Found problems: 4776
Let $ f : \mathbf {R} \to \mathbf {R} $ be a smooth function such that $ f'(x)^2 = f(x) f''(x) $ for all $x$. Suppose $f(0)=1$ and $f^{(4)} (0) = 9$. Find all possible values of $f'(0)$.
Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that for all integers $m,n$,
\[f(m - n + f(n)) = f(m) + f(n).\]
The ratio of the length to the width of a rectangle is $4:3$. If the rectangle has diagonal of length $d$, then the area may be expressed as $kd^2$ for some constant $k$. What is $k$?
$\textbf{(A) }\dfrac27\qquad\textbf{(B) }\dfrac37\qquad\textbf{(C) }\dfrac{12}{25}\qquad\textbf{(D) }\dfrac{16}{25}\qquad\textbf{(E) }\dfrac34$
Suppose $A\subset \{(a_1,a_2,\dots,a_n)\mid a_i\in \mathbb{R},i=1,2\dots,n\}$. For any $\alpha=(a_1,a_2,\dots,a_n)\in A$ and $\beta=(b_1,b_2,\dots,b_n)\in A$, we define
\[ \gamma(\alpha,\beta)=(|a_1-b_1|,|a_2-b_2|,\dots,|a_n-b_n|), \] \[ D(A)=\{\gamma(\alpha,\beta)\mid\alpha,\beta\in A\}. \] Please show that $|D(A)|\geq |A|$.
Let $ABC$ and $DEF$ be acute-angled triangles. Write $d = EF, e = FD, f = DE.$ Show that there exists a point $P$ in the interior of $ABC$ for which the value of the expression $X=d \cdot AP +e \cdot BP +f \cdot CP$ attains a minimum.
Let $(a_n)_{n \geq 1}$ be a sequence of real numbers. We define a sequence of real functions $(f_n)_{n \geq 0}$ such that for all $x \in \mathbb{R}$, the following holds:
\[
f_0(x) = 1 \quad \text{and} \quad f_n(x) = \int_{a_n}^{x} f_{n-1}(t) \, dt \quad \text{for } n \geq 1.
\]
Find all possible sequences $(a_n)_{n \geq 1}$ such that $f_n(0) = 0$ for all $n \geq 2$.\\
[b]Note:[/b] It is not necessarily true that $f_1(0) = 0$.
Let $ABC$ be a triangle and $M,N,P$ be the midpoints of sides $BC, CA, AB$. The lines $AM, BN, CP$ meet the circumcircle of $ABC$ in the points $A_{1}, B_{1}, C_{1}$. Show that the area of triangle $ABC$ is at most the sum of areas of triangles $BCA_{1}, CAB_{1}, ABC_{1}$.
Is there a continuous function $ f: \mathbb {R} ^ {2} \rightarrow \mathbb {R} $ for every $ d \in \mathbb {R} $ we have $ g_{m,d}(x) = f (x, m x + d) $ is strictly monotonic on $ \mathbb {R} $ if $ m \ge 0, $ and not monotone on any non-empty open interval if $ m <0? $
A calculator has a key which replaces the displayed entry with its square, and another key which replaces the displayed entry with its reciprocal. Let $y$ be the final result if one starts with an entry $x \neq 0$ and alternately squares and reciprocates $n$ times each. Assuming the calculator is completely accurate (e.g., no roundoff or overflow), then $y$ equals
A. $x^{((-2)^n)}$
B. $x^{2n}$
C. $x^{-2n}$
D. $x^{-(2^n)}$
E. $x^{((-1)^n 2n)}$
Find all pairs $(x,y)$ of positive real numbers such that $xy$ is an integer and $x+y = \lfloor x^2 - y^2 \rfloor$.
Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying
\[ f(f(x \plus{} y)) \equal{} f(x \plus{} y) \plus{} f(x)f(y) \minus{} xy\]
for all real numbers $x$ and $y$.
Do there exist functions $p(x)$ and $q(x)$, such that $p(x)$ is an even function while $p(q(x))$ is an odd function (different from 0)?
[i](3 points)[/i]
Let $ G$ be a locally compact solvable group, let $ c_1,\ldots, c_n$ be complex numbers, and assume that the complex-valued functions $ f$ and $ g$ on $ G$ satisfy \[ \sum_{k=1}^n c_k f(xy^k)=f(x)g(y) \;\textrm{for all} \;x,y \in G \ \ .\] Prove that if $ f$ is a bounded function and \[ \inf_{x \in G} \textrm{Re} f(x) \chi(x) >0\] for some continuous (complex) character $ \chi$ of $ G$, then $ g$ is continuous.
