Found problems: 85335
Does there exist a natural number $n$, for which $n.2^{2^{2014}}-81-n$ is a perfect square?
Solve the equation $\cos^2{x}+\cos^2{2x}+\cos^2{3x}=1$
Let $n$ be a positive integer. Answer the following questions.
(1) Find the maximum value of $f_n(x)=x^{n}e^{-x}$ for $x\geq 0$.
(2) Show that $\lim_{x\to\infty} f_n(x)=0$.
(3) Let $I_n=\int_0^x f_n(t)\ dt$. Find $\lim_{x\to\infty} I_n(x)$.
Let A be a symmetric matrix such that the sum of elements of any row is zero.
Show that all elements in the main diagonal of cofator matrix of A are equal.
Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?
$\textbf{(A)}\ 37\qquad\textbf{(B)}\ 63\qquad\textbf{(C)}\ 117\qquad\textbf{(D)}\ 143\qquad\textbf{(E)}\ 163$
The Fibonacci sequence $f_n$ is defined by $f_1=f_2=1$ and $f_{n+2}=f_{n+1}+f_n$ for $n\in\mathbb N$.
(a) Show that $f_{1005}$ is divisible by $10$.
(b) Show that $f_{1005}$ is not divisible by $100$.
An alien from the planet Math came to Earth on monday and said: A. On Tuesday he said AY, on Wednesday AYYA, on Thursday AYYAYAAY. What will he say on saturday?
If $ \left (a\plus{}\frac{1}{a} \right )^2\equal{}3$, then $ a^3\plus{}\frac{1}{a^3}$ equals:
$ \textbf{(A)}\ \frac{10\sqrt{3}}{3} \qquad
\textbf{(B)}\ 3\sqrt{3} \qquad
\textbf{(C)}\ 0 \qquad
\textbf{(D)}\ 7\sqrt{7} \qquad
\textbf{(E)}\ 6\sqrt{3}$
Let $a$ and $b$ be real numbers, and the polynomial $P(x) =ax + b$ such that $P(2)- P(1)= 3$:
Compute the value of $P(5)- P(0)$.
[b]A.[/b] $11$ [b]B.[/b] $13$ [b]C.[/b] $15$ [b]D.[/b] $17$ [b]E.[/b] $19$
In triangle $ABC$, the interior and exterior angle bisectors of $ \angle BAC$ intersect the line $BC$ in $D $ and $E$, respectively. Let $F$ be the second point of intersection of the line $AD$ with the circumcircle of the triangle $ ABC$. Let $O$ be the circumcentre of the triangle $ ABC $and let $D'$ be the reflection of $D$ in $O$. Prove that $ \angle D'FE =90.$
A sequence of vertices $v_1,v_2,\ldots,v_k$ in a graph, where $v_i=v_j$ only if $i=j$ and $k$ can be any positive integer, is called a $\textit{cycle}$ if $v_1$ is attached by an edge to $v_2$, $v_2$ to $v_3$, and so on to $v_k$ connected to $v_1$. Rotations and reflections are distinct: $A,B,C$ is distinct from $A,C,B$ and $B,C,A$. Supposed a simple graph $G$ has $2013$ vertices and $3013$ edges. What is the minimal number of cycles possible in $G$?
Let $A$ be a fixed point in the interior of a circle $\omega$ with center $O$ and radius $r$, where $0<OA<r$. Draw two perpendicular chords $BC,DE$ such that they pass through $A$. For which position of these cords does $BC+DE$ maximize?
Let $f = aX^2 + bX+ c \in Z[X]$ be a polynomial such that for every positive integer $n$,$ f(n )$ is a perfect square. Prove that $f = g^2$ for some polynomial $g \in Z[X]$.
Several boys and girls are dancing a waltz at a ball. Is it possible that each girl can always get to dance the next dance with either a more handsome or more clever boy than for the previous dance, and that each time at least $80\%$ of the girls get to dance the next dance with a boy who is more handsome and more clever? (The numbers of boys and girls are equal and all are dancing.)
