Found problems: 85335
The sequence $(a_n)$ of complex numbers is considered in the complex plane, in which is: $$a_0 = 1, \,\,\, a_n = a_{n-1} +\frac{1}{n}(\cos 45^o + i \sin 45^o )^n.$$
Prove that the sequence of the real parts of the terms of $(a_n)$ is convergent and its limit is a number between $0.85$ and $1.15$.
For which positive integers $b > 2$ do there exist infinitely many positive integers $n$ such that $n^2$ divides $b^n+1$?
[i]Evan Chen and Ankan Bhattacharya[/i]
Let $(t_n)_n$ a convergent sequence of real numbers, $t_n\in (0,1),\ (\forall)n\in \mathbb{N}$ and $\lim_{n\to \infty} t_n\in (0,1)$. Define the sequences $(x_n)_n$ and $(y_n)_n$ by
\[x_{n+1}=t_nx_n+(1-t_n)y_n,\ y_{n+1}=(1-t_n)x_n+t_n y_n,\ (\forall)n\in \mathbb{N}\]
and $x_0,y_0$ are given real numbers.
a) Prove that the sequences $(x_n)_n$ and $(y_n)_n$ are convergent and have the same limit.
b) Prove that if $\lim_{n\to \infty} t_n\in \{0,1\}$, then the question is false.
If $a,b,x,y$ are real numbers then find all solutions of $a,b$ such that $\left \lfloor ax+by \right \rfloor + \left \lfloor bx+ay \right \rfloor = (a+b) \left \lfloor x+y \right \rfloor$
Let $S$ be a set not empty of positive integers and $AB$ a segment with, initially, only points $A$ and $B$ colored by red. An operation consists of choosing two distinct points $X, Y$ colored already by red and $n\in S$ an integer, and painting in red the $n$ points $A_1, A_2,..., A_n$ of segment $XY$ such that $XA_1= A_1A_2= A_2A_3=...= A_{n-1}A_n= A_nY$ and $XA_1<XA_2<...<XA_n$. Find the least positive integer $m$ such exists a subset $S$ of $\{1,2,.., m\}$ such that, after a finite number of operations, we can paint in red the point $K$ in the segment $AB$ defined by $\frac{AK}{KB}= \frac{2709}{2022}$. Also, find the number of such subsets for such a value of $m$.
Let $E$ be a point outside of square $ABCD$. If the distance of $E$ to $AC$ is $6$, to $BD$ is $17$, and to the nearest vertex of the square is $10$, what is the area of the square?
$ \textbf{(A)}\ 200
\qquad\textbf{(B)}\ 196
\qquad\textbf{(C)}\ 169
\qquad\textbf{(D)}\ 162
\qquad\textbf{(E)}\ 144
$
A straight line tangent to a circle circumscribed about an isosceles triangle $ABC$ ($AB = AC$) at point $B$ intersects straight line $AC$ at point $P$, $E$ is the midpoint of $AB$ (fig.). What is the projection of $DE$ onto $AB$ if $PA = a$?
[img]https://cdn.artofproblemsolving.com/attachments/e/3/59c67e8f5eb3d399656d86613bc699c8baf1c1.png[/img]
Say that a positive integer is [i]sweet[/i] if it uses only the digits 0, 1, 2, 4, and 8. For instance, 2014 is sweet. There are sweet integers whose squares are sweet: some examples (not necessarily the smallest) are 1, 2, 11, 12, 20, 100, 202, and 210. There are sweet integers whose cubes are sweet: some examples (not necessarily the smallest) are 1, 2, 10, 20, 200, 202, 281, and 2424. Prove that there exists a sweet positive integer $n$ whose square and cube are both sweet, such that the sum of all the digits of $n$ is 2014.
Let $x_1, x_2, \dots, x_n$ be different real numbers. Prove that
\[\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}=\left\{\begin{array}{ll}
0, & \text { if } n \text { is even; } \\
1, & \text { if } n \text { is odd. }
\end{array}\right.\]
Let $S$ the set of natural numbers from $1$ up to $1001$ , $S=\{1,2,...,1001\}$. Lisandro thinks of a number $N$ of $S$ , and Carla has to find out that number with the following procedure. She gives Lisandro a list of subsets of $S$,
Lisandro reads it and tells Carla how many subsets of her list contain $N$ . If Carla wishes, she can repeat the same thing with a second list, and then with a third, but no more than $3$ are allowed. What is the smallest total number of subsets that allow Carla to find $N$ for sure?
Let $AL$ be the bisector of triangle $ABC$, $D$ be its midpoint, and $E$ be the projection of $D$ to $AB$. It is known that $AC = 3AE$. Prove that $CEL$ is an isosceles triangle.
Three straight lines $\ell_1$, $\ell_2$ and $\ell_3$, forming a triangle, divide the plane into $7$ parts. Each of the points $M_1$, $M_2$ and $M_3$ lies in one of the angles, vertical to some angle of the triangle. The distance from $M_1$ to straight lines $\ell_1$, $\ell_2$ and $\ell_3$ are equal to $7,3$ and $1$ respectively The distance from $M_2$ to the same lines are $4$, $1$ and $3$ respectively. For $M_3$ these distances are $3$, $5$ and $2$. What is the radius of the circle inscribed in the triangle?
