Found problems: 4776
Let $a$ be a real number and let $f : \mathbb{R}\rightarrow \mathbb{R}$ be a function satisfying $f(0)=\frac{1}{2}$ and
\[f(x+y)=f(x)f(a-y)+f(y)f(a-x), \quad \forall x,y \in \mathbb{R}.\]
Prove that $f$ is constant.
Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that
\[ f(x+y) = f(x)f(y) - c \sin{x} \sin{y} \] for all reals $x,y$ where $c> 1$ is a given constant.
$ k$ is a given natural number. Find all functions $ f: \mathbb{N}\rightarrow\mathbb{N}$ such that for each $ m,n\in\mathbb{N}$ the following holds: \[ f(m)\plus{}f(n)\mid (m\plus{}n)^k\]
Let $0 < \alpha, \beta, \gamma < \frac{\pi}{2}$ and $\sin^{3} \alpha + \sin^{3} \beta + \sin^3 \gamma = 1$. Prove that
\[ \tan^{2} \alpha + \tan^{2} \beta + \tan^{2} \gamma \geq \frac{3 \sqrt{3}}{2} . \]
Suppose that $(f_n)_{n\in N}$ be the sequence from all functions $f_n:[0,1]\rightarrow \mathbb{R^+}$ s.t. $f_0$ be the continuous function and $\forall x\in [0,1] , \forall n\in \mathbb {N} , f_{n+1}(x)=\int_0^x \frac {1}{1+f_n (t)}dt$. Prove that for every $x\in [0,1]$ the sequence of $(f_n(x))_{n\in N}$ be the convergent sequence and calculate the limitation.
Determine all functions $f:(0;\infty)\to \mathbb{R}$ that satisfy
$$f(x+y)=f(x^2+y^2)\quad \forall x,y\in (0;\infty)$$
Let $f\left(x\right)=x^2-14x+52$ and $g\left(x\right)=ax+b$, where $a$ and $b$ are positive. Find $a$, given that $f\left(g\left(-5\right)\right)=3$ and $f\left(g\left(0\right)\right)=103$.
$\text{(A) }2\qquad\text{(B) }5\qquad\text{(C) }7\qquad\text{(D) }10\qquad\text{(E) }17$
For the give functions in $\mathbb{N}$:
[b](a)[/b] Euler's $\phi$ function ($\phi(n)$- the number of natural numbers smaller than $n$ and coprime with $n$);
[b](b)[/b] the $\sigma$ function such that the $\sigma(n)$ is the sum of natural divisors of $n$.
solve the equation $\phi(\sigma(2^x))=2^x$.
Let $p$, $q$ be two integers, $q\geq p\geq 0$. Let $n \geq 2$ be an integer and $a_0=0, a_1 \geq 0, a_2, \ldots, a_{n-1},a_n = 1$ be real numbers such that \[ a_{k} \leq \frac{ a_{k-1} + a_{k+1} } 2 , \ \forall \ k=1,2,\ldots, n-1 . \] Prove that \[ (p+1) \sum_{k=1}^{n-1} a_k^p \geq (q+1) \sum_{k=1}^{n-1} a_k^q . \]
For any positive integer $ n$, let \[f(n) \equal{} \begin{cases} \log_8{n}, & \text{if }\log_8{n}\text{ is rational,} \\
0, & \text{otherwise.} \end{cases}\] What is $ \sum_{n \equal{} 1}^{1997}{f(n)}$?
$ \textbf{(A)}\ \log_8{2047}\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ \frac {55}{3}\qquad \textbf{(D)}\ \frac {58}{3}\qquad \textbf{(E)}\ 585$
Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be a function satisfying $f(x)f(y)=f(x-y)$. Find all possible values of $f(2017)$.
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 $n\geq 3$ be an integer. Find the number of functions $f:\{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$ such that
\[ f(f(k)) = f^3(k) - 6f^2(k) + 12f(k) - 6 , \ \textrm{ for all } k \geq 1 . \]
Find a necessary and sufficient condition of $a,b,n\in\mathbb{N^*}$ such that for $S=\{a+bt\mid t=0,1,2,\cdots,n-1\}$, there exists a one-to-one mapping $f: S\to S$ such that for all $x\in S$, $\gcd(x,f(x))=1$.
Let $ I $ be a nondegenerate interval, and let $ F $ be a primitive of a function $ f:I\longrightarrow\mathbb{R} . $ Show that for any distinct $ a,b\in I, $ the tangents to the graph of $ F $ at the points $ (a,F(a)) ,(b,F(b)) $ are concurrent at a point whose abscisa is situated in the interval $ (a,b). $
[i]Nicolae Bourbăcuț[/i]
Consider an arc of a planar curve; let the radius of curvature at any point of the arc be a differentiable function of the arc length and its derivative be everywhere different from zero; moreover, let the total curvature be less than $ \frac{\pi}{2}$. Let $ P_1,P_2,P_3,P_4,P_5$ and $ P_6$ be any points on this arc, subject to the only condition that the radius of curvature at $ P_k$ is greater than at $ P_j$ if $ j<k$.
