Found problems: 4776
Find the least positive integer $k$ for which the equation $\lfloor \frac{2002}{n}\rfloor = k$ has no integer solutions for $n.$ (The notation $\lfloor x \rfloor$ means the greatest integer less than or equal to $x.$)
[b]two variable ploynomial[/b]
$P(x,y)$ is a two variable polynomial with real coefficients. degree of a monomial means sum of the powers of $x$ and $y$ in it. we denote by $Q(x,y)$ sum of monomials with the most degree in $P(x,y)$.
(for example if $P(x,y)=3x^4y-2x^2y^3+5xy^2+x-5$ then $Q(x,y)=3x^4y-2x^2y^3$.)
suppose that there are real numbers $x_1$,$y_1$,$x_2$ and $y_2$ such that
$Q(x_1,y_1)>0$ , $Q(x_2,y_2)<0$
prove that the set $\{(x,y)|P(x,y)=0\}$ is not bounded.
(we call a set $S$ of plane bounded if there exist positive number $M$ such that the distance of elements of $S$ from the origin is less than $M$.)
time allowed for this question was 1 hour.
Determine whether there exists a polynomial $f(x_1, x_2)$ with two variables, with integer coefficients, and two points $A=(a_1, a_2)$ and $B=(b_1, b_2)$ in the plane, satisfying the following conditions:
(i) $A$ is an integer point (i.e $a_1$ and $a_2$ are integers);
(ii) $|a_1-b_1|+|a_2-b_2|=2010$;
(iii) $f(n_1, n_2)>f(a_1, a_2)$ for all integer points $(n_1, n_2)$ in the plane other than $A$;
(iv) $f(x_1, x_2)>f(b_1, b_2)$ for all integer points $(x_1, x_2)$ in the plane other than $B$.
[i]Massimo Gobbino, Italy[/i]
Find all functions $f\colon \mathbb{R}^+ \to \mathbb{R}^+$ such that
\[
f(xf(x)+y^2) = x^2+yf(y)
\]
for any positive reals $x,y$.
The graph shows velocity as a function of time for a car. What was the acceleration at time = $90$ seconds?
[asy]
size(275);
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
draw((0,0)--(6,0));
draw((0,1)--(6,1));
draw((0,2)--(6,2));
draw((0,3)--(6,3));
draw((0,4)--(6,4));
draw((0,0)--(0,4));
draw((1,0)--(1,4));
draw((2,0)--(2,4));
draw((3,0)--(3,4));
draw((4,0)--(4,4));
draw((5,0)--(5,4));
draw((6,0)--(6,4));
label("$0$",(0,0),S);
label("$30$",(1,0),S);
label("$60$",(2,0),S);
label("$90$",(3,0),S);
label("$120$",(4,0),S);
label("$150$",(5,0),S);
label("$180$",(6,0),S);
label("$0$",(0,0),W);
label("$10$",(0,1),W);
label("$20$",(0,2),W);
label("$30$",(0,3),W);
label("$40$",(0,4),W);
draw((0,0.6)--(0.1,0.55)--(0.8,0.55)--(1.2,0.65)--(1.9,1)--(2.2,1.2)--(3,2)--(4,3)--(4.45,3.4)--(4.5,3.5)--(4.75,3.7)--(5,3.7)--(5.5,3.45)--(6,3));
label("Time (s)", (7.5,0),S);
label("Velocity (m/s)",(-1,3),W);
[/asy]
$ \textbf{(A)}\ 0.2\text{ m/s}^2\qquad\textbf{(B)}\ 0.33\text{ m/s}^2\qquad\textbf{(C)}\ 1.0\text{ m/s}^2\qquad\textbf{(D)}\ 9.8\text{ m/s}^2\qquad\textbf{(E)}\ 30\text{ m/s}^2 $
Let $x,y,z$ be positive real numbers. Prove the following inequality: \[ \frac 1{x^2+yz} + \frac 1{y^2+zx } + \frac 1{z^2+xy} \leq \frac 12 \left( \frac 1{xy} + \frac 1{yz} + \frac 1{zx} \right). \]
Be $R$ a positive real number. If $R, 1, R+\frac12$ are triangle sides, call $\theta$ the angle between $R$ and $R+\frac12$ (in rad).
Prove $2R\theta$ is between $1$ and $\pi$.
Find all functions $f: \mathbb R \rightarrow \mathbb R$ such that
\[f(x+xy+f(y)) = \left(f(x)+\frac{1}{2}\right) \left(f(y)+\frac{1}{2}\right)\]
holds for all real numbers $x,y$.
Let $F:\mathbb R^{\mathbb R}\to\mathbb R^{\mathbb R}$ be a function (from the set of real-valued functions to itself) such that
$$F(F(f)\circ g+g)=f\circ F(g)+F(F(F(g)))$$
for all $f,g:\mathbb R\to\mathbb R$.
Prove that there exists a function $\sigma:\mathbb R\to\mathbb R$ such that
$$F(f)=\sigma\circ f\circ\sigma$$
for all $f:\mathbb R\to\mathbb R$.
Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$, there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$. Is it possible that for some integer $n \ge 0$, there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$?
