Found problems: 85335
We define a relation between subsets of $\mathbb R ^n$. $A \sim B\Longleftrightarrow$ we can partition $A,B$ in sets $A_1,\dots,A_n$ and $B_1,\dots,B_n$(i.e $\displaystyle A=\bigcup_{i=1} ^n A_i,\ B=\bigcup_{i=1} ^n B_i,
A_i\cap A_j=\emptyset,\ B_i\cap B_j=\emptyset$) and $A_i\simeq B_i$.
Say the the following sets have the relation $\sim$ or not ?
a) Natural numbers and composite numbers.
b) Rational numbers and rational numbers with finite digits in base 10.
c) $\{x\in\mathbb Q|x<\sqrt 2\}$ and $\{x\in\mathbb Q|x<\sqrt 3\}$
d) $A=\{(x,y)\in\mathbb R^2|x^2+y^2<1\}$ and $A\setminus \{(0,0)\}$
Let $a = \sqrt[401]{4} - 1$ and for each $n \ge 2$, let $b_n = \binom{n}{1} + \binom{n}{2} a + \ldots + \binom{n}{n} a^{n-1}$. Find $b_{2006} - b_{2005}$.
Let $d_k$ be the greatest odd divisor of $k$ for $k = 1, 2, 3, \ldots$. Find $d_1 + d_2 + d_3 + \ldots + d_{1024}$.
Prove that three discs of radius $1$ cannot cover entirely a square surface of side $2$, but they can cover more than $99.75\%$ of it.
Let $a_1,a_2,...$ be an arithmetic sequence with the common difference between terms is positive. Assume there are $k$ terms of this sequence creates an geometric sequence with common ratio $d$. Prove that $n\ge 2^{k-1}$.
Prove that for each $x,y,z \in R$ it holds that
$$\frac{x^2-y^2}{2x^2+1} +\frac{y^2-z^2}{2y^2+1}+\frac{z^2-x^2}{2z^2+1}\le 0$$
Consider the hexagon which is formed by the vertices of two equilateral triangles (not necessarily equal) when the triangles intersect. Prove that the area of the hexagon is unchanged when one of the triangles is translated (without rotating) relative to the other in such a way that the hexagon continues to be defined.
(V Proizvolov)
I centered incircle of triangle $ABC$ $(m(\hat{B})=90^\circ)$ touches $\left[AB\right], \left[BC\right], \left[AC\right]$ respectively at $F, D, E$. $\left[CI\right]\cap\left[EF\right]={L}$ and $\left[DL\right]\cap\left[AB\right]=N$. Prove that $\left[AI\right]=\left[ND\right]$.
For a given real number $a$, define the sequence $(a_n)$ by $a_1 = a$ and
$$a_{n+1} =\begin{cases}
\dfrac12 \left(a_n -\dfrac{1}{a_n}\right) \,\,\, if \,\,\, a_n \ne 0, \\
0 \,\,\, if \,\,\, a_n = 0 \end{cases}$$
Prove that the sequence $(a_n)$ contains infinitely many nonpositive terms.
In a square $ABCD$, $K$ is a point on the side $BC$ and the bisector of $\angle KAD$ cuts the side $CD$ at the point $M$. Prove that the length of segment $AK$ is equal to the sum of the lengths of segments $DM$ and $BK$.
(Folklore)
A frog is positioned at the origin in the coordinate plane. From the point $(x,y)$, the frog can jump to any of the points $(x+1, y), (x+2, y), (x, y+1),$ or $(x, y+2)$. Find the number of distinct sequences of jumps in which the frog begins at $(0,0)$ and ends at $(4,4)$.
Two concentric circles of radii $1$ and $2$ have centere the point $O$. The vertex $A$ of the equilateral triangle $ABC$ lies at the largest circle, while the midpoint of side $BC$ lies on the smaller circle. If$ B$,$O$ and $C$ are not collinear, what measure can the angle $\angle BOC$ have?
A ternary sequence is one whose terms all lie in the set $\{0, 1, 2\}$. Let $w$ be a length $n$ ternary sequence $(a_1,\ldots,a_n)$. Prove that $w$ can be extended leftwards and rightwards to a length $m=6n$ ternary sequence \[(d_1,\ldots,d_m) = (b_1,\ldots,b_p,a_1,\ldots,a_n,c_1,\ldots,c_q), \quad p,q\geqslant 0,\]containing no length $t > 2n$ palindromic subsequence.
(A sequence is called palindromic if it reads the same rightwards and leftwards. A length $t$ subsequence of $(d_1,\ldots,d_m)$ is a sequence of the form $(d_{i_1},\ldots,d_{i_t})$, where $1\leqslant i_1<\cdots<i_t \leqslant m$.)
