Found problems: 4776
Find all real numbers $ x$ such that:
$ [x^2\plus{}2x]\equal{}{[x]}^2\plus{}2[x]$
(Here $ [x]$ denotes the largest integer not exceeding $ x$.)
Suppose $0<m_1<...<m_n$ and $m_i \equiv i (\mod 2)$. Prove that the following polynomial has at most $n$ real roots. ($\forall 1\le i \le n: a_i \in \mathbb R$).
\[a_0+a_1x^{m_1}+a_2x^{m_2}+...+a_nx^{m_n}.\]
Let \[F=\log\dfrac{1+x}{1-x}.\] Find a new function $G$ by replacing each $x$ in $F$ by \[\dfrac{3x+x^3}{1+3x^2},\] and simplify. The simplified expression $G$ is equal to:
$\textbf{(A)}\ -F \qquad
\textbf{(B)}\ F\qquad
\textbf{(C)}\ 3F \qquad
\textbf{(D)}\ F^3 \qquad
\textbf{(E)}\ F^3-F$
A sequence of real numbers $a_1,a_2,\ldots$ satisfies the relation
$$a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all}\quad n>2017.$$
Prove that the sequence is bounded, i.e., there is a constant $M$ such that $|a_n|\leq M$ for all positive integers $n$.
Let $\varphi$ denote the Euler phi-function. Prove that for every positive integer $n$
$$2^{n(n+1)} | 32 \cdot \varphi \left( 2^{2^n} - 1 \right).$$
Let $\mathbb N$ be the set of positive integers. Let $f: \mathbb N \to \mathbb N$ be a function satisfying the following two conditions:
(a) $f(m)$ and $f(n)$ are relatively prime whenever $m$ and $n$ are relatively prime.
(b) $n \le f(n) \le n+2012$ for all $n$.
Prove that for any natural number $n$ and any prime $p$, if $p$ divides $f(n)$ then $p$ divides $n$.
Antonio and Bernardo play the following game. They are given two piles of chips, one with $m$ and the other with $n$ chips. Antonio starts, and thereafter they make in turn one of the following moves:
(i) take a chip from one pile;
(ii) take a chip from each of the piles;
(ii) remove a chip from one of the piles and put it onto the other.
Who cannot make any more moves, loses. Decide, as a function of $m$ and $n$ if one of the players has a winning strategy, and in the case of the affirmative answer describe that strategy.
Suppose $A$, $B$, and $C$ are points in the plane with $AB=40$ and $AC=42$, and let $x$ be the length of the line segment from $A$ to the midpoint of $\overline{BC}$. Define a function $f$ by letting $f(x)$ be the area of $\triangle ABC$. Then the domain of $f$ is an open interval $(p,q)$, and the maximum value $r$ of $f(x)$ occurs at $x=s$. What is $p+q+r+s$?
$
\textbf{(A) }909\qquad
\textbf{(B) }910\qquad
\textbf{(C) }911\qquad
\textbf{(D) }912\qquad
\textbf{(E) }913\qquad
$
Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$.
[i]Carl Lian and Brian Hamrick.[/i]
Does there exist a function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for every $n \ge 2$,
\[f (f (n - 1)) = f (n + 1) - f (n)?\]
Given an integer $k$. $f(n)$ is defined on negative integer set and its values are integers. $f(n)$ satisfies \[ f(n)f(n+1)=(f(n)+n-k)^2, \] for $n=-2,-3,\cdots$. Find an expression of $f(n)$.
Show that $8x^4 - 16x^3 + 16x^2 - 8x + k = 0$ has at least one real root for all real $k$. Find the sum of the non-real roots.
Assume $\Omega(n),\omega(n)$ be the biggest and smallest prime factors of $n$ respectively . Alireza and Amin decided to play a game. First Alireza chooses $1400$ polynomials with integer coefficients. Now Amin chooses $700$ of them, the set of polynomials of Alireza and Amin are $B,A$ respectively . Amin wins if for all $n$ we have :
$$\max_{P \in A}(\Omega(P(n))) \ge \min_{P \in B}(\omega(P(n)))$$
Who has the winning strategy.
