Found problems: 85335
Let $S$ be a list of positive integers - not necessarily distinct - in which the number 68 appears. The average (arithmetic mean) of the numbers in $S$ is 56. However, if 68 is removed, the average of the remaining numbers drops to 55. What is the largest number that can appear in $S$?
In a coordinate plane, consider the set of points with integer cooedinates at least one of which is not divisible by $4$. Prove that these points cannot be partitioned into pairs such that the distance between points in each pair equals $1$.
In other words, an infinite chessboard, whose cells with both coordinates divisible by $4$ are cut out, cannot be tiled by dominoes.
Given a quadrilateral $ABCD$ in which$$\sqrt{2}(BC-BA)=AC.$$Let $X$ be the midpoint of $AC$. Prove that $2\angle BXD=\angle DAB - \angle DCB.$
For positive $a,b,c$, \[a^{2}+b^{2}+c^{2}+abc=4\] Prove $a+b+c \leq3$
Triangle $ABC$ has circumcircle $\omega$. Let $D$ be the foot of the altitude from $A$ to $BC$ and let $AD$ intersect $\omega$ at $E \neq A$. Let $M$ be the midpoint of $AD$. If $\angle{BMC} = 90^\circ$, $AB = 9$ and $AE = 10$, the area of $\triangle{ABC}$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a,b,c$ are positive integers and $b$ is square-free. Find $a+b+c$.
[i]Proposed by Andy Xu[/i]
Let $\mathcal{P}$ be a set of monic polynomials with integer coefficients of the least degree, with root $k \cdot \cos\left(\frac{4\pi}{7}\right)$, as $k$ spans over the positive integers. Let $P(x) \in \mathcal{P}$ be the polynomial so that $|P(1)|$ is minimized. Find the remainder when $P(2017)$ is divided by $1000$.
[i]Proposed by [b] eisirrational [/b][/i]
A disk with radius $1$ is externally tangent to a disk with radius $5$. Let $A$ be the point where the disks are tangent, $C$ be the center of the smaller disk, and $E$ be the center of the larger disk. While the larger disk remains fixed, the smaller disk is allowed to roll along the outside of the larger disk until the smaller disk has turned through an angle of $360^\circ$. That is, if the center of the smaller disk has moved to the point $D$, and the point on the smaller disk that began at $A$ has now moved to point $B$, then $\overline{AC}$ is parallel to $\overline{BD}$. Then $\sin^2(\angle BEA)=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
If $x(y + 1) = 41$ and $x^2(y^2 + 1) = 881$, determine all possible pairs of real numbers $(x,y)$.
Find all positive integers n such that $ (n ^{ 2} + 11n - 4) n! + 33.13 ^ n + 4 $ is the perfect square
Prove that $9$-digit number, that contains all the decimal digits except zero and does not ends with $5$ can not be exact square.
Let $n$ be an integer greater than $6$.Show that if $n+1$ is a prime number,than
$\left\lceil \frac{(n-1)!}{n(n+1)}\right \rceil$ is $ODD.$
Call a positive integer $N \ge 2$ [i]special [/i] if for every k such that $2 \le k \le N, N$ can be expressed as a sum of $k$ positive integers that are relatively prime to $N$ (although not necessarily relatively prime to each other). Find all special positive integers.
A $8\times 8$ board is given. Seven out of $64$ unit squares are painted black. Suppose that there exists a positive $k$ such that no matter which squares are black, there exists a rectangle (with sides parallel to the sides of the board) with area $k$ containing no black squares. Find the maximum value of $k$.
Let $n>100$ be a positive integer and originally the number $1$ is written on the blackboard. Petya and Vasya play the following game: every minute Petya represents the number of the board as a sum of two distinct positive fractions with coprime nominator and denominator and Vasya chooses which one to delete. Show that Petya can play in such a manner, that after $n$ moves, the denominator of the fraction left on the board is at most $2^n+50$, no matter how Vasya acts.