(AY Belov)
Compute the unique positive integer $n$ such that $\frac{n^3-1989}{n}$ is a perfect square.
Consider the square grid with $A=(0,0)$ and $C=(n,n)$ at its diagonal ends. Paths from $A$ to $C$ are composed of moves one unit to the right or one unit up. Let $C_n$ (n-th catalan number) be the number of paths from $A$ to $C$ which stay on or below the diagonal $AC$. Show that the number of paths from $A$ to $C$ which cross $AC$ from below at most twice is equal to $C_{n+2}-2C_{n+1}+C_n$
Let $n$ be a fixed positive integer.
- Show that there exist real polynomials $p_1, p_2, p_3, \cdots, p_k \in \mathbb{R}[x_1, \cdots, x_n]$ such that
\[(x_1 + x_2 + \cdots + x_n)^2 + p_1(x_1, \cdots, x_n)^2 + p_2(x_1, \cdots, x_n)^2 + \cdots + p_k(x_1, \cdots, x_n)^2 = n(x_1^2 + x_2^2 + \cdots + x_n^2)\]
- Find the least natural number $k$, depending on $n$, such that the above polynomials $p_1, p_2, \cdots, p_k$ exist.
In a round robin tournament of $7$ people, each person plays every other person exactly once in a game of table tennis. For each game played, the winner is given $2$ points, the loser is given $0$ points, and in the event of a tie, each player gets $1$ point. At the end of the tournament, what is the average score of the $7$ people?
Given a cube with a side of $4$. Is it possible to completely cover $3$ of its faces, which have a common vertex, with sixteen rectangular paper strips measuring $1 \times3$?
Let $ABC$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $BC, CA, AB$ respectively. One of the intersection points of the line $EF$ and the circumcircle is $P.$ The lines $BP$ and $DF$ meet at point $Q.$ Prove that $AP = AQ.$
[i]Proposed by Christopher Bradley, United Kingdom[/i]
In isosceles triangle $ABC$ ($AC=BC$), angle bisector $\angle BAC$ and altitude $CD$ from point $C$ intersect at point $O$, such that $CO=3 \cdot OD$. In which ratio does altitude from point $A$ on side $BC$ divide altitude $CD$ of triangle $ABC$
The number $1$ is special. The number $2$ is special because it is relatively prime to $1$. The number $3$ is not special because it is not relatively prime to the sum of the special numbers less than it, $1 + 2$. The number $4$ is special because it is relatively prime to the sum of the special numbers less than it. So, a number bigger than $1$ is special only if it is relatively prime to the sum of the special numbers less than it. Find the twentieth special number.
The triangle $ABC$ is inscribed in a circle $w_1$. Inscribed in a triangle circle touchs the sides $BC$ in a point $N$. $w_2$ — the circle inscribed in a segment $BAC$ circle of $w_1$, and passing through a point $N$. Let points $O$ and $J$ — the centers of circles $w_2$ and an extra inscribed circle (touching side $BC$) respectively. Prove, that lines $AO$ and $JN$ are parallel.
Line $DE$ cuts through triangle $ABC$, with $DF$ parallel to $BE$. Given that $BD =DF = 10$ and $AD = BE = 25$, find $BC$.
[img]https://cdn.artofproblemsolving.com/attachments/0/e/d6e3d7c1f9bd15f4573ccd5fc67c190b9cf7e9.png[/img]
Let \(A\) be a square matrix with entries in the field \(\mathbb Z / p \mathbb Z\) such that \(A^n - I\) is invertible for every positive integer \(n\). Prove that there exists a positive integer \(m\) such that \(A^m = 0\).
[i](A matrix having entries in the field \(\mathbb Z / p \mathbb Z\) means that two matrices are considered the same if each pair of corresponding entries differ by a multiple of \(p\).)[/i]
[i]Proposed by Tony Wang[/i]