Found problems: 85335
Andrew writes down all of the prime numbers less than $50$. How many times does he write the digit $2$?
Evaluate $\displaystyle\sum_{n=1}^\infty \dfrac{1}{n^2+2n}$.
I was wondering if anyone had a sol for this. I am probably just going to bash it out.
Given a ball $K$. Find the locus of the vertices $A$ of all parallelograms $ABCD$ such that $ AC \leq BD$, and the diagonal $BD$ lies completely inside the ball $K$.
Let acute scalene triangle $ABC$ have orthocenter $H$ and altitude $AD$ with $D$ on side $BC$. Let $M$ be the midpoint of side $BC$, and let $D'$ be the reflection of $D$ over $M$. Let $P$ be a point on line $D'H$ such that lines $AP$ and $BC$ are parallel, and let the circumcircles of $\triangle AHP$ and $\triangle BHC$ meet again at $G \neq H$. Prove that $\angle MHG = 90^\circ$.
[i]Proposed by Daniel Hu.[/i]
Let $\alpha > 1$ be an irrational number and $L$ be a integer such that $L > \frac{\alpha^2}{\alpha - 1}$. A sequence $x_1, x_2, \cdots$ satisfies that $x_1 > L$ and for all positive integers $n$, \[ x_{n+1} = \begin{cases} \left \lfloor \alpha x_n \right \rfloor & \textup{if} \; x_n \leqslant L \\\left \lfloor \frac{x_n}{\alpha} \right \rfloor & \textup{if} \; x_n > L \end{cases}. \]
Prove that
(i) $\left\{x_n\right\}$ is eventually periodic.
(ii) The eventual fundamental period of $\left\{x_n\right\}$ is an odd integer which doesn't depend on the choice of $x_1$.
Find all groups of positive integers $ (a,x,y,n,m)$ that satisfy $ a(x^n \minus{} x^m) \equal{} (ax^m \minus{} 4) y^2$ and $ m \equiv n \pmod{2}$ and $ ax$ is odd.
The sequence of integers $\{a_i\}_{i = 0}^{\infty}$ satisfies $a_0 = 3$, $a_1 = 4$, and
\[a_{n+2} = a_{n+1} a_n + \left\lceil \sqrt{a_{n+1}^2 - 1} \sqrt{a_n^2 - 1}\right\rceil\]
for $n \ge 0$. Evaluate the sum
\[\sum_{n = 0}^{\infty} \left(\frac{a_{n+3}}{a_{n+2}} - \frac{a_{n+2}}{a_n} + \frac{a_{n+1}}{a_{n+3}} - \frac{a_n}{a_{n+1}}\right).\]
There is a game. The magician let the participant think up a positive integer (at least two digits). For example, an integer $ \displaystyle\overline{a_1a_2 \cdots a_n}$ is rearranged as $ \overline{a_{i_1}a_{i_2} \cdots a_{i_n}}$, that is, $ i_1, i_2, \cdots, i_n$ is a permutation of $ 1,2, \cdots, n$. Then we get $ n!\minus{}1$ integers. The participant is asked to calculate the sum of the $ n!\minus{}1$ numbers, then tell the magician the sum $ S$. The magician claims to be able to know the original number when he is told the sum $ S$. Try to decide whether the magician can be successful or not.
Prove that for each $k$ points in the plane, no three collinear and having integral distances from each other. If we have an infinite set of points with integral distances from each other, then all points are collinear.
[i](Anonymous314)[/i]
PS. wording needs to be fixed , [url=http://www.mathcenter.net/forum/showthread.php?t=7288]source[/url]
Find all positive integers $n$ such that the number $\frac{(2n)!+1}{n!+1}$ is positive integer.
Find a positive integrer number $n$ such that, if yor put a number $2$ on the left and a number $1$ on the right, the new number is equal to $33n$.
Let $a, b, c$ be real numbers such that $0 < a, b, c < 1/2$ and $a + b + c= 1$. Prove that for all real numbers $x,y,z$,
$$abc(x + y + z)^2 \ge ayz( 1- 2a) + bxz( 1 - 2b) + cxy( 1 - 2c)$$. When does equality hold?
In triangle $ABC$, the difference between angles $B$ and $C$ is equal to $90^o$, and $AL$ is the angle bisector of triangle $ABC$. The bisector of the exterior angle $A$ of the triangle $ABC$ intersects the line $BC$ at the point $F$. Prove that $AL = AF$.