Proposed by [i]Alireza Haghi[/i]
Let $a$ be real number with $\sqrt{2}<a<2$, and let $ABCD$ be a convex cyclic quadrilateral whose circumcentre $O$ lies in its interior. The quadrilateral's circumcircle $\omega$ has radius $1$, and the longest and shortest sides of the quadrilateral have length $a$ and $\sqrt{4-a^2}$, respectively. Lines $L_A,L_B,L_C,L_D$ are tangent to $\omega$ at $A,B,C,D$, respectively.
Let lines $L_A$ and $L_B$, $L_B$ and $L_C$,$L_C$ and $L_D$,$L_D$ and $L_A$ intersect at $A',B',C',D'$ respectively. Determine the minimum value of $\frac{S_{A'B'C'D'}}{S_{ABCD}}$.
Let $P(x)$ be a polynomial of degree $2012$ with real coefficients satisfying the condition \[P(a)^3 + P(b)^3 + P(c)^3 \geq 3P(a)P(b)P(c),\] for all real numbers $a,b,c$ such that $a+b+c=0$. Is it possible for $P(x)$ to have exactly $2012$ distinct real roots?
Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[yf(x+1)=f(x+y-f(x))+f(x)f(f(y))\]
for all $x,y \in \mathbb{R}$.
Let $f$ be a bijection from $\mathbb N$ to itself. Prove that one can always find three natural number $a,b,c$ such that $a<b<c$ and $f(a)+f(c)=2f(b)$.
Find all functions $ f\colon \mathbb{R} \to \mathbb{R} $ such that $ f(x^3+y^3+xy)=x^2f(x)+y^2f(y)+f(xy) $, for all $ x,y \in \mathbb{R} $.
Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying the following conditions:
(1) $f(1)=1$;
(2) $\forall n\in \mathbb{N}$, $3f(n) f(2n+1) =f(2n) ( 1+3f(n) )$;
(3) $\forall n\in \mathbb{N}$, $f(2n) < 6 f(n)$.
Find all solutions of equation $f(k) +f(l)=293$, where $k<l$.
($\mathbb{N}$ denotes the set of all natural numbers).
A function $f$ satisfies $f(\cos x) = \cos (17x)$ for every real $x$. Show that $f(\sin x) =\sin (17x)$ for every $x \in \mathbb{R}.$
Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a countinuous and periodic function, of period $ T. $ If $ F $ is a primitive of $ f, $ show that:
[b]a)[/b] the function $ G:\mathbb{R}\longrightarrow\mathbb{R}, G(x)=F(x)-\frac{x}{T}\int_0^T f(t)dt $ is periodic.
[b]b)[/b] $ \lim_{n\to\infty}\sum_{i=1}^n\frac{F(i)}{n^2+i^2} =\frac{\ln 2}{2T}\int_0^T f(x)dx. $
Prove that for any interval $[a,b]$ of real numbers and any positive integer $n$ there exists a positive integer $k$ and a partition of the given interval
\[a = x (0) < x (1) < x (2) < \cdots < x (k-1) < x (k) = b\]
such that
\[\int_{x(0)}^{x(1)}f(x)dx+\int_{x(2)}^{x(3)}f(x)dx+\cdots=\int_{x(1)}^{x(2)}f(x)dx+\int_{x(3)}^{x(4)}f(x)dx+\cdots\]
for all polynomials $f$ with real coefficients and degree less than $n$.
Let $\mathbb{F}_{p}$ denote the $p$-element field for a prime $p>3$ and let $S$ be the set of functions from $\mathbb{F}_{p}$ to $\mathbb{F}_{p}$. Find all functions $D\colon S\to S$ satisfying
\[D(f\circ g)=(D(f)\circ g)\cdot D(g)\]
for all $f,g \in {S}$. Here, $\circ$ denotes the function composition, so $(f\circ g)(x)$ is the function $f(g(x))$, and $\cdot$ denotes multiplication, so $(f\cdot g)=f(x)g(x)$.
Find all functions $f:\mathbb{R}\to\mathbb{R}$ for which \[f(x+y)=f(x-y)+f(f(1-xy))\] holds for all real numbers $x$ and $y$
Let $\mathbb N$ be the set of positive integers. Find all functions $f\colon\mathbb N\to\mathbb N$ such that $$\frac{f(x)-f(y)+x+y}{x-y+1}$$ is an integer, for all positive integers $x,y$ with $x>y$.