Found problems: 4776
The function $f(x)$ satisfies $f(2+x) = f(2-x)$ for all real numbers $x$. If the equation $f(x) = 0$ has exactly four distinct real roots, then the sum of these roots is
A. 0
B. 2
C. 4
D. 6
E. 8
Find all functions from $\mathbb{N}\cup\{0\}\to\mathbb{N}\cup\{0\}$ such that $f(m^2+mf(n))=mf(m+n)$, for all $m,n\in \mathbb{N}\cup\{0\}$.
Find all functions $f:\mathbb{R}\mapsto \mathbb{R}$ satisfy:
$$f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)$$
for all real numbers $x,y$.
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all real numbers $x$ and $y$,
$$f(f(x-y)-yf(x))=xf(y).$$
Let $A{}$ be a set with $n{}$ $(n\geq3)$ elements. Iterations $f^2,f^2,\ldots$ of the function $f:A\rightarrow A$ are defined as $f^2(x)=f(f(x)), f^{i+1}=f(f^i(x)), \forall i\geq2$. Find the number of functions $f:A\rightarrow A$ with the property: the function $f^{n-2}$ is constant, but $f^{n-3}$ is not.
Let $a_1,a_2,a_3,\ldots,a_n$ be a sequence of non-negative integers, where $n$ is a positive integer. Let
\[ A_n={a_1+a_2+\cdots+a_n\over n}\ . \]
Prove that
\[ a_1!a_2!\ldots a_n!\ge\left(\lfloor A_n\rfloor !\right)^n \]
where $\lfloor A_n\rfloor$ is the greatest integer less than or equal to $A_n$, and $a!=1\times 2\times\cdots\times a$ for $a\ge 1$(and $0!=1$). When does equality hold?
A $40$ feet high screen is put on a vertical wall $10$ feet above your eye-level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye?
Let $f \colon [0,1] \to \mathbb{R}$ be a continuous function which is differentiable on $(0,1)$. Prove that either $f(x) = ax + b$ for all $x \in [0,1]$ for some constants $a,b \in \mathbb{R}$ or there exists $t \in (0,1)$ such that $|f(1) - f(0)| < |f'(t)|$.
If $x$ is a real number, then the quantity $(1-|x|)(1+x)$ is positive if and only if
$\textbf{(A) }|x|<1\qquad\textbf{(B) }|x|>1\qquad\textbf{(C) }x<-1\text{ or }-1<x<1\qquad$
$\textbf{(D) }x<1\qquad \textbf{(E) }x<-1$
Let $x$ and $y$ be positive real numbers with $x + y =1 $. Prove that
$$\frac{(3x-1)^2}{x}+ \frac{(3y-1)^2}{y} \ge1.$$ For which $x$ and $y$ equality holds?
(K. Czakler, GRG 21, Vienna)
Let $ f(x)$ be a continuous function defined in the interval $ 0\leq x\leq 1.$
Prove that $ \int_0^1 xf(x)f(1\minus{}x)\ dx\leq \frac{1}{4}\int_0^1 \{f(x)^2\plus{}f(1\minus{}x)^2\}\ dx.$
The definite integrals between $0$ and $1$ of the squares of the continuous real functions $f(x)$ and $g(x)$ are both equal to $1$.
Prove that there is a real number $c$ such that
\[f(c)+g(c)\leq 2\]
Find the minimum value of $f(x)=\int_0^{\frac{\pi}{4}} |\tan t-x|dt.$
Let $ k \equal{} \sqrt[3]{3}$.
a, Find all polynomials $ p(x)$ with rationl coefficients whose degree are as least as possible such that $ p(k \plus{} k^2) \equal{} 3 \plus{} k$.
b, Does there exist a polynomial $ p(x)$ with integer coefficients satisfying $ p(k \plus{} k^2) \equal{} 3 \plus{} k$
Let $k$ be a positive integer. For each nonnegative integer $n$, let $f(n)$ be the number of solutions $(x_1,\ldots,x_k)\in\mathbb{Z}^k$ of the inequality $|x_1|+...+|x_k|\leq n$. Prove that for every $n\ge1$, we have $f(n-1)f(n+1)\leq f(n)^2$.
(Proposed by Esteban Arreaga, Renan Finder and José Madrid, IMPA, Rio de Janeiro)
Consider the operation "minus the reciprocal of," defined by $a\diamond b=a-\frac{1}{b}$. What is $((1\diamond2)\diamond3)-(1\diamond(2\diamond3))$?
$\textbf{(A) } -\dfrac{7}{30}
\qquad\textbf{(B) } -\dfrac{1}{6}
\qquad\textbf{(C) } 0
\qquad\textbf{(D) } \dfrac{1}{6}
\qquad\textbf{(E) } \dfrac{7}{30}
$
Let $f(x)$ and $g(x)$ be mutually inverse decreasing functions on the interval $(0,\infty)$. Can it hold that $f(x)>g(x)$ for all $x\in(0,\infty)$?
In a culturing of bacteria, there are two species of them: red and blue bacteria.
When two red bacteria meet, they transform into one blue bacterium.
When two blue bacteria meet, they transform into four red bacteria.
When a red and a blue bacteria meet, they transform into three red bacteria.
Find, in function of the amount of blue bacteria and the red bacteria initially in the culturing,
all possible amounts of bacteria, and for every possible amount, the possible amounts of red and blue bacteria.
$f$ is a function defined on integers and satisfies $f(x)+f(x+3)=x^2$ for every integer $x$. If $f(19)=94$, then calculate $f(94)$.
Find the twice-differentiable functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that have the property that
$$ f'(x)+F(x)=2f(x)+x^2/2, $$
for any real numbers $ x; $ where $ F $ is a primitive of $ f. $
[i]Carmen Botea[/i]
Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$. Prove that $\prod_{i=1}^{n}\frac {r_i s_i
t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.
Suppose that $f(x,y)$ is a continuous real-valued function on the unit square $0\le x\le1,0\le y\le1.$ Show that
$\int_0^1\left(\int_0^1f(x,y)dx\right)^2dy + \int_0^1\left(\int_0^1f(x,y)dy\right)^2dx$
$\le\left(\int_0^1\int_0^1f(x,y)dxdy\right)^2 + \int_0^1\int_0^1\left[f(x,y)\right]^2dxdy.$
Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be a function that satisfies for all $x>y>0$
\[f(x+y)-f(x-y)=4\sqrt{f(x)f(y)}\]
a) Prove that $f(2x)=4f(x)$ for all $x>0$;
b) Find all such functions.
[i]Nikolai Nikolov, Oleg Mushkarov [/i]
Find all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ such that $ \forall x,y,z \in \mathbb{R}$ we have: If
\[ x^3 \plus{} f(y) \cdot x \plus{} f(z) \equal{} 0,\]
then
\[ f(x)^3 \plus{} y \cdot f(x) \plus{} z \equal{} 0.\]
Let $f$ be the unique function defined on the positive integers such that \[\sum_{d\mid n}d\cdot f\left(\frac{n}{d}\right)=1\] for all positive integers $n$, where the sum is taken over all positive divisors of $n$. What is $f(2023)$?
$\textbf{(A)}~-1536\qquad\textbf{(B)}~96\qquad\textbf{(C)}~108\qquad\textbf{(D)}~116\qquad\textbf{(E)}~144$