Found problems: 260
Let the Fourier series \[ \frac{a_0}{2}+ \sum _{k\geq 1}(a_k\cos kx+b_k \sin kx)\] of a function $ f(x)$ be
absolutely convergent, and let \[ a^2_k+b^2_k \geq a_{k+1}^2+b_{k+1}^2 \;(k=1,2,...)\ .\] Show that \[ \frac1h \int_0^{2\pi} (f(x+h)-f(x-h))^2dx \;(h>0)\] is uniformly bounded in $ h$. [K. Tandori]
show that $\frac1{15} < \frac12\cdot\frac34\cdots\frac{99}{100} < \frac1{10}$
Show that there exists a bijective function $ f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $ m,n\in \mathbb{N}_{0}$:
\[ f(3mn+m+n)=4f(m)f(n)+f(m)+f(n). \]
Let $a,b$ be coprime integers. Show that the equation $ax^2 + by^2 =z^3$ has an infinite set of solutions $(x,y,z)$ with $\{x,y,z\}\in\mathbb{Z}$ and each pair of $x,y$ mutually coprime.
Let $ a,b,c,d$ be nonnegative real numbers. Prove that
\[ a^4\plus{}b^4\plus{}c^4\plus{}d^4\plus{}2abcd \ge a^2b^2\plus{}a^2c^2\plus{}a^2d^2\plus{}b^2c^2\plus{}b^2d^2\plus{}c^2d^2.\]
The integers $a$ and $b$ have the property that for every nonnegative integer $n$ the number of $2^n{a}+b$ is the square of an integer. Show that $a=0$.
Let $ABC$ be an acute triangle. Determine the locus of points $M$ in the interior of the triangle such that $AB-FG=\frac{MF \cdot AG+MG \cdot BF}{CM}$, where $F$ and $G$ are the feet of the perpendiculars from $M$ to lines $BC$ and $AC$, respectively.
Consider the sequence $ \{a_n\}_{n\geq1}$ defined as follows: $ a_1 \equal{} 1$, $ a_{2k} \equal{} 1 \plus{} a_k$ and $ a_{2k \plus{} 1} \equal{} \frac {1}{a_{2k}}$ for every $ k\geq 1$. Prove that every positive rational number appears on the sequence $ \{a_n\}$ exactly once.
Determine all polynomials $p(x)$ with real coefficients such that $p((x+1)^3)=(p(x)+1)^3$ and $p(0)=0$.
Consider sequences that consist entirely of $ A$'s and $ B$'s and that have the property that every run of consecutive $ A$'s has even length, and every run of consecutive $ B$'s has odd length. Examples of such sequences are $ AA$, $ B$, and $ AABAA$, while $ BBAB$ is not such a sequence. How many such sequences have length 14?
For each positive integer n, let $ f(n) \equal{} \sum_{k \equal{} 1}^{100} \lfloor \log_{10} (kn) \rfloor$. Find the largest value of n for which $ f(n) \le 300$.
[b]Note:[/b] $ \lfloor x \rfloor$ is the greatest integer less than or equal to $ x$.
Show that there exist infinitely many square-free positive integers $n$ that divide $2005^n-1$.
Find all monic polynomials $f(x)$ in $\mathbb Z[x]$ such that $f(\mathbb Z)$ is closed under multiplication.
[i]By Mohsen Jamali[/i]
Prove that there are real numbers $ a_1, a_2, ..$ such that:
i) For all real numbers x, the serie $ f(x) \equal{} \sum_{n \equal{} 1}^\infty a_nx^n$ converge;
ii) f is a bijection of R to R;
iii) f'(x) >0;
iv) f(Q) = A, where Q is the set of rational numbers and A is the set of algebraic numbers.
Let $A$ and $B$ be positions of two ships $M$ and $N$, respectively, at the moment when $N$ saw $M$ moving with constant speed $v$ following the line $Ax$. In search of help, $N$ moves with speed $kv$ ($k < 1$) along the line $By$ in order to meet $M$ as soon as possible. Denote by $C$ the point of meeting of the two ships, and set
\[AB = d, \angle BAC = \alpha, 0 \leq \alpha < \frac{\pi}{2}.\]
Determine the angle $\angle ABC = \beta$ and time $t$ that $N$ needs in order to meet $M$.
For any positive integer $n>1$, let $p(n)$ be the greatest prime divisor of $n$. Prove that there are infinitely many positive integers $n$ with \[p(n)<p(n+1)<p(n+2).\]
Let a cyclic quadrilateral $ ABCD$, $ AC \cap BD \equal{} E$ and let a circle $ \Gamma$ internally tangent to the arch $ BC$ (that not contain $ D$) in $ T$ and tangent to $ BE$ and $ CE$. Call $ R$ the point where the angle bisector of $ \angle ABC$ meet the angle bisector of $ \angle BCD$ and $ S$ the incenter of $ BCE$. Prove that $ R$, $ S$ and $ T$ are collinear.