Yuri and Vlad are playing a game on the table $4 \times 100$. Firstly, Yuri chooses $73$ squares $2 \times 2$ (squares can intersect, but cannot be equal). Then Vlad colours the cells of the table in $4$ colours such that in any row and in any column, and in any square chosen by Yuri, there were cells of all 4 colours. After that Vlad pays 2 rubles for every square $2 \times 2$, not chosen by Yuri, which cells of all 4 colours.
What is the maximum possible number of rubles Yuri can get regardless of Vlad's actions
[i]M. Shutro[/i]
$\bullet$ Determine a natural $n$ such that the constant sum $S$ of a magic square of $ n \times n$ (that is, the sum of its elements in any column, or the diagonal) differs as little as possible from $1992$.
$\bullet$ Construct or describe the construction of this magic square.
Three distinct vertices are chosen at random from the vertices of a given regular polygon of $ (2n\plus{}1)$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen random points?
$$Problem 1$$ ;Find all composite numbers $n$ with the following property: For every proper divisor $d$ of $n$ (i.e. $1 < d < n$), it holds that $n-12 \geq d \geq n-20$.
Amy made a list of every possible distinct five-digit positive integer that can be formed using each of the digits $1, 2, 3, 4,$ and $5$ exactly once in each integer. What is the sum of the integers on Amy's list?
$\mathrm{(A) \ } 3000000 \qquad \mathrm{(B) \ } 3600000 \qquad \mathrm {(C) \ } 3999960 \qquad \mathrm{(D) \ } 3999990 \qquad \mathrm{(E) \ } 5999940$
Find the least positive integer $n$ such that \[ \frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}. \]
Consider the numbers $ a_n=1-\binom{n}{3} +\binom{n}{6} -\cdots, b_n= -\binom{n}{1} +\binom{n}{4}-\binom{n}{7} +\cdots $ and $ c_n=\binom{n}{2} -\binom{n}{5} +\binom{n}{8} -\cdots , $ for a natural number $ n\ge 2. $ Prove that
$$ a_n^2+b_n^2+c_n^2-a_nb_n-b_nc_n-c_na_n =3^{n-1}. $$
Find the number of strictly increasing sequences of nonnegative integers with the following properties:
• The first term is $0$ and the last term is $12$. In particular, the sequence has at least two terms.
• Among any two consecutive terms, exactly one of them is even.
Let $a, b, c$ be positive reals, and let $\sqrt[2013]{\frac{3}{a^{2013}+b^{2013}+c^{2013}}}=P$. Prove that
\[\prod_{\text{cyc}}\left(\frac{(2P+\frac{1}{2a+b})(2P+\frac{1}{a+2b})}{(2P+\frac{1}{a+b+c})^2}\right)\ge \prod_{\text{cyc}}\left(\frac{(P+\frac{1}{4a+b+c})(P+\frac{1}{3b+3c})}{(P+\frac{1}{3a+2b+c})(P+\frac{1}{3a+b+2c})}\right).\][i]Proposed by David Stoner[/i]
Let $a_1,a_2,a_3,\ldots$ be an infinite sequence where for all positive integers $i$, $a_i$ is chosen to be a random positive integer between $1$ and $2016$, inclusive. Let $S$ be the set of all positive integers $k$ such that for all positive integers $j<k$, $a_j\neq a_k$. (So $1\in S$; $2\in S$ if and only if $a_1\neq a_2$; $3\in S$ if and only if $a_1\neq a_3$ and $a_2\neq a_3$; and so on.) In simplest form, let $\dfrac{p}{q}$ be the expected number of positive integers $m$ such that $m$ and $m+1$ are in $S$. Compute $pq$.
Any positive integer $n$ can be written in the form $n = 2^aq$, where $a \ge 0$ and $q$ is odd. We call $q$ the [i]odd part[/i] of $n$. Define the sequence $a_0,a_1,...$ as follows: $a_0 = 2^{2011}-1$ and for $m > 0, a_{m+i}$ is the odd part of $3a_m + 1$. Find $a_{2011}$.