Found problems: 4776
Find all functions $ f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that:
$ f(x\plus{}f(y))\equal{}y\plus{}f(x)$ for all $ x,y \in \mathbb{Q}$.
Let $k$ be an integer. Determine all functions $f\colon \mathbb{R}\to\mathbb{R}$ with $f(0)=0$ and \[f(x^ky^k)=xyf(x)f(y)\qquad \mbox{for } x,y\neq 0.\]
Let $a_1,a_2,\ldots,a_n$ are real numbers, prove that;
\[\sum_{i=1}^na_i^2-\sum_{i=1}^n a_i a_{i+1} \le \left\lfloor \frac{n}{2}\right\rfloor(M-m)^2.\]
where $a_{n+1}=a_1,M=\max_{1\le i\le n} a_i,m=\min_{1\le i\le n} a_i$.
Find all integer $c\in\{0,1,...,2016\}$ such that the number of $f:\mathbb{Z}\rightarrow\{0,1,...,2016\}$ which satisfy the following condition is minimal:\\
(1) $f$ has periodic $2017$\\
(2) $f(f(x)+f(y)+1)-f(f(x)+f(y))\equiv c\pmod{2017}$\\
Proposed by William Chao
Find the values of real numbers $a,\ b$ for which the function $f(x)=a|\cos x|+b|\sin x|$ has local minimum at $x=-\frac{\pi}{3}$ and satisfies $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\{f(x)\}^{2}dx=2$.
Find all functions $f: \mathbb R \to \mathbb R$ such that for all reals $x$ and $y$,
\[f(x+y)+f(x)f(y)=f(xy)+f(x)+f(y).\]
Let $a,b,c,$ and $d$ be real numbers such that at least one of $c$ and $d$ is non-zero. Let $ f:\mathbb{R}\to\mathbb{R}$ be a function defined as $f(x)=\frac{ax+b}{cx+d}$. Suppose that for all $x\in\mathbb{R}$, we have $f(x) \neq x$. Prove that if there exists some real number $a$ for which $f^{1387}(a)=a$, then for all $x$ in the domain of $f^{1387}$, we have $f^{1387}(x)=x$. Notice that in this problem,
\[f^{1387}(x)=\underbrace{f(f(\cdots(f(x)))\cdots)}_{\text{1387 times}}.\]
[i]Hint[/i]. Prove that for every function $g(x)=\frac{sx+t}{ux+v}$, if the equation $g(x)=x$ has more than $2$ roots, then $g(x)=x$ for all $x\in\mathbb{R}-\left\{\frac{-v}{u}\right\}$.
Find all nondecreasing functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the relation
$$ f\left( f\left( x^2 \right) +y+f(y) \right) =x^2+2f(y) , $$
for any real numbers $ x,y. $
The sequence $ (x_n)_{n \in \mathbb{N}}$ is defined by $ x_1\equal{}2, x_2\equal{}3,$ and
$ x_{2m\plus{}1}\equal{}x_{2m}\plus{}x_{2m\minus{}1}$ for $ m \ge 1;$
$ x_{2m}\equal{}x_{2m\minus{}1}\plus{}2x_{2m\minus{}2}$ for $ m \ge 2.$
Determine $ x_n$ as a function of $ n$.
Let $n$ be a positive integer. Prove that
\[ \binom{n}{0}^{-1} + \binom{n}{1}^{-1} + \cdots + \binom{n}{n}^{-1} = \frac{n+1}{2^{n+1}} \left( \frac{2}{1} + \frac{2^2}{2} + \cdots + \frac{2^{n+1}}{n+1} \right). \]
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
Let $\mathbb{N}_0$ denote the set of non-negative integers. Determine all non-negative integers $k$ for which there exists a function $f: \mathbb{N}_0 \to \mathbb{N}_0$ such that $f(2024) = k$ and $f(f(n)) \leq f(n+1) - f(n)$ for all non-negative integers $n$.
Find all functions $f:\mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$ such that the conditions
$\quad a) \quad a-b \mid f(a)-f(b)$ for all $a\neq b$ and $a,b \in \mathbb{Z}^{+}$
$\quad b) \quad f(\varphi(a))=\varphi(f(a))$ for all $a \in \mathbb{Z}^{+}$ where $\varphi$ is the Euler's totient function.
holds
Let $a < b$ be real numbers and let $f : (a, b) \to \mathbb{R}$ be a function such that the functions $g : (a, b) \to \mathbb{R}$, $g(x) = (x - a) f(x)$ and $h : (a, b) \to \mathbb{R}$, $h(x) = (x - b) f(x)$ are increasing. Show that the function $f$ is continuous on $(a, b)$.
