Found problems: 348
Let $f : \mathbb{R} \to \mathbb{R}$ be a function whose second derivative is continuous. Suppose that $f$ and $f''$ are bounded. Show that $f'$ is also bounded.
Prove that for any real number $ x$ the following inequality is true:
$ \max\{|\sin x|, |\sin(x\plus{}2010)|\}>\dfrac1{\sqrt{17}}$
Let $P(x)$ and $Q(x)$ be polynomials with nonnegative coefficients. We denote by $P'(x)$ the derivative of $P(x)$. Suppose that $P(0)=Q(0)=0$ and $Q(1) \leq 1 \leq P'(0)$.
$(1)$ Prove that $0 \leq Q(x) \leq x \leq P(x)$ for all $0 \leq x \leq 1$.
$(2)$ Prove that $P(Q(x)) \leq Q(P(x))$ for all $0 \leq x \leq 1$.
[i]Proposed by Otgonbayar Uuye.[/i]
Let $a, b, c, d, e$ be distinct real numbers. Prove that the equation
\[(x - a)(x - b)(x - c)(x - d) + (x - a)(x - b)(x - c)(x - e)\]
\[+(x - a)(x - b)(x - d)(x - e) + (x - a)(x - c)(x - d)(x - e)\]
\[+(x - b)(x - c)(x - d)(x - e) = 0\]
has four distinct real solutions.
Find the greatest $n$ such that $(z+1)^n = z^n + 1$ has all its non-zero roots in the unitary circumference, e.g. $(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0$ implies $|\alpha| = 1.$
A rectangular piece of of paper measures 4 units by 5 units. Several lines are drawn parallel to the edges of the paper. A rectangle determined by the intersections of some of these lines is called [i]basic [/i]if
(i) all four sides of the rectangle are segments of drawn line segments, and
(ii) no segments of drawn lines lie inside the rectangle.
Given that the total length of all lines drawn is exactly 2007 units, let $N$ be the maximum possible number of basic rectangles determined. Find the remainder when $N$ is divided by 1000.
Let $ f$ be an infinitely differentiable real-valued function defined on the real numbers. If $ f(1/n)\equal{}\frac{n^{2}}{n^{2}\plus{}1}, n\equal{}1,2,3,...,$ Compute the values of the derivatives of $ f^{k}(0), k\equal{}0,1,2,3,...$
Let $ A,B$ be different points on a parabola. Prove that we can find $ P_1,P_2,\dots,P_{n}$ between $ A,B$ on the parabola such that area of the convex polygon $ AP_1P_2\dots P_nB$ is maximum. In this case prove that the ratio of $ S(AP_1P_2\dots P_nB)$ to the sector between $ A$ and $ B$ doesn't depend on $ A$ and $ B$, and only depends on $ n$.
Show that $ \pi$ is irrational.
Let $p(x)$ be a nonzero polynomial of degree less than $1992$ having no nonconstant factor in common with
$x^3 -x$. Let
$$ \frac{d^{1992}}{dx^{1992}} \left( \frac{p(x)}{x^3 -x } \right) =\frac{f(x)}{g(x)}$$
for polynomials $f(x)$ and $g(x).$ Find the smallest possible degree of $f(x)$.
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 $
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 $\pi$.
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]
Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=Q(x)P^{\prime\prime}(x)$, where $Q(x)$ is a quadratic polynomial and $P^{\prime\prime}(x)$ is the second derivative of $P(x)$. Show that if $P(x)$ has at least two distinct roots then it must have $n$ distinct roots.
Express $\int_0^2 f(x)dx$ for any quadratic functions $f(x)$ in terms of $f(0),\ f(1)$ and $f(2).$
Find all $f: \mathbb R \to\mathbb R$ such that for all real numbers $x$, $f(x) \geq 0$ and for all real numbers $x$ and $y$, \[ f(x+y)+f(x-y)-2f(x)-2y^2=0. \]
Zeros of a fourth-degree polynomial $f (x)$ form an arithmetic progression. Prove that the zeros of $f '(x)$ also form an arithmetic progression.
