Found problems: 85335
Let $F_0 = 0, F_{1} = 1$, and for $n \ge 1, F_{n+1} = F_n + F_{n-1}$. Define $a_n = \left(\frac{1 + \sqrt{5}}{2}\right)^n \cdot F_n$ . Then there are rational numbers $A$ and $B$ such that $\frac{a_{30} + a_{29}}{a_{26} + a_{25}} = A + B \sqrt{5}$. Find $A + B$.
Set $A$ consists of $2016$ positive integers. All prime divisors of these numbers are smaller than $30.$ Prove that there are four distinct numbers $a, b, c$ and $d$ in $A$ such that $abcd$ is a perfect square.
Let $ABC$ be a triangle. Draw circles $\omega_A$, $\omega_B$, and $\omega_C$ such that $\omega_A$ is tangent to $AB$ and $AC$, and $\omega_B$ and $\omega_C$ are defined similarly. Let $P_A$ be the insimilicenter of $\omega_B$ and $\omega_C$. Define $P_B$ and $P_C$ similarly. Prove that $AP_A$, $BP_B$, and $CP_C$ are concurrent.
[i]Tom Lu.[/i]
Given a triangle $ABC$ and a point $P_0$ on the side $AB$. Construct points $P_i, Q_i, R_i $ as follows. $Q_i$ is the foot of the perpendicular from $P_i$ to $BC, R_i$ is the foot of the perpendicular from $Q_i$ to $AC$ and $P_i$ is the foot of the perpendicular from $R_{i-1}$ to $AB$. Show that the points $P_i$ converge to a point $P$ on $AB$ and show how to construct $P$.
A particle is located on the coordinate plane at $ (5,0)$. Define a [i]move[/i] for the particle as a counterclockwise rotation of $ \pi/4$ radians about the origin followed by a translation of $ 10$ units in the positive $ x$-direction. Given that the particle's position after $ 150$ moves is $ (p,q)$, find the greatest integer less than or equal to $ |p|\plus{}|q|$.
Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$.
[i]Proposed by Evan Chen[/i]
Let $A_1, A_2, ... , A_{100}$ be subsets of a line, each a union of $100$ pairwise disjoint closed segments. Prove that the intersection of all hundred sets is a union of at most $9901$ disjoint closed segments.
At a party there are eight guests, and each participant can't talk with at most three persons. Prove that we can group the persons in four pairs such that in every pair a conversation can take place.
A boy at point $A$ wants to get water at a circular lake and carry it to point $B$. Find the point $C$ on the lake such that the distance walked by the boy is the shortest possible given that the line $AB$ and the lake are exterior to each other.
Let $N, P$ be the centers of the faces A$BB'A'$ and $ADD'A'$, respectively, of a right parallelepiped $ABCDA'B'C'D'$ and $M \in (A'C)$ such that $A'M= \frac13 A' C$. Prove that $MN \perp AB'$ and $ MP \perp AD' $ if and only if the parallelepiped is a cube.
In the accompanying figure, the outer square has side length 40. A second square S' of side length 15 is constructed inside S with the same center as S and with sides parallel to those of S. From each midpoint of a side of S, segments are drawn to the two closest vertices of S'. The result is a four-pointed starlike figure inscribed in S. The star figure is cut out and then folded to form a pyramid with base S'. Find the volume of this pyramid.
[asy]
draw((0,0)--(8,0)--(8,8)--(0,8)--(0,0));
draw((2.5,2.5)--(4,0)--(5.5,2.5)--(8,4)--(5.5,5.5)--(4,8)--(2.5,5.5)--(0,4)--(2.5,2.5)--(5.5,2.5)--(5.5,5.5)--(2.5,5.5)--(2.5,2.5));
[/asy]
$\boxed{\text{N4}}$ Find all primes $p,q$ such that $2p^3-q^2=2(p+q)^2$.