[i](Gabriel Giorgieri)[/i]
Police want to arrest on the famous criminals of the country whose name is Kaiser. Kaiser is in one of the streets of a square shaped city with $ n$ vertical streets and $ n$ horizontal streets. In the following cases how many police officers are needed to arrest Kaiser?
[img]http://i38.tinypic.com/2i1icec_th.png[/img] [img]http://i34.tinypic.com/28rk4s3_th.png[/img]
a) Each police officer has the same speed as Kaiser and every police officer knows the location of Kaiser anytime.
b) Kaiser has an infinite speed (finite but with no bound) and police officers can only know where he is only when one of them see Kaiser.
Everybody in this problem (including police officers and Kaiser) move continuously and can stop or change his path.
What is the hundreds digit of $2011^{2011}$?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 9 $
Prove that there exist infinitely many positive integers $n$ such that the number $\frac{1^2+2^2+\cdots+n^2}{n}$ is a perfect square. Obviously, $1$ is the least integer having this property. Find the next two least integers having this property.
The digits of the three-digit integers $a, b,$ and $c$ are the nine nonzero digits $1,2,3,\cdots 9$ each of them appearing exactly once. Given that the ratio $a:b:c$ is $1:3:5$, determine $a, b,$ and $c$.
I tried to search SRMC problems,but i didn't find them(I found only SRMC 2006).Maybe someone know where on this site i could find SRMC problems?I have all SRMC problems,if someone want i could post them, :wink:
Here is one of them,this is one nice inequality from first SRMC:
Let $ n$ be an integer with $ n>2$ and $ a_{1},a_{2},\dots,a_{n}\in R^{\plus{}}$.Given any positive integers $ t,k,p$ with $ 1<t<n$,set $ m\equal{}k\plus{}p$,prove the following inequalities:
a)
$ \frac{a_{1}^{p}}{a_{2}^{k}\plus{}a_{3}^{k}\plus{}\dots\plus{}a_{t}^{k}}\plus{}\frac{a_{2}^{p}}{a_{3}^{k}\plus{}a_{4}^{k}\plus{}\dots\plus{}a_{t\plus{}1}^{k}}\plus{}\dots\plus{}\frac{a_{n\minus{}1}^{p}}{a_{n}^{k}\plus{}a_{1}^{k}\plus{}\dots\plus{}a_{t\minus{}2}^{k}}\plus{}\frac{a_{n}^{p}}{a_{1}^{k}\plus{}a_{2}^{k}\plus{}\dots\plus{}a_{t\minus{}1}^{k}}\geq\frac{(a_{1}^{p}\plus{}a_{2}^{p}\dots\plus{}a_{n}^{p})^{2}}{(t\minus{}1) ( a_{1}^{m}\plus{}a_{2}^{m}\plus{}\dots\plus{}a_{n}^{m})}$
b)$ \frac{a_{2}^{k}\plus{}a_{3}^{k}\dots\plus{}a_{t}^{k}}{a_{1}^{p}}\plus{}\frac{a_{3}^{k}\plus{}a_{4}^{k}\dots\plus{}a_{t\plus{}1}^{k}}{a_{2}^{p}}\plus{}\dots\plus{}\frac{a_{1}^{k}\plus{}a_{2}^{k}\dots\plus{}a_{t\minus{}1}^{k}}{a_{n}^{p}}\geq\frac{(t\minus{}1)(a_{1}^{k}\plus{}a_{2}^{k}\dots\plus{}a_{n}^{k})^{2}}{( a_{1}^{m}\plus{}a_{2}^{m}\plus{}\dots\plus{}a_{n}^{m})}$ :wink:
Given that $x^2 + y^2 = 14x + 6y + 6$, what is the largest possible value that $3x + 4y$ can have?
$\text{(A)}\ 72 \qquad \text{(B)}\ 73 \qquad \text{(C)}\ 74 \qquad \text{(D)}\ 75\qquad \text{(E)}\ 76$
Let $ABC$ be a triangle such that $|AC|>|AB|$. Let $X$ be on line $AB$ (closer to $A$) such that $|BX|=|AC|$ and let $Y$ be on the segment $AC$ such that $|CY|=|AB|$. Intersection of lines $XY$ and bisector of $BC$ is point $P$. Prove that $\angle BPC+\angle BAC = 180^\circ$.
A convex quadrilateral $ABCD$ is inscribed in a circle with center $O$ and circumscribed to a circle with center $I$. Its diagonals meet at $P$. Prove that points $O, I$ and $P$ lie on a line.