Problem of the Day. This is how I'll conduct this: 1. In each post (not including this one since it has rules, etc) everyday, I'll post the problem. I may post another thread after it to give hints though. 2. Level of problem.. This is VERY important. All problems in this thread will be all AHSME or problems similar to this level. No AIME. Some AHSME problems, however, that involve tough insight or skills will not be posted. The chosen problems will be usually ones that everyone can solve after working. Calculators are allowed when you solve problems but it is NOT necessary. 3. Response.. All you have to do is simply solve the problem and post the solution. There is no credit given or taken away if you get the problem wrong. This isn't like other threads where the number of problems you get right or not matters. As for posting, post your solutions here in this thread. Do NOT PM me. Also, here are some more restrictions when posting solutions: A. No single answer post. It doesn't matter if you put hide and say "Answer is ###..." If you don't put explanation, it simply means you cheated off from some other people. I've seen several posts that went like "I know the answer" and simply post the letter. What is the purpose of even posting then? Huh? B. Do NOT go back to the previous problem(s). This causes too much confusion. C. You're FREE to give hints and post different idea, way or answer in some cases in problems. If you see someone did wrong or you don't understand what they did, post here. That's what this thread is for. 4. Main purpose.. This is for anyone who visits this forum to enjoy math. I rememeber when I first came into this forum, I was poor at math compared to other people. But I kindly got help from many people such as JBL, joml88, tokenadult, and many other people that would take too much time to type. Perhaps without them, I wouldn't be even a moderator in this forum now. This site clearly made me to enjoy math more and more and I'd like to do the same thing. That's about the rule.. Have fun problem solving! Next post will contain the Day 1 Problem. You can post the solutions until I post one. :D

Suppose that a function $f:\mathbb R\to\mathbb R$ is such that for any real $h$ there exist at most $19990509$ different values of $x$ for which $f(x)\ne f(x+h)$. Prove that there is a set of at most $9995256$ real numbers such that $f$ is constant outside of this set.

Let $a_0$ be a positive integer and $a_{n + 1} =\sqrt{a_n^2 + 1}$, for all $n \ge 0$. 1) Prove that for all $a_0$ the sequence contains infinitely many integers and infinitely many irrational numbers. 2) Is there an $a_0$ for which $a_{2010}$ is an integer?

Find all the polynomials with real coefficients which satisfy $ (x^2-6x+8)P(x)=(x^2+2x)P(x-2)$ for all $x\in \mathbb{R}$.

9. The real quartic $P x^{4}+U x^{3}+M x^{2}+A x+C$ has four different positive real roots. Find the square of the smallest real number $z$ for which the expression $M^{2}-2 U A+z P C$ is always positive, regardless of what the roots of the quartic are.

Let $n \geq 2, n \in \mathbb{N},$ find the least positive real number $\lambda$ such that for arbitrary $a_i \in \mathbb{R}$ with $i = 1, 2, \ldots, n$ and $b_i \in \left[0, \frac{1}{2}\right]$ with $i = 1, 2, \ldots, n$, the following holds: \[\sum^n_{i=1} a_i = \sum^n_{i=1} b_i = 1 \Rightarrow \prod^n_{i=1} a_i \leq \lambda \sum^n_{i=1} a_i b_i.\]

Consider the polynomial $ P(x) \equal{} x^3 \plus{} 153x^2 \minus{} 111x \plus{} 38$. (a) Prove that there are at least nine integers $ a$ in the interval $ [1, 3^{2000}]$ for which $ P(a)$ is divisible by $ 3^{2000}$. (b) Find the number of integers $ a$ in $ [1, 3^{2000}]$ with the property from (a).