Find all functions $f : \mathbb{Z}_{+} \rightarrow \mathbb{Z}_{+}$ that satisfy $f(mf(n)) = n+f(2015m)$ for all $m,n \in \mathbb{Z}_{+}$.

Let $S = \{1,2,3,\ldots,n\}$. Consider a function $f\colon S\to S$. A subset $D$ of $S$ is said to be invariant if for all $x\in D$ we have $f(x)\in D$. The empty set and $S$ are also considered as invariant subsets. By $\deg (f)$ we define the number of invariant subsets $D$ of $S$ for the function $f$. [b]i)[/b] Show that there exists a function $f\colon S\to S$ such that $\deg (f)=2$. [b]ii)[/b] Show that for every $1\leq k\leq n$ there exists a function $f\colon S\to S$ such that $\deg (f)=2^{k}$.

For every positive integer $n$, let $\operatorname{mod_5}(n)$ be the remainder obtained when $n$ is divided by $5$. Define a function $f : \{0, 1, 2, 3, \dots\} \times \{0, 1, 2, 3, 4\} \to \{0, 1, 2, 3, 4\}$ recursively as follows: \[f(i, j) = \begin{cases} \operatorname{mod_5}(j+1) & \text{if }i=0\text{ and }0\leq j\leq 4 \\ f(i-1, 1) & \text{if }i\geq 1\text{ and }j=0 \text{, and}\\ f(i-1, f(i, j-1)) & \text{if }i\geq 1\text{ and }1\leq j\leq 4 \end{cases}\] What is $f(2015, 2)$? $\textbf{(A) }0 \qquad\textbf{(B) }1 \qquad\textbf{(C) }2 \qquad\textbf{(D) }3 \qquad\textbf{(E) }4$

Find the largest real number $k$, such that for any positive real numbers $a,b$, $$(a+b)(ab+1)(b+1)\geq kab^2$$

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

[i]In this round, problems will depend on the answers to other problems. A bolded letter is used to denote a quantity whose value is determined by another problem's answer.[/i] [u]Part I[/u] [b]p1.[/b] Let F be the answer to problem number $6$. You want to tile a nondegenerate square with side length $F$ with $1\times 2$ rectangles and $1 \times 1$ squares. The rectangles can be oriented in either direction. How many ways can you do this? [b]p2.[/b] Let [b]A[/b] be the answer to problem number $1$. Triangle $ABC$ has a right angle at $B$ and the length of $AC$ is [b]A[/b]. Let $D$ be the midpoint of $AB$, and let $P$ be a point inside triangle $ABC$ such that $PA = PC = \frac{7\sqrt5}{4}$ and $PD = \frac74$ . The length of $AB^2$ is expressible as $m/n$ for $m, n$ relatively prime positive integers. Find $m$. [b]p3.[/b] Let [b]B[/b] be the answer to problem number $2$. Let $S$ be the set of positive integers less than or equal to [b]B[/b]. What is the maximum size of a subset of $S$ whose elements are pairwise relatively prime? [b]p4.[/b] Let [b]C[/b] be the answer to problem number $3$. You have $9$ shirts and $9$ pairs of pants. Each is either red or blue, you have more red shirts than blue shirts, and you have same number of red shirts as blue pants. Given that you have [b]C[/b] ways of wearing a shirt and pants whose colors match, find out how many red shirts you own. [b]p5.[/b] Let [b]D[/b] be the answer to problem number $4$. You have two odd positive integers $a, b$. It turns out that $lcm(a, b) + a = gcd(a, b) + b =$ [b]D[/b]. Find $ab$. [b]p6.[/b] Let [b]E[/b] be the answer to problem number $5$. A function $f$ defined on integers satisfies $f(y)+f(12-y) = 10$ and $f(y) + f(8 - y) = 4$ for all integers $y$. Given that $f($ [b]E[/b] $) = 0$, compute $f(4)$. [u]Part II[/u] [b]p7.[/b] Let [b]L[/b] be the answer to problem number $12$. You want to tile a nondegenerate square with side length [b]L[/b] with $1\times 2$ rectangles and $7\times 7$ squares. The rectangles can be oriented in either direction. How many ways can you do this? [b]p8.[/b] Let [b]G[/b] be the answer to problem number $7$. Triangle $ABC$ has a right angle at $B$ and the length of $AC$ is [b]G[/b]. Let $D$ be the midpoint of $AB$, and let $P$ be a point inside triangle $ABC$ such that $PA = PC = \frac12$ and $PD = \frac{1}{2010}$ . The length of $AB^2$ is expressible as $m/n$ for $m, n$ relatively prime positive integers. Find $m$. [b]p9.[/b] Let [b]H[/b] be the answer to problem number $8$. Let $S$ be the set of positive integers less than or equal to [b]H[/b]. What is the maximum size of a subset of $S$ whose elements are pairwise relatively prime? [b]p10.[/b] Let [b]I[/b] be the answer to problem number $9$. You have $391$ shirts and $391$ pairs of pants. Each is either red or blue, you have more red shirts than blue shirts, and you have same number of red shirts as red pants. Given that you have [b]I[/b] ways of wearing a shirt and pants whose colors match, find out how many red shirts you own. [b]p11.[/b] Let [b]J[/b] be the answer to problem number $10$. You have two odd positive integers $a, b$. It turns out that $lcm(a, b) + 2a = 2 gcd(a, b) + b = $ [b]J[/b]. Find $ab$. [b]p12.[/b] Let [b]K[/b] be the answer to problem number $11$. A function $f$ defined on integers satisfies $f(y)+f(7-y) = 8$ and $f(y) + f(5 - y) = 4$ for all integers $y$. Given that $f($ [b]K[/b] $) = 453$, compute $f(2)$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The time-varying temperature of a certain body is given by a polynomial in the time of degree at most three. Show that the average temperature of the body between $9$ am and $3$ pm can always be found by taking the average of the temperatures at two fixed times, which are independent of the polynomial. Also, show that these two times are $10\colon \! 16$ am and $1\colon \!44$ pm to the nearest minute.

Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that $ f(x+y)+f(x)f(y)=f(xy)+2xy+1$ for all real numbers $ x$ and $ y$. [i]Proposed by B.J. Venkatachala, India[/i]

Find all functions $ f:\mathbb{R^{\ast }}\rightarrow \mathbb{ R^{\ast }}$ satisfying $f(\frac{f(x)}{f(y)})=\frac{1}{y}f(f(x))$ for all $x,y\in \mathbb{R^{\ast }}$ and are strictly monotone in $(0,+\infty )$

For all positive real numbers $a, b,c.$ Prove the folllowing inequality$$\frac{a}{c+5b}+\frac{b}{a+5c}+\frac{c}{b+5a}\geq\frac{1}{2}.$$

Let $a, b, c,$ be nonnegative reals with $ a+b+c=3 $, find the largest positive real $ k $ so that for all $a,b,c,$ we have $$ a^2+b^2+c^2+k(abc-1)\ge 3 $$

A number of runners competed in a race. When Ammar finished, there were half as many runners who had finished before him compared to the number who finished behind him. Julia was the 10th runner to finish behind Ammar. There were twice as many runners who had finished before Julia compared to the number who finished behind her. How many runners were there in the race?

Let $x$- minimal root of equation $x^2-4x+2=0$. Find two first digits of number $ \{x+x^2+....+x^{20} \}$ after $0$, where $\{a\}$- fractional part of $a$.

Solve the system of equations: $ y^2\equal{}(x\plus{}8)(x^2\plus{}2),$ $ y^2\minus{}(8\plus{}4x)y\plus{}(16\plus{}16x\minus{}5x^2)\equal{}0.$

Find real roots $x_1$, $x_2$ of equation $x^5-55x+21=0$, if we know $x_1\cdot x_2=1$

Let $A$ and $B$ be nonempty sets and let $f : A \to B$. Prove that the following statements are equivalent: $\text{(i) }$ $f$ is surjective. $\text{(ii)} $ For every set $C$ and and every functions $g, h : B \to C$, if $g\circ f = h \circ f$ then $g = h$.

Find all possible values of $a\in \mathbb{R}$ and $n\in \mathbb{N^*}$ such that $f(x)=(x-1)^n+(x-2)^{2n+1}+(1-x^2)^{2n+1}+a$ is divisible by $\phi (x)=x^2-x+1$

Find all triples $(a, b, c)$ of real numbers such that $ab + bc + ca = 1$ and $$a^2b + c = b^2c + a = c^2a + b.$$

What is the largest integer not exceeding $8x^3 +6x - 1$, where $x =\frac12 \left(\sqrt[3]{2+\sqrt5} + \sqrt[3]{2-\sqrt5}\right)$ ? (A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.

Let $n\geqslant 2$ be a positive integer and $a_1,a_2, \ldots ,a_n$ be real numbers such that \[a_1+a_2+\dots+a_n=0.\] Define the set $A$ by \[A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}\] Prove that, if $A$ is not empty, then \[\sum_{(i, j) \in A} a_{i} a_{j}<0.\]

Suppose that for two real numbers $x$ and $y$ the following equality is true: $$(x+ \sqrt{1+ x^2})(y+\sqrt{1+y^2})=1$$ Find (with proof) the value of $x+y$.

Let $r$, $s$, $t$ be the roots of the polynomial $x^3+2x^2+x-7$. Then \[ \left(1+\frac{1}{(r+2)^2}\right)\left(1+\frac{1}{(s+2)^2}\right)\left(1+\frac{1}{(t+2)^2}\right)=\frac{m}{n} \] for relatively prime positive integers $m$ and $n$. Compute $100m+n$. [i]Proposed by Justin Stevens[/i]

Natural number $k$ has $n$ digits in its decimal notation. It was rounded up to tens, then the obtained number was rounded up to hundreds, and so on $(n-1)$ times. Prove that the obtained number $m$ satisfies inequality $m < \frac{18k}{13}$. (Examples of rounding: $191\to190\to 200, 135\to140\to 100$.)

Determine all $4$-tuples $(a,b, c, d)$ of positive real numbers satisfying $a + b +c + d = 1$ and $\max (\frac{a^2}{b},\frac{b^2}{a}) \cdot \max (\frac{c^2}{d},\frac{d^2}{c}) = (\min (a + b, c + d))^4$

For the system of equations $x^2 + x^2y^2 + x^2y^4 = 525$ and $x + xy + xy^2 = 35$, the sum of the real $y$ values that satisfy the equations is $ \mathrm{(A) \ } 2 \qquad \mathrm{(B) \ } \frac{5}{2} \qquad \mathrm {(C) \ } 5 \qquad \mathrm{(D) \ } 20 \qquad \mathrm{(E) \ } \frac{55}{2}$