A $5\times5$ grid of squares is filled with integers. Call a rectangle [i]corner-odd[/i] if its sides are grid lines and the sum of the integers in its four corners is an odd number. What is the maximum possible number of corner-odd rectangles within the grid?
Note: A rectangles must have four distinct corners to be considered [i]corner-odd[/i]; i.e. no $1\times k$ rectangle can be [i]corner-odd[/i] for any positive integer $k$.
Circle $I$ is the incircle of $\triangle ABC$. Circle $I$ is tangent to sides $BC$ and $AC$ at $M,N$ respectively. $E,F$ are midpoints of sides $AB$ and $AC$ respectively. Lines $EF, BI$ intersect at $D$. Show that $M,N,D$ are collinear.
Consider an $N\times N$ matrix, where $N$ is an odd positive integer, such that all its entries are $-1,0$ or $1$. Consider the sum of the numbers in every line and every column. Prove that at least two of the $2N$ sums are equal.
How many subsets of two elements can be removed from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$ so that the mean (average) of the remaining numbers is 6?
$\textbf{(A)}\text{ 1}\qquad\textbf{(B)}\text{ 2}\qquad\textbf{(C)}\text{ 3}\qquad\textbf{(D)}\text{ 5}\qquad\textbf{(E)}\text{ 6}$
Find the value of $2(\sin2^{\circ}\tan1^{\circ}+\sin4^{\circ}\tan1^{\circ}+\cdots+\sin178^{\circ}\tan 1^{\circ})$
Let $A_k$ be the number of pairs $(a_1, a_2, ..., a_{2k})$ for $k\leq 50$, where $a_1, a_2, ..., a_{2k}$ are all different positive integers that satisfy the following.
[b]$\cdot$[/b] $a_1, a_2, ..., a_{2k} \leq 100$
[b]$\cdot$[/b] For an odd number less or equal than $2k-1$, we have $a_i > a_{i+1}$
[b]$\cdot$[/b] For an even number less or equal than $2k-2$, we have $a_i < a_{i+1}$
Prove that $A_1 \leq A_2 \leq \cdots \leq A_{49}$.
This is the one with the 8 circles?
I made each circle into the square in which the circle is inscribed, then calculated it with that. It got the right answer but I don't think that my method is truly valid...
There are $n$ rooms in a sauna, each has unlimited capacity. No room may be attended by a female and a male simultaneously. Moreover, males want to share a room only with males that they don't know and females want to share a room only with females that they know. Find the biggest number $k$ such that any $k$ couples can visit the sauna at the same time, given that two males know each other if and only if their wives know each other.
Show that there are nine distinct nonzero integers such that their sum is a perfect square and the sum of any eight of them is a perfect cube.
In a book with page numbers from $ 1$ to $ 100$ some pages are torn off. The sum of the numbers on the remaining pages is $ 4949$. How many pages are torn off?
Given a triangle $ABC$ with circumcircle $\Gamma$. Points $E$ and $F$ are the foot of angle bisectors of $B$ and $C$, $I$ is incenter and $K$ is the intersection of $AI$ and $EF$. Suppose that $T$ be the midpoint of arc $BAC$. Circle $\Gamma$ intersects the $A$-median and circumcircle of $AEF$ for the second time at $X$ and $S$. Let $S'$ be the reflection of $S$ across $AI$ and $J$ be the second intersection of circumcircle of $AS'K$ and $AX$. Prove that quadrilateral $TJIX$ is cyclic.
[i]Proposed by Alireza Dadgarnia and Amir Parsa Hosseini[/i]
Let $ a $ be a positive real number. Calculate $ \lim_{n\to\infty} \frac{a^n}{(1+a)(1+a^2)\cdots (1+a^n)} . $
We consider an $n \times n$ ($n \in N, n \ge 2$) square divided into $n^2$ unit squares. Determine all the values of $k \in N$ for which we can write a real number in each of the unit squares such that the sum of the $n^2$ numbers is a positive number, while the sum of the numbers from the unit squares of any $k \times k$ square is a negative number.