Found problems: 85335
Let $\{F_n\}^\infty_{n=1}$ be the Fibonacci sequence and let $f$ be a polynomial of degree $1006$ such that $f(k) = F_k$ for all $k \in \{1008, \dots , 2014\}$. Prove that $$233\mid f(2015)+1.$$
[i]Note: $F_1=F_2=1$ and $F_{n+2}=F_{n+1}+F_n$ for all $n\geq 1$.[/i]
Let $ABC$ be a triangle satisfying $AB<AC$. Let $M$ be the midpoint of $BC$. A point $P$ lies on the segment $AB$ with $AP>PB$. A point $Q$ lies on the segment $AC$ with $\angle MPA = \angle AQM$. The perpendicular bisectors of $BC$ and $PQ$ intersect at $S$. Prove that $\angle BAC + \angle QSP = \angle QMP$.
In a television series about incidents in a conspicuous town there are $n$ citizens staging in it, where $n$ is an integer greater than $3$. Each two citizens plan together a conspiracy against one of the other citizens. Prove that there exists a citizen, against whom at least $\sqrt{n}$ other citizens are involved in the conspiracy.
The circles $k_1$ and $k_2$ intersect at points $D$ and $P$. The common tangent of the two circles on the side of $D$ touches $k_1$ at $A$ and $k_2$ at $B$. The straight line $AD$ intersects $k_2$ for a second time at $C$. Let $M$ be the center of the segment $BC$. Show that $ \angle DPM = \angle BDC$ .
Hello,
I've been trying to solve this for a while now, but no success! I mean, I have an expression for this but not a closed one. I derived something in terms of Tchebychev Polynomials : cos(nx) = P_n(cos(x)). Any help is appreciated.
Compute the following primitive:
\[ \int \frac{x\sin\left(2004 x\right)}{\tan x}\ dx\]
Let be a nondegenerate and closed interval $ I $ of real numbers, a short map $ m:I\longrightarrow I, $ and a sequence of functions $ \left( x_n \right)_{n\ge 1} :I\longrightarrow\mathbb{R} $ such that $ x_1 $ is the identity map and
$$ 2x_{n+1}=x_n+m\circ x_n , $$
for any natural numbers $ n. $ Prove that:
[b]a)[/b] there exists a nondegenerate interval having the property that any point of it is a fixed point for $ m. $
[b]b)[/b] $ \left( x_n \right)_{n\ge 1} $ is pointwise convergent and that its limit function is a short map.
Find all positive integers $n$ for which there exist distinct positive integers $a_1,a_2,\ldots,a_n$ such that
\[ \frac 1{a_1} + \frac 2{a_2} + \cdots + \frac n { a_n} = \frac { a_1 + a_2 + \cdots + a_n } n. \]
Given a parallelogram $ABCD$, let $\mathcal{P}$ be a plane such that the distance from vertex $A$ to $\mathcal{P}$ is $49$, the distance from vertex $B$ to $\mathcal{P}$ is $25$, and the distance from vertex $C$ to $\mathcal{P}$ is $36$. Find the sum of all possible distances from vertex $D$ to $\mathcal{P}$.
[i]Proposed by [b]HrishiP[/b][/i]
Let $A,B,C$ be points in that order along a line, such that $AB=20$ and $BC=18$. Let $\omega$ be a circle of nonzero radius centered at $B$, and let $\ell_1$ and $\ell_2$ be tangents to $\omega$ through $A$ and $C$, respectively. Let $K$ be the intersection of $\ell_1$ and $\ell_2$. Let $X$ lie on segment $\overline{KA}$ and $Y$ lie on segment $\overline{KC}$ such that $XY\|BC$ and $XY$ is tangent to $\omega$. What is the largest possible integer length for $XY$?
Find all natural numbers $a$, $b$, $c$ and prime numbers $p$ and $q$, such that:
$\blacksquare$ $4\nmid c$
$\blacksquare$ $p\not\equiv 11\pmod{16}$
$\blacksquare$ $p^aq^b-1=(p+4)^c$
Let $a,b,c,d$ are integers such that $(a,b)=(c,d)=1$ and $ad-bc=k>0$. Prove that there are exactly $k$ pairs $(x_{1},x_{2})$ of rational numbers with $0\leq x_{1},x_{2}<1$ for which both $ax_{1}+bx_{2},cx_{1}+dx_{2}$ are integers.
