Found problems: 85335
(a) Prove that a square with sides $1000$ divided into $31$ squares tiles, at least one of which has a side length less than $1$.
(b) Show that a corresponding decomposition into $30$ squares is also possible.
[i](Walther Janous)[/i]
There is a point inside a regular triangle located at distances $5$, $6$ and $7$ from its vertices. Find the area of this regular triangle.
For positive integers $n, m$ define $f(n,m)$ as follows. Write a list of $ 2001$ numbers $a_i$, where $a_1 = m$, and $a_{k+1}$ is the residue of $a_k^2$ $mod \, n$ (for $k = 1, 2,..., 2000$). Then put $f(n,m) = a_1-a_2 + a_3 -a_4 + a_5- ... + a_{2001}$. For which $n \ge 5$ can we find m such that $2 \le m \le n/2$ and $f(m,n) > 0$?
[b]p1.[/b] Prove that $x = 2$ is the only real number satisfying $3^x + 4^x = 5^x$.
[b]p2.[/b] Show that $\sqrt{9 + 4\sqrt5} -\sqrt{9 - 4\sqrt5}$ is an integer.
[b]p3.[/b] Two players $A$ and $B$ play a game on a round table. Each time they take turn placing a round coin on the table. The coin has a uniform size, and this size is at least $10$ times smaller than the table size. They cannot place the coin on top of any part of other coins, and the whole coin must be on the table. If a player cannot place a coin, he loses. Suppose $A$ starts first. If both of them plan their moves wisely, there will be one person who will always win. Determine whether $A$ or $B$ will win, and then determine his winning strategy.
[b]p4.[/b] Suppose you are given $4$ pegs arranged in a square on a board. A “move” consists of picking up a peg, reflecting it through any other peg, and placing it down on the board. For how many integers $1 \le n \le 2013$ is it possible to arrange the $4$ pegs into a [i]larger [/i] square using exactly $n$ moves? Justify your answers.
[b]p5.[/b] Find smallest positive integer that has a remainder of $1$ when divided by $2$, a remainder of $2$ when divided by $3$, a remainder of $3$ when divided by $5$, and a remainder of $5$ when divided by $7$.
[b]p6.[/b] Find the value of $$\sum_{m|496,m>0} \frac{1}{m},$$
where $m|496$ means $496$ is divisible by $m$.
[b]p7.[/b] What is the value of
$${100 \choose 0}+{100 \choose 4}+{100 \choose 8}+ ... +{100 \choose 100}?$$
[b]p8.[/b] An $n$-term sequence $a_0, a_1, ...,a_n$ will be called [i]sweet [/i] if, for each $0 \le i \le n -1$, $a_i$ is the number of times that the number $i$ appears in the sequence. For example, $1, 2, 1,0$ is a sweet sequence with $4$ terms. Given that $a_0$, $a_1$, $...$, $a_{2013}$ is a sweet sequence, find the value of $a^2_0+ a^2_1+ ... + a^2_{2013}.$
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
What sizes of squares with integer sides can be completely covered without overlap by identical tiles consisting of three squares with side $1$ joined together in one $L$ shape?
[center][img]https://cdn.artofproblemsolving.com/attachments/3/f/9fe95b05527857f7e44dfd033e6fb01e5d25a2.png[/img][/center]
$(FRA 3)$ A polygon (not necessarily convex) with vertices in the lattice points of a rectangular grid is given. The area of the polygon is $S.$ If $I$ is the number of lattice points that are strictly in the interior of the polygon and B the number of lattice points on the border of the polygon, find the number $T = 2S- B -2I + 2.$
The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).
Let $n$ be an integer greater than 1. The set $S$ of all diagonals of a $ \left( 4n-1\right) $-gon is partitioned into $k$ sets, $S_{1},S_{2},\ldots ,S_{k},$ so that, for every pair of distinct indices $i$ and $j,$ some diagonal in $S_{i}$ crosses some diagonal in $S_{j};$ that is, the two diagonals share an interior point. Determine the largest possible value of $k $ in terms of $n.$
Given an integer $n\ge 2$, given $n+1$ distinct points $X_0,X_1,\ldots,X_n$ in the plane, and a positive real number $A$, show that the number of triangles $X_0X_iX_j$ of area $A$ does not exceed $4n\sqrt n$.
Find all positive integers $n$ such that the unit segments of an $n \times n$ grid of unit squares can be partitioned into groups of three such that the segments of each group share a common vertex.
If $a$ and $b$ are coprime positive integers and $n$ an integer, prove that the greatest common divisor of $a^2+b^2-nab$ and $a+b$ divides $n+2$.