[i]Proposed by David Yang[/i]
Let $ f : [0, \plus{}\infty) \to \mathbb R$ be a differentiable function. Suppose that $ \left|f(x)\right| \le 5$ and $ f(x)f'(x) \ge \sin x$ for all $ x \ge 0$. Prove that there exists $ \lim_{x\to\plus{}\infty}f(x)$.
$$\begin{array}{ rlrlrl}
f_1(x)=&x^2+x & f_2(x)=&2x^2-x & f_3(x)=&x^2 +x \\
g_1(x)=&x-2 & g_2(x)=&2x \ \ & g_3(x)=&x+2 \\
\end{array}$$
If $h(x)=x$ can be get from $f_i$ and $g_i$ by using only addition, substraction, multiplication defined on those functions where $i\in\{1,2,3\}$, then $F_i=1$. Otherwise, $F_i=0$. What is $(F_1,F_2,F_3)$ ?
$ \textbf{(A)}\ (0,0,0)
\qquad\textbf{(B)}\ (0,0,1)
\qquad\textbf{(C)}\ (0,1,0)
\qquad\textbf{(D)}\ (0,1,1)
\qquad\textbf{(E)}\ \text{None}
$
For every positive integer $n$, form the number $n/s(n)$, where $s(n)$ is the sum of digits of $n$ in base 10. Determine the minimum value of $n/s(n)$ in each of the following cases:
(i) $10 \leq n \leq 99$
(ii) $100 \leq n \leq 999$
(iii) $1000 \leq n \leq 9999$
(iv) $10000 \leq n \leq 99999$
Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for all $m,n\in \mathbb{N}$: [list][*] $f(2)=2$, [*] $f(mn)=f(m)f(n)$, [*] $f(n+1)>f(n)$. [/list]
If for any three distinct numbers $a$, $b$ and $c$ we define \[\boxed{a,b,c} = \frac{c + a}{c - b},\] then $\boxed{1,-2,-3}=$
$ \textbf{(A)}\ -2\qquad\textbf{(B)}\ -\frac{2}{5}\qquad\textbf{(C)}\ -\frac{1}{4}\qquad\textbf{(D)}\ \frac{2}{5}\qquad\textbf{(E)}\ 2 $
A function $ f$ is defined by $ f(z) \equal{} i\bar z$, where $ i \equal{}\sqrt{\minus{}\!1}$ and $ \bar z$ is the complex conjugate of $ z$. How many values of $ z$ satisfy both $ |z| \equal{} 5$ and $ f (z) \equal{} z$?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 8$
Let $\mathbb{N}$ be the set of positive integers. For every $n \in \mathbb{N}$, define $d(n)$ as the number of positive divisors of $n$. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that:
a) $d(f(x)) = x$ for all $x \in \mathbb{N}$
b) $f(xy)$ divides $(x-1)y^{xy-1}f(x)$ for all $x,y \in \mathbb{N}$
For what real values of $k>0$ is it possible to dissect a $1 \times k$ rectangle into two similar, but noncongruent, polygons?
a) Find the value of $\sum_{n=1}^{\infty}\frac{\phi(n)}{2^n-1}$;
b) Show that $\sum_k {m\choose k}{{n+k}\choose m}=\sum_k {m\choose k} {n\choose k} 2^k$ for $m,n\geq 0$;
c) Using the identity $(1-x)^{-\frac 12}(1-x)^{-\frac 12}=(1-x)^{-1}$ derive a combinatorial identity!
d) Express the value of $\sum (2^{a_1}-1)\ldots (2^{a_k}-1)$ where the sum is over all $2^{n-1}$ ways of choosing $(a_1,a_2,\ldots,a_k)$ such that $a_1+a_2+\ldots +a_k=n$, as a function of some Fibonacci term.
Let $ a<a'<b<b'$ be real numbers and let the real function $ f$ be continuous on the interval $ [a,b']$ and differentiable in its interior. Prove that there exist $ c \in (a,b), c'\in (a',b')$ such that \[ f(b)\minus{}f(a)\equal{}f'(c)(b\minus{}a),\] \[ f(b')\minus{}f(a')\equal{}f'(c')(b'\minus{}a'),\] and $ c<c'$.
[i]B. Szokefalvi Nagy[/i]
Find all $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfies $f(0)=0,f(1)=2013$ and
\[(x-y)(f(f^2(x))-f(f^2(y)))=(f(x)-f(y))(f^2(x)-f^2(y))\]
Note: $f^2(x)=(f(x))^2$
For the vertex $ A$ of a triangle $ ABC$, let $ l_a$ be the distance between the projections on $ AB$ and $ AC$ of the intersection of the angle bisector of ∠$ A$ with side $ BC$. Define $ l_b$ and $ l_c$ analogously. If $ l$ is the perimeter of triangle $ ABC$, prove that $ \frac{l_a l_b l_c}{l^3}\le\frac{1}{64}$.
Let $x_1, x_2 , x_3, .... x_n$ be $n$ real numbers such that $0 < x_j < \frac{1}{2}$. Prove that \[ \frac{ \prod\limits_{j=1}^{n} x_j } { \left( \sum\limits_{j=1}^{n} x_j \right)^n} \leq \frac{ \prod\limits_{j=1}^{n} (1-x_j) } { \left( \sum\limits_{j=1}^{n} (1 - x_j) \right)^n} \]
Each face of a tetrahedron can be placed in a circle of radius $ 1$. Show that the tetrahedron can be placed in a sphere of radius $ \frac{3}{2\sqrt2}$.
If $f$ is a continuous real function such that $ f(x-1) + f(x+1) \ge x + f(x) $ for all $x$, what is the minimum possible value of $ \displaystyle\int_{1}^{2005} f(x) \, \mathrm{d}x $?