[hide=second sentence in Russian]Каждая из точек М_1, М_2 и М_з лежит в одном из углов, вертикальном по отношению к какому-то углу треугольника.[/hide]
A circle is inscribed in a triangle $ABC$ with sides $a,b,c$. Tangents to the circle parallel to the sides of the triangle are contructe. Each of these tangents cuts off a triagnle from $\triangle ABC$. In each of these triangles, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of $a,b,c$).
Let $n\geq 2$ be a fixed positive integer. Let $\{a_1,a_2,...,a_n\}$ be fixed positive integers whose sum is $2n-1$. Denote by $S_{\mathbb{A}}$ the sum of elements of a set $A$. Find the minimal and maximal value of $S_{\mathbb{X}}\cdot S_{\mathbb{Y}}$ where $\mathbb{X}$ and $\mathbb{Y}$ are two sets with the property that $\mathbb{X}\cup \mathbb{Y}=\{a_1,a_2,...,a_n\}$ and $\mathbb{X}\cap \mathbb{Y}=\emptyset.$
[i]Note: $\mathbb{X}$ and $\mathbb{Y}$ can have multiple equal elements. For example, when $n=5$ and $a_1=...=a_4=1$ and $a_5=5$, we can consider $\mathbb{X}=\{1,1,1\}$ and $\mathbb{Y}=\{1,5\}$. Moreover, in this case, $S_\mathbb{X}=3$ and $S_{\mathbb{Y}}=6.$[/i]
Let $ABC$ be an acute triangle with $AB>AC$. Prove that there exists a point $D$ with the following property: whenever two distinct points $X$ and $Y$ lie in the interior of $ABC$ such that the points $B$, $C$, $X$, and $Y$ lie on a circle and
$$\angle AXB-\angle ACB=\angle CYA-\angle CBA$$
holds, the line $XY$ passes through $D$.
Let $\triangle ABC$ be a triangle with $BC = 4, CA= 5, AB= 6$, and let $O$ be the circumcenter of $\triangle ABC$. Let $O_b$ and $O_c$ be the reflections of $O$ about lines $CA$ and $AB$ respectively. Suppose $BO_b$ and $CO_c$ intersect at $T$, and let $M$ be the midpoint of $BC$. Given that $MT^2 = \frac{p}{q}$ for some coprime positive integers $p$ and $q$, find $p+q$.
[i]Proposed by Sreejato Bhattacharya[/i]
The numbers $1,2,\ldots,64$ are written in the squares of an $8\times 8$ chessboard, one number to each square. Then $2\times 2$ tiles are placed on the chessboard (without overlapping) so that each tile covers exactly four squares whose numbers sum to less than $100$. Find, with proof, the maximum number of tiles that can be placed on the chessboard, and give an example of a distribution of the numbers $1,2,\ldots,64$ into the squares of the chessboard that admits this maximum number of tiles.
Consider all line segments of length 4 with one end-point on the line $y=x$ and the other end-point on the line $y=2x$. Find the equation of the locus of the midpoints of these line segments.
Sir Alex is coaching a soccer team of $n$ players of distinct heights. He wants to line them up so that for each player $P$, the total number of players that are either to the left of $P$ and taller than $P$ or to the right of $P$ and shorter than $P$ is even. In terms of $n$, how many possible orders are there?
[i]Michael Ren[/i]
Let $S$ be the set of points inside and on the boarder of a regular haxagon with side length 1. Find the least constant $r$, such that there exists one way to colour all the points in $S$ with three colous so that the distance between any two points with same colour is less than $r$.
In a sequence, $x_1=\frac{1}{2}$ and $x_{n+1}=1-x_1x_2x_3...x_n$ for $n\ge 1$. Prove that $0.99<x_{100}<0.991$.
Fresh translation. This problem may be similar to one of the 9th grade problems.
The polynomials $P(z)$ and $Q(z)$ with complex coefficients have the same set of numbers for their zeros but possibly different multiplicities. The same is true for the polynomials
$$P(z)+1 \;\; \text{and} \;\; Q(z)+1.$$
Prove that $P(z)=Q(z).$
Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$.
[i]Proposed by Evan Chen, Taiwan[/i]
Let $\Gamma$ be the circumcircle of $\triangle ABC$. Let $D$ be a point on the side $BC$. The tangent to $\Gamma$ at $A$ intersects the parallel line to $BA$ through $D$ at point $E$. The segment $CE$ intersects $\Gamma$ again at $F$. Suppose $B$, $D$, $F$, $E$ are concyclic. Prove that $AC$, $BF$, $DE$ are concurrent.
The table below gives the percent of students in each grade at Annville and Cleona elementary schools:
\[\begin{tabular}{rccccccc}
& \textbf{\underline{K}} & \textbf{\underline{1}} & \textbf{\underline{2}} & \textbf{\underline{3}} & \textbf{\underline{4}} & \textbf{\underline{5}} & \textbf{\underline{6}} \\
\textbf{Annville:} & 16\% & 15\% & 15\% & 14\% & 13\% & 16\% & 11\% \\
\textbf{Cleona:} & 12\% & 15\% & 14\% & 13\% & 15\% & 14\% & 17\%
\end{tabular}\]
Annville has 100 students and Cleona has 200 students. In the two schools combined, what percent of the students are in grade 6?
$\text{(A)}\ 12\% \qquad \text{(B)}\ 13\% \qquad \text{(C)}\ 14\% \qquad \text{(D)}\ 15\% \qquad \text{(E)}\ 28\%$