There are three colleges in a town. Each college has $n$ students. Any student of any college knows $n+1$ students of the other two colleges. Prove that it is possible to choose a student from each of the three colleges so that all three students would know each other.
For real numbers $x_i$, the statement \[ x_1 + x_2 + x_3 = 0 \Rightarrow x_1x_2 + x_2x_3 + x_3x_1 \leq 0\] is always true. (Prove!)
For which $n\geq 4$ integers, the statement \[x_1 + x_2 + \dots + x_n = 0 \Rightarrow x_1x_2 + x_2x_3 + \dots + x_{n-1}x_n + x_nx_1 \leq 0\] is always true. Justify your answer.
[list=a]
[*]Prove that for every real number $t$ such that $0 < t < \tfrac{1}{2}$ there exists a positive integer $n$ with the following property: for every set $S$ of $n$ positive integers there exist two different elements $x$ and $y$ of $S$, and a non-negative integer $m$ (i.e. $m \ge 0 $), such that \[ |x-my|\leq ty.\]
[*]Determine whether for every real number $t$ such that $0 < t < \tfrac{1}{2} $ there exists an infinite set $S$ of positive integers such that \[|x-my| > ty\] for every pair of different elements $x$ and $y$ of $S$ and every positive integer $m$ (i.e. $m > 0$).
If $p (x) = c_0 + c_1x + c_2x^2 + c_3x^3$ is a polynomial with integer coefficients with $a, b,c$ integers and different from each other, prove that it cannot happen simultaneously that $p (a) = b$, $p (b) = c$ and $p (c) = a$.
A [i]lattice point[/i] is a point whose coordinates are both integers. Suppose Johann walks in a line from the point $(0, 2004)$ to a random lattice point in the interior (not on the boundary) of the square with vertices $(0, 0)$, $(0, 99)$, $(99,99)$, $(99, 0)$. What is the probability that his path, including the endpoints, contains an even number of lattice points?
Let $S$ be the set of [b]discs[/b] $D$ contained completely in the set $\{ (x,y) : y<0\}$ (the region below the $x$-axis) and centered (at some point) on the curve $y=x^2-\frac{3}{4}$. What is the area of the union of the elements of $S$?
Find the least positive number m such that for any polynimial f(x) with real coefficients, there is a polynimial g(x) with real coefficients (degree not greater than m) such that there exist 2017 distinct number $a_1,a_2,...,a_{2017}$ such that $g(a_i)=f(a_{i+1})$ for i=1,2,...,2017 where indices taken modulo 2017.
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.
$ABCD$ - convex quadrilateral. $\angle A+ \angle D=150, \angle B<150, \angle C<150$ Prove, that area $ABCD$ is greater than $\frac{1}{4}(AB*CD+AB*BC+BC*CD)$
Let $O$ be a point on the oriented Euclidean plane and $(\mathbf i, \mathbf j)$ a directly oriented orthonormal basis. Let $C$ be the circle of radius $1$, centered at $O$. For every real number $t$ and non-negative integer$ n$ let $M_n$ be the point on $C$ for which $\langle \mathbf i , \overrightarrow{OM_n} \rangle = \cos 2^n t.$ (or $\overrightarrow{OM_n} =\cos 2^n t \mathbf i +\sin 2^n t \mathbf j$).
Let $k \geq 2$ be an integer. Find all real numbers $t \in [0, 2\pi)$ that satisfy
[b](i)[/b] $M_0 = M_k$, and
[b](ii)[/b] if one starts from $M0$ and goes once around $C$ in the positive direction, one meets successively the points $M_0,M_1, \dots,M_{k-2},M_{k-1}$, in this order.
In triangle $ABC$, on line $CA$ it is given point $D$ such that $CD = 3 \cdot CA$ (point $A$ is between points $C$ and $D$), and on line $BC$ it is given point $E$ ($E \neq B$) such that $CE=BC$. If $BD=AE$, prove that $\angle BAC= 90^{\circ}$
Prove that if the product of $n$ real and positive numbers is equal to $1$, its sum is greater than or equal to $n$.
Express
$$\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^2 n +m n^2 +2mn }$$
as a rational number.
The graph of the degree $2021$ polynomial $P(x)$, which has real coefficients and leading coefficient $1$, meets the $x$-axis at the points $(1,0),\, (2,0),\,(3,0),\dots,\, (2020,0)$ and nowhere else. The mean of all possible values of $P(2021)$ can be written in the form $a!/b$, where $a$ and $b$ are positive integers and $a$ is as small as possible. Compute $a+b$.