Found problems: 85335
Given a sequence of $19$ positive (not necessarily distinct) integers not greater than $93$, and a set of $93$ positive (not necessarily distinct) integers not greater than $19$. Show that we can find non-empty subsequences of the two sequences with equal sum.
Find the number of subsets $A\subset M=\{2^0,\,2^1,\,2^2,\dots,2^{2005}\}$ such that equation $x^2-S(A)x+S(B)=0$ has integral roots, where $S(M)$ is the sum of all elements of $M$, and $B=M\setminus A$ ($A$ and $B$ are not empty).
The numbers $ \log(a^3b^7)$, $ \log(a^5b^{12})$, and $ \log(a^8b^{15})$ are the first three terms of an arithmetic sequence, and the $ 12^\text{th}$ term of the sequence is $ \log{b^n}$. What is $ n$?
$ \textbf{(A)}\ 40 \qquad
\textbf{(B)}\ 56 \qquad
\textbf{(C)}\ 76 \qquad
\textbf{(D)}\ 112 \qquad
\textbf{(E)}\ 143$
The perimeter of an equilateral triangle exceeds the perimeter of a square by $1989 \ \text{cm}$. The length of each side of the triangle exceeds the length of each side of the square by $d \ \text{cm}$. The square has perimeter greater than 0. How many positive integers are NOT possible value for $d$?
$\text{(A)} \ 0 \qquad \text{(B)} \ 9 \qquad \text{(C)} \ 221 \qquad \text{(D)} \ 663 \qquad \text{(E)} \ \text{infinitely many}$
Consider the following sequence of positive real numbers $\dots<a_{-2}<a_{-1}<a_0<a_1<a_2<\dots$ infinite in both directions. For each positive integer $k$ let $b_k$ be the least integer such that the ratio between the sum of $k$ consecutive terms and the greatest of these $k$ terms is less than or equal to $b_k$(This fact occurs for any sequence of $k$ consecutive numbers). Prove that the sequence $b_1,b_2,b_3,...$ coincides with the sequence $1,2,3,...$ or is eventually constant.
Two congruent equilateral triangles $ABC$ and $A'B'C'$ in the plane are given. Show that the midpoints of the segments $AA',BB', CC'$ either are collinear or form an equilateral triangle.
In acute triangle $\triangle ABC$, $M$ is the midpoint of segment $BC$. Point $P$ lies in the interior of $\triangle ABC$ such that $AP$ bisects $\angle BAC$. Line $MP$ intersects the circumcircles of $\triangle ABP,\triangle ACP$ at $D,E$ respectively. Prove that if $DE=MP$, then $BC=2BP$.
Consider $0<a<T$, $D=\mathbb{R}\backslash \{ kT+a\mid k\in \mathbb{Z}\}$, and let $f:D\to \mathbb{R}$ a $T-$periodic and differentiable function which satisfies $f' > 1$ on $(0, a)$ and
$$f(0)=0,\lim_{\substack{x\to a\\x<a}}f(x)=+\infty \text{ and }\lim_{\substack{x\to a\\ x<a}}\frac{f'(x)}{f^2(x)}=1.$$
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[*]Prove that for every $n\in \mathbb{N}^*$, the equation $f(x)=x$ has a unique solution in the interval $(nT, nT+a)$ , denoted $x_n$.[/*]
[*]Let $y_n=nT+a-x_n$ and $z_n=\int_0^{y_n}f(x)\text{d}x$. Prove that $\lim_{n\to \infty}{y_n}=0$ and study the convergence of the series $\sum_{n=1}^{\infty}{y_n}$ and $\sum_{n=1}^{n}{z_n}$.
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$N$ is a $50-$digit number (in the decimal scale). All digits except the $26^{\text{th}}$ digit (from the left) are $1$. If $N$ is divisible by $13$, what is the $26^{\text{th}}$ digit?
$ \textbf{(A)}\ 1
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ 8
\qquad\textbf{(E)}\ \text{More information is needed}
$
Let $C_1$ and $C_2$ be the incircle and the circumcircle of the triangle $ABC$, respectively. Prove that, for any point $A'$ on $C_2$, there exist points $B'$ and $C'$ such that $C_1$ and $C_2$ are the incircle and the circumcircle of triangle $A'B'C'$, respectively.
Let $b$ be the probability that the cards are from different suits. Compute $\lfloor1000b\rfloor$.
On a plane, Bob chooses 3 points $A_0$, $B_0$, $C_0$ (not necessarily distinct) such that $A_0B_0+B_0C_0+C_0A_0=1$. Then he chooses points $A_1$, $B_1$, $C_1$ (not necessarily distinct) in such a way that $A_1B_1=A_0B_0$ and $B_1C_1=B_0C_0$.
Next he chooses points $A_2$, $B_2$, $C_2$ as a permutation of points $A_1$, $B_1$, $C_1$. Finally, Bob chooses points $A_3$, $B_3$, $C_3$ (not necessarily distinct) in such a way that $A_3B_3=A_2B_2$ and $B_3C_3=B_2C_2$. What are the smallest and the greatest possible values of $A_3B_3+B_3C_3+C_3A_3$ Bob can obtain?
Prove that there are no positive integers $a,b$ such that $(15a +b)(a +15b)$ is a power of $3.$
a) Can you pose the numbers $0,1,...,9$ on the circumference in such a way, that the difference between every two neighbours would be either $3$ or $4$ or $5$?
b) The same question, but about the numbers $0,1,...,13$.