(Alexander Dzyunyak)
Temerant is a spherical planet with radius $1000$ kilometers. The government wants to build twelve towers of the same height on the equator of Temerant, so that every point on the equator can be seen from at least one tower. The minimum possible height of the towers can be written, in kilometers, as a $\sqrt{b} - c\sqrt{d} - e$ for positive integers $a, b, c, d$, and $e$ (with $b$ and $d$ not divisible by the square of any prime). Compute $a + b + c + d + e$.
One side of a triangle is equal to one third of the sum of the other two. Prove that the angle opposite the first side is the smallest angle of the triangle.
(AK Tolpygo)
Hexagonal pieces numbered by positive integers are placed on the cells of a hexagonal board with side $n$. Two adjacent cells are left empty, and thanks to it some pieces can be moved. Two pieces with common sides exchanged places (see an example in the attachment 2).
Prove that if $n \ge 3$ the second arrangement cannot be obtained from the first one by moving piece
Note. Moving a piece a requires two adjacent empty cells. For instance, if they are on the right of a (attachment 1, left figure), a can be moved right till it touches an angle (attachment 1, middle figure), and then it can be moved upward right or downward right (attachment 1, right figure)
$(USS 3)$ $(a)$ Prove that if $0 \le a_0 \le a_1 \le a_2,$ then $(a_0 + a_1x - a_2x^2)^2 \le (a_0 + a_1 + a_2)^2\left(1 +\frac{1}{2}x+\frac{1}{3}x^2+\frac{1}{2}x^3+x^4\right)$
$(b)$ Formulate and prove the analogous result for polynomials of third degree.
If \[2011^{2011^{2012}} = x^x\] for some positive integer $x$, how many positive integer factors does $x$ have?
[i]Author: Alex Zhu[/i]
In a trapezoid $ABCD$ such that the internal angles are not equal to $90^{\circ}$, the diagonals $AC$ and $BD$ intersect at the point $E$. Let $P$ and $Q$ be the feet of the altitudes from $A$ and $B$ to the sides $BC$ and $AD$ respectively. Circumscribed circles of the triangles $CEQ$ and $DEP$ intersect at the point $F\neq E$. Prove that the lines $AP$, $BQ$ and $EF$ are either parallel to each other, or they meet at exactly one point.
Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\displaystyle\frac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$.
[i]Evan O' Dorney.[/i]
Let $ABC$ be a triangle with angles $\angle A = 80^\circ, \angle B = 70^\circ, \angle C = 30^\circ$. Let $P$ be a point on the bisector of $\angle BAC$ satisfying $\angle BPC =130^\circ$. Let $PX, PY, PZ$ be the perpendiculars drawn from $P$ to the sides $BC, AC, AB$, respectively.
Prove that the following equation with segment lengths is satisfied
$$AY^3+BZ^3+CX^3=AZ^3+BX^3+CY^3.$$
There is a list of seven numbers. The average of the first four numbers is $5$, and the average of the last four numbers is $8$. If the average of all seven numbers is $6\frac{4}{7}$, then the number common to both sets of four numbers is
$\text{(A)}\ 5\frac{3}{7} \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 6\frac{4}{7} \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 7\frac{3}{7}$
A rectangular table with 2001 rows and 2002 columns is partitioned into $1\times 2$ rectangles. It is known that any other partition of the table into $1\times 2$ rectangles contains a rectangle belonging to the original partition.
Prove that the original partition contains two successive columns covered by 2001 horizontal rectangles.
[i]Proposed by S. Volchenkov[/i]
[b](i)[/b] Solve the equation:
\[ \sin^3(x) + \sin^3\left( \frac{2 \pi}{3} + x\right) + \sin^3\left( \frac{4 \pi}{3} + x\right) + \frac{3}{4} \cos {2x} = 0.\]
[b](ii)[/b] Supposing the solutions are in the form of arcs $AB$ with one end at the point $A$, the beginning of the arcs of the trigonometric circle, and $P$ a regular polygon inscribed in the circle with one vertex in $A$, find:
1) The subsets of arcs having the other end in $B$ in one of the vertices of the regular dodecagon.
2) Prove that no solution can have the end $B$ in one of the vertices of polygon $P$ whose number of sides is prime or having factors other than 2 or 3.