Found problems: 85335
Let be a natural number $ a\ge 2. $ Prove that for any choice of primes which has the property that none of them divides any of the numbers $ N_n=1+a+a^2+a^3+\cdots +a^{2n} , $ with natural $ n, $ there is another prime not among this choice which doesn't divide any of the numbers $ N_n. $
If the graph is a graph of POSITION vs. TIME, then the squirrel has the greatest speed at what time(s) or during what time interval(s)?
(A) From A to B
(B) From B to C only
(C) From B to D
(D) From C to D only
(E) From D to E
Let $C$ and $D$ be two distinct points on a semicircle of diameter $AB$. Let $E$ be the intersection of $AC$ and $BD$, $F$ be the intersection of $AD$ and $BC$ and $X, Y$, and $Z$ are the midpoints of $AB, CD$, and $EF$, respectively. Prove that the points $X, Y,$ and $Z$ are collinear.
In the inscribed quadrilateral $ABCD$, $P$ is the intersection point of diagonals and $M$ is the midpoint of arc $AB$. Prove that line $MP$ passes through the midpoint of segment $CD$, if and only if lines $AB, CD$ are parallel.
Let $P_1$, $P_2$, $P_3$ be pairwise distinct parabolas in the plane. Find the maximum possible number of intersections between two or more of the $P_i$. In other words, find the maximum number of points that can lie on two or more of the parabolas $P_1$, $P_2$, $P_3$ .
Let $n$ and $k$ be arbitrary non-negative integers. Suppose we have drawn $2kn+1$ (different) diagonals of a convex $n$-gon. Show that there exists a broken line formed by $2k+1$ of these diagonals that passes through no point more than once. Prove also that this is not necessarily true when we draw only $kn$ diagonals.
As shown in the figure, $BQ$ is a diameter of the circumcircle of $ABC$, and $D$ is the midpoint of arc $BC$ (excluding point $A$) . The bisector of the exterior angle of $\angle BAC$ intersects and the extension of $BC$ at point $E$. The ray $EQ$ intersects $\odot (ABC)$ at point $P$. Point $S$ lies on $PQ$ so that $SA = SP$. Point $T$ lies on $BC$ such that $TB = TD$. Prove that $TS \perp SE$.
[img]https://cdn.artofproblemsolving.com/attachments/c/4/01460565e70b32b29cddb65d92e041bea40b25.png[/img]
Determine the positive integers $n$ such that the set of all positive divisors of $30^n$ can be divided into groups of three so that the product of the three numbers in each group is the same.
The circumradius of acute triangle $ABC$ is twice of the distance of its circumcenter to $AB$. If $|AC|=2$ and $|BC|=3$, what is the altitude passing through $C$?
$
\textbf{(A)}\ \sqrt {14}
\qquad\textbf{(B)}\ \dfrac{3}{7}\sqrt{21}
\qquad\textbf{(C)}\ \dfrac{4}{7}\sqrt{21}
\qquad\textbf{(D)}\ \dfrac{1}{2}\sqrt{21}
\qquad\textbf{(E)}\ \dfrac{2}{3}\sqrt{14}
$
How many ordered pairs of real numbers $ (a, b) $ satisfy equality $\lim_{x \to 0} \frac{\sin^2x}{e^{ax}-2bx-1}= \frac{1}{2}$?
Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0), (2,0), (2,1),$ and $(0,1)$. $R$ can be divided into two unit squares, as shown; the resulting figure has seven edges.
[asy]
size(3cm);
draw((0,0)--(2,0)--(2,1)--(0,1)--cycle); draw((1,0)--(1,1));
[/asy]
Compute the number of ways to choose one or more of the seven edges such that the resulting figure is traceable without lifting a pencil. (Rotations and reflections are considered distinct.)
Five points $A$, $B$, $C$, $D$ and $E$ lie on a circle $\tau$ clockwise in that order such that $AB \parallel CE$ and $\angle ABC > 90^{\circ}$. Let $k$ be a circle tangent to $AD$, $CE$ and $\tau$ such that $k$ and $\tau$ touch on the arc $\widehat{DE}$ not containing $A$, $B$ and $C$. Let $F \neq A$ be the intersection of $\tau$ and the tangent line to $k$ passing through $A$ different from $AD$.
Prove that there exists a circle tangent to $BD$, $BF$, $CE$ and $\tau$.
Do there exist $ 14$ positive integers, upon increasing each of them by $ 1$,their product increases exactly $ 2008$ times?
Find all positive integers $n$ such that $4^{n}+2007$ is a perfect square.
A domino has a left end and a right end, each of a certain color. Alice has four dominos, colored red-red, red-blue, blue-red, and blue-blue. Find the number of ways to arrange the dominos in a row end-to-end such that adjacent ends have the same color. The dominos cannot be rotated.
Compute the smallest positive integer $a$ for which $$\sqrt{a +\sqrt{a +...}} - \frac{1}{a +\frac{1}{a+...}}> 7$$
Given are two distinct sequences of positive integers $(a_n)$ and $(b_n)$, such that their first two members are coprime and smaller than $1000$, and each of the next members is the sum of the previous two.
8-9 grade Prove that if $a_n$ is divisible by $b_n$, then $n<50$
10-11 grade Prove that if $a_n^{100}$ is divisible by $b_n$ then $n<5000$
[b](a)[/b] A $2 \times n$-table (with $n > 2$) is filled with numbers so that the sums in all the columns are different. Prove that it is possible to permute the numbers in the table so that the sums in the columns would still be different and the sums in the rows would also be different.