For how many integers $n$, there are four distinct real numbers satisfying the equation $ |x^2-4x-7|=n$?
$
\textbf{(A)}\ 12
\qquad\textbf{(B)}\ 10
\qquad\textbf{(C)}\ 8
\qquad\textbf{(D)}\ 7
\qquad\textbf{(E)}\ 5
$
Prove that for any positive integer $n$, the number of odd integers among the binomial coefficients $\binom nh \ ( 0 \leq h \leq n)$ is a power of 2.
Jordi stands 20 m from a wall and Diego stands 10 m from the same wall. Jordi throws a ball at an angle of 30 above the horizontal, and it collides elastically with the wall. How fast does Jordi need to throw the ball so that Diego will catch it? Consider Jordi and Diego to be the same height, and both are on the same perpendicular line from the wall.
$\textbf{(A) } 11 \text{ m/s}\\
\textbf{(B) } 15 \text{ m/s}\\
\textbf{(C) } 19 \text{ m/s}\\
\textbf{(D) } 30 \text{ m/s}\\
\textbf{(E) } 35 \text{ m/s}$
Karlson and Smidge divide a cake in a shape of a square in the following way. First, Karlson places a candle on the cake (chooses some interior point). Then Smidge makes a straight cut from the candle to the boundary in the direction of his choice. Then Karlson makes a straight cut from the candle to the boundary in the direction perpendicular to Smidge's cut. As a result, the cake is split into two pieces; Smidge gets the smaller one. Smidge wants to get a piece which is no less than a quarter of the cake. Can Karlson prevent Smidge from getting the piece of that size?
Let $B$ be a bounded closed convex symmetric (with respect to the origin) set in $\mathbb{R}^{2}$ with boundary $\Gamma$. Let $B$ have the property that the ellipse of maximal area contained in $B$ is the disc $D$ of radius $1$ centered at the origin with boundary $C$. Prove that $A \cap \Gamma \ne \emptyset$ for any arc $A$ of $C$ of length $l(A)\geq \frac{\pi}{2}$.
Provided a convex equilateral pentagon. On every side of the pentagon We construct equilateral triangles which run through the interior of the pentagon. Prove that at least one of the triangles does not protrude the pentagon's boundary.
Let $0\leq a_1\leq a_2\leq \cdots\leq a_{n-1}\leq a_n $ and $a_1+a_2+\cdots+a_n=1.$ Prove that: For any non-negative numbers $x_1,x_2,\cdots,x_n ; y_1, y_2,\cdots, y_n$ , have
$$\left(\sum_{i=1}^n a_ix_i - \prod_{i=1}^n x_i^{a_i}\right)
\left(\sum_{i=1}^n a_iy_i - \prod_{i=1}^n y_i^{a_i}\right) \leq
a_n^2\left(n\sqrt{\sum_{i=1}^n x_i\sum_{i=1}^n y_i} - \sum_{i=1}^n\sqrt{x_i} \sum_{i=1}^n\sqrt{y_i}\right)^2.$$
Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.
How many ways are there to color the $8$ regions of a three-set Venn Diagram with $3$ colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and/or reflection.
Show that the right circular cylinder of volume $V$ which has the least surface area is the one whose diameter is equal to its altitude. (The top and bottom are part of the surface.)
Ryan uses $91$ puzzle pieces to make a rectangle. Each of them is identical to one of the tiles shown. Given that pieces can be flipped or rotated, find the number of pieces that are red in the puzzle. (He is not allowed to join two ``flat sides'' together.)
$a_1,a_2,...,a_{100}$ are permutation of $1,2,...,100$. $S_1=a_1, S_2=a_1+a_2,...,S_{100}=a_1+a_2+...+a_{100}$Find the maximum number of perfect squares from $S_i$
Let $A, E, H, L, T,$ and $V$ be chosen independently and at random from the set $\{0, \tfrac{1}{2}, 1\}.$ Compute the probability that $\lfloor T \cdot H \cdot E \rfloor = L \cdot A \cdot V \cdot A.$
Find all triples of integers $(x, y, z)$ such that
$$x^2 + y^2 + z^2 = 16(x + y + z).$$