Let $S$ be the sum of the greatest odd divisors of the natural numbers $1$ through $2^n$. Prove that $3S = 4^n + 2$.
The set of all numbers x for which \[x+\sqrt{x^{2}+1}-\frac{1}{x+\sqrt{x^{2}+1}}\] is a rational number is the set of all:
$\textbf{(A)}\ \text{ integers } x \qquad
\textbf{(B)}\ \text{ rational } x \qquad
\textbf{(C)}\ \text{ real } x\qquad
\textbf{(D)}\ x \text{ for which } \sqrt{x^2+1} \text{ is rational} \qquad
\textbf{(E)}\ x \text{ for which } x+\sqrt{x^2+1} \text{ is rational }$
[b]6.[/b] Let $H_{n}(x)$ be the [i]n[/i]th Hermite polynomial. Find
$ \lim_{n \to \infty } (\frac{y}{2n})^{n} H_{n}(\frac{n}{y})$
For an arbitrary real y. [b](S.5)[/b]
$H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}\left(e^{{-x^2}}\right)$
Find all real numbers $x, y,z,t \in [0, \infty)$ so that
$$x + y + z \le t, \,\,\, x^2 + y^2 + z^2 \ge t \,\,\, and \,\,\,x^3 + y^3 + z^3 \le t.$$
How many ways are there to choose distinct positive integers $a, b, c, d$ dividing $15^6$ such that none of $a, b, c,$ or $d$ divide each other? (Order does not matter.)
[i]Proposed by Miles Yamner and Andrew Wu[/i]
(Note: wording changed from original to clarify)
a) The real number $a$ and the continuous function $f : [a, \infty) \rightarrow [a, \infty)$ are such that $|f(x)-f(y)| < |x–y|$ for every two different $x, y \in [a, \infty)$. Is it always true that the equation $f(x)=x$ has only one solution in the interval $[a, \infty)$?
b) The real numbers $a$ and $b$ and the continuous function $f : [a, b] \rightarrow [a, b]$ are such that $|f(x)-f(y)| < |x–y|$, for every two different $x, y \in [a, b]$. Is it always true that the equation $f(x)=x$ has only one solution in the interval $[a, b]$?
In a group of people, every two of them have exactly one mutual friend in the group. Prove that there is one person who is friends with all the other people in the group.
Note: the friendship is mutual, that is, if $X$ is friends with $Y$, then $Y$ is friends with $X$.
In the rectangle $ABCD, BC = 5, EC = 1/3 CD$ and $F$ is the point where $AE$ and $BD$ are cut. The triangle $DFE$ has area $12$ and the triangle $ABF$ has area $27$. Find the area of the quadrilateral $BCEF$ .
[img]https://1.bp.blogspot.com/-4w6e729AF9o/XNY9hqHaBaI/AAAAAAAAKL0/eCaNnWmgc7Yj9uV4z29JAvTcWCe21NIMgCK4BGAYYCw/s400/may%2B2011%2Bl1.png[/img]
Find all functions $f : R \to R$ such that $$2f(x) =f(x+y)+f(x+2y)$$, for all $x \in R$ and for all $y \ge 0$.
Determine $ Q \in \mathbb{R}$ which is so big that a sequence with non-negative reals elements $ a_1 ,a_2, \ldots$ which satisfies the following two conditions:
[b](i)[/b] $ \forall m,n \geq 1$ we have $ a_{m \plus{} n} \leq 2 \left(a_m \plus{} a_n \right)$
[b](ii)[/b] $ \forall k \geq 0$ we have $ a_{2^k} \leq \frac {1}{(k \plus{} 1)^{2008}}$
such that for each sequence element we have the inequality $ a_n \leq Q.$
Let $\omega$ be a $n$ -th primitive root of unity. Given complex numbers $a_1,a_2,\cdots,a_n$, and $p$ of them are non-zero. Let
$$b_k=\sum_{i=1}^n a_i \omega^{ki}$$
for $k=1,2,\cdots, n$. Prove that if $p>0$, then at least $\tfrac{n}{p}$ numbers in $b_1,b_2,\cdots,b_n$ are non-zero.
Billy is standing at $(1,0)$ in the coordinate plane as he watches his Aunt Sydney go for her morning jog starting at the origin. If Aunt Sydney runs into the First Quadrant at a constant speed of $1$ meter per second along the graph of $x=\frac25y^2$, find the rate, in radians per second, at which Billy’s head is turning clockwise when Aunt Sydney passes through $x=1$.