[i]($2$ points)[/i]
[b](b)[/b] A $100 \times 100$-table is filled with numbers such that the sums in all the columns are different. Is it always possible to permute the numbers in the table so that the sums in the columns would still be different and the sums in the rows would also be different?
[i]($6$ points)[/i]
Let $n$ be an odd positive integer, and consider an infinite square grid. Prove that it is impossible to fill in one of $1,2$ or $3$ in every cell, which simultaneously satisfies the following conditions:
(1) Any two cells which share a common side does not have the same number filled in them.
(2) For any $1\times 3$ or $3\times 1$ subgrid, the numbers filled does not contain $1,2,3$ in that order be it reading from top to bottom, bottom to top, or left to right, or right to left.
(3) The sum of numbers of any $n\times n$ subgrid is the same.
[b]p1.[/b] Find all $x$ such that $(\ln (x^4))^2 = (\ln (x))^6$.
[b]p2.[/b] On a piece of paper, Alan has written a number $N$ between $0$ and $2010$, inclusive. Yiwen attempts to guess it in the following manner: she can send Alan a positive number $M$, which Alan will attempt to subtract from his own number, which we will call $N$. If $M$ is less than or equal $N$, then he will erase $N$ and replace it with $N -M$. Otherwise, Alan will tell Yiwen that $M > N$. What is the minimum number of attempts that Yiwen must make in order to determine uniquely what number Alan started with?
[b]p3.[/b] How many positive integers between $1$ and $50$ have at least $4$ distinct positive integer divisors? (Remember that both $1$ and $n$ are divisors of $n$.)
[b]p4.[/b] Let $F_n$ denote the $n^{th}$ Fibonacci number, with $F_0 = 0$ and $F_1 = 1$. Find the last digit of $$\sum^{97!+4}_{i=0}F_i.$$
[b]p5.[/b] Find all prime numbers $p$ such that $2p + 1$ is a perfect cube.
[b]p6.[/b] What is the maximum number of knights that can be placed on a $9\times 9$ chessboard such that no two knights attack each other?
[b]p7.[/b] $S$ is a set of $9$ consecutive positive integers such that the sum of the squares of the $5$ smallest integers in the set is the sum of the squares of the remaining $4$. What is the sum of all $9$ integers?
[b]p8.[/b] In the following infinite array, each row is an arithmetic sequence, and each column is a geometric sequence. Find the sum of the infinite sequence of entries along the main diagonal.
[img]https://cdn.artofproblemsolving.com/attachments/5/1/481dd1e496fed6931ee2912775df630908c16e.png[/img]
[b]p9.[/b] Let $x > y > 0$ be real numbers. Find the minimum value of $\frac{x}{y} + \frac{4x}{x-y}$ .
[b]p10.[/b] A regular pentagon $P = A_1A_2A_3A_4A_5$ and a square $S = B_1B_2B_3B_4$ are both inscribed in the unit circle. For a given pentagon $P$ and square $S$, let $f(P, S)$ be the minimum length of the minor arcs $A_iB_j$ , for $1 \le i \le 5$ and $1 \le j \le4$. Find the maximum of $f(P, S)$ over all pairs of shapes.
[b]p11.[/b] Find the sum of the largest and smallest prime factors of $9^4 + 3^4 + 1$.
[b]p12.[/b] A transmitter is sending a message consisting of $4$ binary digits (either ones or zeros) to a receiver. Unfortunately, the transmitter makes errors: for each digit in the message, the probability that the transmitter sends the correct digit to the receiver is only $80\%$. (Errors are independent across all digits.) To avoid errors, the receiver only accepts a message if the sum of the first three digits equals the last digit modulo $2$. If the receiver accepts a message, what is the probability that the message was correct?
[b]p13.[/b] Find the integer $N$ such that $$\prod^{8}_{i=0}\sec \left( \frac{\pi}{9}2^i \right)= N.$$
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Triangle $ABC$ has side lengths in arithmetic progression, and the smallest side has length $6.$ If the triangle has an angle of $120^\circ,$ what is the area of $ABC$?
$\textbf{(A)}\ 12\sqrt 3 \qquad\textbf{(B)}\ 8\sqrt 6 \qquad\textbf{(C)}\ 14\sqrt 2 \qquad\textbf{(D)}\ 20\sqrt 2 \qquad\textbf{(E)}\ 15\sqrt 3$
An acute-angle non-isosceles triangle $ ABC $ is given. The point $ H $ is its orthocenter, the points $ O $ and $ I $ are the centers of its circumscribed and inscribed circles, respectively. The circumcircle of the triangle $ OIH $ passes through the vertex $ A $. Prove that one of the angles of the triangle is $ 60^\circ $.
Of the students attending a school athletic event, $80\%$ of the boys were dressed in the school colors, $60\%$ of the girls were dressed in the school colors, and $45\% $ of the students were girls. Find the percentage of students attending the event who were wearing the school colors.
The graph $y=x^2+ax+b$ intersects any of the two axes at points $A$, $B$, and $C$. The incenter of triangle $ABC$ lies on the line $y=x$. Prove that $a+b+1=0$.
Let $b = \tfrac 12 (-1 + 3\sqrt{5})$. Determine the number of rational numbers which can be written in the form \[ a_{2014}b^{2014} + a_{2013}b^{2013} + \dots + a_1b + a_0 \] where $a_0, a_1, \dots, a_{2014}$ are nonnegative integers less than $b$.
[i]Proposed by Michael Kural and Evan Chen[/i]