Let $AXYBZ$ be a convex pentagon inscribed in a circle with diameter $\overline{AB}$. The tangent to the circle at $Y$ intersects lines $BX$ and $BZ$ at $L$ and $K$, respectively. Suppose that $\overline{AY}$ bisects $\angle LAZ$ and $AY=YZ$. If the minimum possible value of \[ \frac{AK}{AX} + \left( \frac{AL}{AB} \right)^2 \] can be written as $\tfrac{m}{n} + \sqrt{k}$, where $m$, $n$ and $k$ are positive integers with $\gcd(m,n)=1$, compute $m+10n+100k$.
[i]Proposed by Evan Chen[/i]
Find all the functions $ f :\mathbb{R}\mapsto\mathbb{R} $ with the following property: $ \forall x$ $f(x)= f(x/2) + (x/2)f'(x)$
Let $ k$ be a positive integer. Prove that there exist polynomials $ P_0(n),P_1(n),\dots,P_{k\minus{}1}(n)$ (which may depend on $ k$) such that for any integer $ n,$
\[ \left\lfloor\frac{n}{k}\right\rfloor^k\equal{}P_0(n)\plus{}P_1(n)\left\lfloor\frac{n}{k}\right\rfloor\plus{} \cdots\plus{}P_{k\minus{}1}(n)\left\lfloor\frac{n}{k}\right\rfloor^{k\minus{}1}.\]
($ \lfloor a\rfloor$ means the largest integer $ \le a.$)
Let $A=(a_{ij})\in \mathcal{M}_p(\mathbb{C})$ such that $a_{12}=a_{23}=\ldots=a_{p-1,p}=1$ and $a_{ij}=0$ for any other entry.
a)Prove that $A^{p-1}\neq O_p$ and $A^p=O_p$.
b)If $X\in \mathcal{M}_{p}(\mathbb{C})$ and $AX=XA$, prove that there exist $a_1,a_2,\ldots,a_p\in \mathbb{C}$ such that:
\[X=\left( \begin{array}{ccccc} a_1 & a_2 & a_3 & \ldots & a_p \\ 0 & a_1 & a_2 & \ldots & a_{p-1} \\ 0 & 0 & a_1 & \ldots & a_{p-2} \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & a_1 \end{array} \right)\]
c)If there exist $B,C\in \mathcal{M}_p(\mathbb{C})$ such that $(I_p+A)^n=B^n+C^n,\ (\forall)n\in \mathbb{N}^*$, prove that $B=O_p$ or $C=O_p$.
Let $ f: [0,1]^2\to\mathbb{R}$ be a continuous function on the closed unit square such that $ \frac{\partial f}{\partial x}$ and $ \frac{\partial f}{\partial y}$ exist and are continuous on the interior of $ (0,1)^2.$ Let $ a\equal{}\int_0^1f(0,y)\,dy,\ b\equal{}\int_0^1f(1,y)\,dy,\ c\equal{}\int_0^1f(x,0)\,dx$ and $ d\equal{}\int_0^1f(x,1)\,dx.$ Prove or disprove: There must be a point $ (x_0,y_0)$ in $ (0,1)^2$ such that
$ \frac{\partial f}{\partial x}(x_0,y_0)\equal{}b\minus{}a$ and $ \frac{\partial f}{\partial y}(x_0,y_0)\equal{}d\minus{}c.$
In a triangle $ABC$ with $BC = CA + \frac 12 AB$, point $P$ is given on side $AB$ such that $BP : PA = 1 : 3$. Prove that $\angle CAP = 2 \angle CPA.$
Let $ a$ and $ b$ be positive real numbers with $ a\ge b$. Let $ \rho$ be the maximum possible value of $ \frac{a}{b}$ for which the system of equations
\[ a^2\plus{}y^2\equal{}b^2\plus{}x^2\equal{}(a\minus{}x)^2\plus{}(b\minus{}y)^2\]has a solution in $ (x,y)$ satisfying $ 0\le x<a$ and $ 0\le y<b$. Then $ \rho^2$ can be expressed as a fraction $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.
Let $ x,y,z$ be positive real numbers,satisfying equality $ x^{2}\plus{}y^{2}\plus{}z^{2}\equal{}25$. Find the minimal possible value of the expression $ \frac{xy}{z} \plus{} \frac{yz}{x} \plus{} \frac{zx}{y}$.