Prove that the radius of the circle passing through the points $ P_1,P_3$ and $ P_5$ is less than the radius of the circle through $ P_2,P_4$ and $ P_6$
a) Let $a\in \mathbb{R}$ and $f \colon \mathbb{R} \to \mathbb{R}$ be a continuous function for which there exists an antiderivative $F \colon \mathbb{R} \to \mathbb{R} $, such that $F(x)+a\cdot f(x) \geq 0$, for any $x \in \mathbb{R}$, and$ \lim_{|x| \to \infty} \frac{F(x)}{e^{|\alpha \cdot x|}}=0$ holds for any $\alpha \in \mathbb{R}^*$. Prove that $F(x) \geq 0$ for all $x \in \mathbb{R}$.
b) Let $n\geq 2$ be a positive integer, $g \in \mathbb{R}[X]$, $g = X^n + a_1X^{n-1}+ \dots + a_{n-1}X+a_n$ be a polynomial with all of its roots being real, and $f \colon \mathbb{R} \to \mathbb{R}$ a polynomial function such that $f(x)+a_1\cdot f'(x)+a_2\cdot f^{(2)}(x)+\dots+a_n\cdot f^{(n)}(x) \geq 0$ for any $x \in \mathbb{R}$. Prove that $f(x) \geq 0$ for all $x \in \mathbb{R}$.
Thirty people sit at a round table. Each of them is either smart or dumb. Each of them is asked: "Is your neighbor to the right smart or dumb?" A smart person always answers correctly, while a dumb person can answer both correctly and incorrectly. It is known that the number of dumb people does not exceed $F$. What is the largest possible value of $F$ such that knowing what the answers of the people are, you can point at at least one person, knowing he is smart?
Let $C$ be a class of functions $f : \mathbb N \to \mathbb N$ that contains the functions $S(x) = x + 1$ and $E(x) = x - [\sqrt x]^2$ for every $x \in \mathbb N$. ($[x]$ is the integer part of $x$.) If $C$ has the property that for every $f, g \in C, f + g, fg, f \circ g \in C$, show that the function $\max(f(x) - g(x), 0)$ is in $C$, for all $f; g \in C$.
If $a$, $b$, $c$ are positive reals such that $a+b+c=1$, prove that
\[\frac{1+a}{1-a}+\frac{1+b}{1-b}+\frac{1+c}{1-c}\leq 2\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right).\]
Consider the graphs of all the functions $y = x^2 + px + q$ having 3 different intersection points with the coordinate axes. For every such graph we pick these 3 intersection points and draw a circumcircle through them. Prove that all these circles have a common point!
Consider in the plane a finite set of segments such that the sum of their lengths is less than $\sqrt{2}$. Prove that there exists an infinite unit square grid covering the plane such that the lines defining the grid do not intersect any of the segments.
[i]Vasile Pop[/i]
Provided that the roots of the polynom $ X^n+a_1X^{n-1} +a_2X^{n-2} +\cdots +a_{n-1}X +a_n:\in\mathbb{R}[X] , $ of degree $ n\ge 2, $ are all real and pairwise distinct, prove that there exists is a neighbourhood $ \mathcal{V} $ of $ \left(
a_1,a_2,\ldots ,a_n \right) $ in $ \mathbb{R}^n $ and $ n $ functions $ x_1,x_2,\ldots ,x_n\in\mathcal{C}^{\infty } \left(
\mathcal{V} \right) $ whose values at $ \left( a_1,a_2,\ldots ,a_n \right) $ are roots of the mentioned polynom.
Let $f$ be the function defined by $f(x) = 4x(1 - x)$. Let $n$ be a positive integer. Prove that there exist distinct real numbers $x_1$, $x_2$, $\ldots\,$, $x_n$ such that $x_{i + 1} = f(x_i)$ for each integer $i$ with $1 \le i \le n - 1$, and such that $x_1 = f(x_n)$.
A finite set $\mathcal{S}$ of positive integers is called cardinal if $\mathcal{S}$ contains the integer $|\mathcal{S}|$ where $|\mathcal{S}|$ denotes the number of distinct elements in $\mathcal{S}$. Let $f$ be a function from the set of positive integers to itself such that for any cardinal set $\mathcal{S}$, the set $f(\mathcal{S})$ is also cardinal. Here $f(\mathcal{S})$ denotes the set of all integers that can be expressed as $f(a)$ where $a \in \mathcal{S}$. Find all possible values of $f(2024)$
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Proposed by Sutanay Bhattacharya