[i]L. Szekelyhidi[/i]
In a television series about incidents in a conspicuous town there are $n$ citizens staging in it, where $n$ is an integer greater than $3$. Each two citizens plan together a conspiracy against one of the other citizens. Prove that there exists a citizen, against whom at least $\sqrt{n}$ other citizens are involved in the conspiracy.
Let be a nondegenerate and closed interval $ I $ of real numbers, a short map $ m:I\longrightarrow I, $ and a sequence of functions $ \left( x_n \right)_{n\ge 1} :I\longrightarrow\mathbb{R} $ such that $ x_1 $ is the identity map and
$$ 2x_{n+1}=x_n+m\circ x_n , $$
for any natural numbers $ n. $ Prove that:
[b]a)[/b] there exists a nondegenerate interval having the property that any point of it is a fixed point for $ m. $
[b]b)[/b] $ \left( x_n \right)_{n\ge 1} $ is pointwise convergent and that its limit function is a short map.
A $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ where $h_i\left(x_1,x_2, \cdots , x_n\right)$ are $n$ variable polynomials with real coefficients is called [i]good[/i] if the following condition holds:
For any $n$ functions $f_1,f_2, \cdots ,f_n : \mathbb R \to \mathbb R$ if for all $1 \le i \le n+1$, $P_i(x)=h_i \left(f_1(x),f_2(x), \cdots, f_n(x) \right)$ is a polynomial with variable $x$, then $f_1(x),f_2(x), \cdots, f_n(x)$ are polynomials.
$a)$ Prove that for all positive integers $n$, there exists a [i]good[/i] $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ such that the degree of all $h_i$ is more than $1$.
$b)$ Prove that there doesn't exist any integer $n>1$ that for which there is a [i]good[/i] $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ such that all $h_i$ are symmetric polynomials.
[i]Proposed by Alireza Shavali[/i]
Find $A^2$, where $A$ is the sum of the absolute values of all roots of the following equation: \begin{eqnarray*}x &=& \sqrt{19} + \frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{x}}}}}}}}}\end{eqnarray*}
Let $f(k)$ be the number of integers $n$ satisfying the following conditions:
(i) $0\leq n < 10^k$ so $n$ has exactly $k$ digits (in decimal notation), with leading zeroes allowed;
(ii) the digits of $n$ can be permuted in such a way that they yield an integer divisible by $11$.
Prove that $f(2m) = 10f(2m-1)$ for every positive integer $m$.
[i]Proposed by Dirk Laurie, South Africa[/i]
Determine all $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that $f(m)\geq m$ and $f(m+n) \mid f(m)+f(n)$ for all $m,n\in \mathbb{Z}^+$
$f$ and $g$ are real-valued functions defined on the real line. For all $x$ and $y, f(x+y)+f(x-y)=2f(x)g(y)$. $f$ is not identically zero and $|f(x)|\le1$ for all $x$. Prove that $|g(x)|\le1$ for all $x$.
For real numbers $a,b,c,d$ not all equal to $0$ , define a real function $f(x) = a +b\cos{2x} + c\sin{5x} +d \cos{8x}$. Suppose $f(t) = 4a$ for some real $t$. prove that there exist a real number $s$ s.t. $f(s)<0$
Let $n$ be a positive integer and let $\alpha_n $ be the number of $1$'s within binary representation of $n$.
Show that for all positive integers $r$,
\[2^{2n-\alpha_n}\phantom{-1} \bigg|^{\phantom{0}}_{\phantom{-1}} \sum_{k=-n}^{n} \binom{2n}{n+k} k^{2r}.\]
Let $f \in \mathbb Z[X]$. For an $n \in \mathbb N$, $n \geq 2$, we define $f_n : \mathbb Z / n \mathbb Z \to \mathbb Z / n \mathbb Z$ through $f_n \left( \widehat x \right) = \widehat{f \left( x \right)}$, for all $x \in \mathbb Z$.
(a) Prove that $f_n$ is well defined.
(b) Find all polynomials $f \in \mathbb Z[X]$ such that for all $n \in \mathbb N$, $n \geq 2$, the function $f_n$ is surjective.
[i]Bogdan Enescu[/i]
Which of the following are true?
$\textbf{(A)}~\exists f:\mathbb N\to\mathbb Z\text{ onto and increasing}$
$\textbf{(B)}~\exists f:\mathbb Z\to\mathbb Q\text{ onto and increasing}$
$\textbf{(C)}~\exists f:\mathbb Q\to\mathbb Z\text{ onto and increasing and bounded}$
$\textbf{(D)}~\text{None of the above}$
Find all functions $f$ from positive integers to themselves such that:
1)$f(mn)=f(m)f(n)$ for all positive integers $m, n$
2)$\{1, 2, ..., n\}=\{f(1), f(2), ... f(n)\}$ is true for infinitely many positive integers $n$.