(1) Evaluate $ \int_0^1 xe^{x^2}dx$.
(2) Let $ I_n\equal{}\int_0^1 x^{2n\minus{}1}e^{x^2}dx$. Express $ I_{n\plus{}1}$ in terms of $ I_n$.
For a group $\left( G, \star \right)$ and $A, B$ two non-void subsets of $G$, we define $A \star B = \left\{ a \star b : a \in A \ \text{and}\ b \in B \right\}$.
(a) Prove that if $n \in \mathbb N, \, n \geq 3$, then the group $\left( \mathbb Z \slash n \mathbb Z,+\right)$ can be writen as $\mathbb Z \slash n \mathbb Z = A+B$, where $A, B$ are two non-void subsets of $\mathbb Z \slash n \mathbb Z$ and $A \neq \mathbb Z \slash n \mathbb Z, \, B \neq \mathbb Z \slash n \mathbb Z, \, \left| A \cap B \right| = 1$.
(b) If $\left( G, \star \right)$ is a finite group, $A, B$ are two subsets of $G$ and $a \in G \setminus \left( A \star B \right)$, then prove that function $f : A \to G \setminus B$ given by $f(x) = x^{-1}\star a$ is well-defined and injective. Deduce that if $|A|+|B| > |G|$, then $G = A \star B$.
[hide="Question."]Does the last result have a name?[/hide]
Let $p(x, y)$ and $q(x, y)$ be polynomials in two variables such that for $x \ge 0, y \ge 0$ the following conditions hold:
$(i) p(x, y)$ and $q(x, y)$ are increasing functions of $x$ for every fixed $y$.
$(ii) p(x, y)$ is an increasing and $q(x)$ is a decreasing function of $y$ for every fixed $x$.
$(iii) p(x, 0) = q(x, 0)$ for every $x$ and $p(0, 0) = 0$.
Show that the simultaneous equations $p(x, y) = a, q(x, y) = b$ have a unique solution in the set $x \ge 0, y \ge 0$ for all $a, b$ satisfying $0 \le b \le a$ but lack a solution in the same set if $a < b$.
If $ f$ is a nonnegative, continuous, concave function on the closed interval $ [0,1]$ such that $ f(0)=1$, then \[ \int_0^1 xf(x)dx \leq \frac 23 \left[ %Error. "diaplaymath" is a bad command.
\int_0^1 f(x)dx \right]^2.\]
[i]Z. Daroczy[/i]
Find all positive integers $k$ for which there exists a function $f: \mathbb{N}\to\mathbb{Z}$ satisfying $f(1997)=1998$ and $f(ab)=f(a)+f(b)+kf(\gcd{(a,b)})\forall a,b$.
Prove that for all natural $n$ there exists $a,b,c$ such that $n=\gcd (a,b)(c^2-ab)+\gcd (b,c)(a^2-bc)+\gcd (c,a)(b^2-ca)$.
Let $k<<n$ denote that $k<n$ and $k\mid n$. Let $f:\{1,2,...,2013\}\rightarrow \{1,2,...,M\}$ be such that, if $n\leq 2013$ and $k<<n$, then $f(k)<<f(n)$. What’s the least possible value of $M$?
Find all strictly ascending functions $f$ such that for all $x\in \mathbb R$,
\[f(1-x)=1-f(f(x)).\]
is there a function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that
$i) \exists n\in \mathbb{N}:f(n)\neq n$
$ii)$ the number of divisors of $m$ is $f(n)$ if and only if the number of divisors of $f(m)$ is $n$
Let $S(x) = x+1$ and $V(x) = x^2-1$. Find the sum of the squares of all real solutions to $S(V(S(V(x)))) = 1$.
[i]Individual #2[/i]
Given are $10$ distinct real numbers. Kyle wrote down the square of the difference for each pair of those numbers in his notebook, while Peter wrote in his notebook the absolute value of the differences of the squares of these numbers. Is it possible for the two boys to have the same set of $45$ numbers in their notebooks?