Prove that there exists a real $c<\frac{3}{4}$, such that for each sequence $x_1, x_2, \ldots$ satisfying $0 \leq x_i \leq 1$ for all $i$, there exist infinitely many $(m, n)$ with $m>n$, such that $$|x_m-x_n|\leq \frac{c} {m}.$$
Omar has four fair standard six-sided dice. Omar invented a game where he rolls his four dice over and over again until the number 1 does not appear on the top face of any of the dice. Omar wins the game if on that roll the top faces of his dice show at least one 2 and at least one 5. The probability that Omar wins the game is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Let $D$ be the point different from $B$ on the hypotenuse $AB$ of a right triangle $ABC$ such that $|CB| = |CD|$. Let $O$ be the circumcenter of triangle $ACD$. Rays $OD$ and $CB$ intersect at point $P$, and the line through point $O$ perpendicular to side AB and ray $CD$ intersect at point $Q$. Points $A, C, P, Q$ are concyclic. Does this imply that $ACPQ$ is a square?
For a positive integer $n$, we de ne $D_n$ as the largest integer that is a divisor of $a^n + (a + 1)^n + (a + 2)^n$ for all positive integers $a$.
1. Show that for all positive integers $n$, the number $D_n$ is of the form $3^k$ with $k \ge 0$ an integer.
2. Show that for all integers $k \ge 0$ there exists a positive integer n such that $D_n = 3^k$.
[b]Black hole thermodynamics [/b]
The goal of this problem is to explore some interesting properties of Black Holes. The following equation was obtained by L. Smarr in 1973:
\[ M^2 = \frac{1}{16\pi} A + \frac{4\pi}{A}\left(J^2 + \frac{1}{4}Q^4\right)+\frac{1}{2}Q^2\]
where $M$, $J$, $Q$ and $A$ are the mass, angular momentum, charge and area of the event horizon of a black hole.
To make contact with thermodynamics we write for the entropy of the Black Hole,
\[S = \frac{1}{4}k_B A\]
where $k_B$ is the Boltzmann constant.
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[*] Work in natural units $G = \hbar = c = 1$ and show that the equation for the entropy is dimensionally correct. [/*]
[*] Take $k_B = 1/8\pi$ (by choosing units) and derive an expression for $S(M,J,Q)$. Is this expression unique? (Hint: What is the entropy of the Schwarzschild Black Hole which corresponds to $J=Q=0$?)
\item We suppose the mass-energy $M$ (since $c=1$) plays the role of internal energy. Show that $T,\Omega,\Phi$ defined via,
\[ dM = T dS + \Omega dJ + \Phi dQ\]
are given by,
\begin{eqnarray*}
& T = \frac{1}{M} \left[1- \frac{1}{16S^2}\left(J^2 + \frac{1}{4}Q^4\right)\right] \\
& \Omega = \frac{J}{8MS}\\
& \Phi = \frac{Q}{2M}\left[1+\frac{Q^2}{8S}\right].
\end{eqnarray*}
This is the analog of the first law of thermodynamics. [/*]
[*]Look at the expression for $M(S,J,Q)$ closely and derive the analog of the Gibbs-Duhem Relation familiar from Thermodynamics. [/*]
[*] Show that,
\[ S \to \frac{1}{4}M^2 - \frac{1}{8}Q^2 \]
as $T \to 0$. What does this say about the third law of thermodynamics? Give reasons to support your answer.
\item An alternative statement to the third law is that "it is impossible to reach absolute-zero in a finite number of steps". What can we conclude from part (e)? [/*]
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a) Pupils of the $8$-th form are standing in a row. There is the pupil of the $7$-th form in before each, and he is smaller (in height) than the elder. Prove that if you will sort the pupils in each of rows with respect to their height, every 8-former will still be taller than the $7$-former standing before him.
b) An infantry detachment soldiers stand in the rectangle, being arranged in columns with respect to their height. Prove that if you rearrange them with respect to their height in every separate row, they will still be staying with respect to their height in columns.
Sokal da tries to find out the largest positive integer n such that if n transforms to base-7, then it looks like twice of base-10. $156$ is such a number because $(156)_{10}$ = $(312)_7$ and 312 = 2$\times$156. Find out Sokal da's number.
Is it true that for each natural number $n$ there exist a circle, which contains exactly $n$ points with integer coordinates?
Let $n$ be a fixed positive integer. Find the minimum value of $$\frac{x_1^3+\dots+x_n^3}{x_1+\dots+x_n}$$ where $x_1,x_2,\dots,x_n$ are distinct positive integers.
Find the real numbers $ x,y,z $ that satisfy the following:
$ \text{(i)} -2\le x\le y\le z $
$ \text{(ii)} x+y+z=2/3 $
$ \text{(iii)} \frac{1}{x^2} +\frac{1}{y^2} +\frac{1}{z^2} =\frac{1}{x} +\frac{1}{y} +\frac{1}{z} +\frac{3}{8} $
[i]Cristinel Mortici[/i]
We call the set $A\in \mathbb R^n$ CN if and only if for every continuous $f:A\to A$ there exists some $x\in A$ such that $f(x)=x$.
a) Example: We know that $A = \{ x\in\mathbb R^n | |x|\leq 1 \}$ is CN.
b) The circle is not CN.
Which one of these sets are CN?
1) $A=\{x\in\mathbb R^3| |x|=1\}$
2) The cross $\{(x,y)\in\mathbb R^2|xy=0,\ |x|+|y|\leq1\}$
3) Graph of the function $f:[0,1]\to \mathbb R$ defined by
\[f(x)=\sin\frac 1x\ \mbox{if}\ x\neq0,\ f(0)=0\]