Found problems: 85335
Find all value of $ n$ for which there are nonzero real numbers $ a, b, c, d$ such that after expanding and collecting similar terms, the polynomial $ (ax \plus{} b)^{100} \minus{} (cx \plus{} d)^{100}$ has exactly $ n$ nonzero coefficients.
Two distinct numbers $ a$ and $ b$ are chosen randomly from the set $ \{ 2, 2^2, 2^3, \ldots, 2^{25} \}$. What is the probability that $ \log_{a} b$ is an integer?
$ \textbf{(A)}\ \frac {2}{25} \qquad \textbf{(B)}\ \frac {31}{300} \qquad \textbf{(C)}\ \frac {13}{100} \qquad \textbf{(D)}\ \frac {7}{50} \qquad \textbf{(E)}\ \frac {1}{2}$
A convex quadrilateral $ABCD$ has perpendicular diagonals. The perpendicular bisectors of the sides $AB$ and $CD$ meet at a unique point $P$ inside $ABCD$. Prove that the quadrilateral $ABCD$ is cyclic if and only if triangles $ABP$ and $CDP$ have equal areas.
Let be a continuous function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ that has a root, and for which the line $ y=0 $ in the Cartesian plane is an horizontal asymptote. Show that $ f $ is bounded and touches its boundaries.
[i]Mihai Piticari[/i] and [i]Vladimir Cerbu[/i]
A line initially $ 1$ inch long grows according to the following law, where the first term is the initial length.
\[ 1 \plus{} \frac {1}{4}\sqrt {2} \plus{} \frac {1}{4} \plus{} \frac {1}{16}\sqrt {2} \plus{} \frac {1}{16} \plus{} \frac {1}{64}\sqrt {2} \plus{} \frac {1}{64} \plus{} \cdots.
\]If the growth process continues forever, the limit of the length of the line is:
$ \textbf{(A)}\ \infty \qquad\textbf{(B)}\ \frac {4}{3} \qquad\textbf{(C)}\ \frac {8}{3} \qquad\textbf{(D)}\ \frac {1}{3}(4 \plus{} \sqrt {2}) \qquad\textbf{(E)}\ \frac {2}{3}(4 \plus{} \sqrt {2})$
For a convex hexagon $ ABCDEF$ with an area $ S$, prove that:
\[ AC\cdot(BD\plus{}BF\minus{}DF)\plus{}CE\cdot(BD\plus{}DF\minus{}BF)\plus{}AE\cdot(BF\plus{}DF\minus{}BD)\geq 2\sqrt{3}S
\]
A wooden cube $ n$ units on a side is painted red on all six faces and then cut into $ n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $ n$?
$ \textbf{(A)}\ 3\qquad
\textbf{(B)}\ 4\qquad
\textbf{(C)}\ 5\qquad
\textbf{(D)}\ 6\qquad
\textbf{(E)}\ 7$
$a,b$ are positive integers and $(a!+b!)|a!b!$.Prove that $3a\ge 2b+2$.
Let $f(n)=\displaystyle\sum_{k=1}^n \dfrac{1}{k}$. Then there exists constants $\gamma$, $c$, and $d$ such that \[f(n)=\ln(x)+\gamma+\dfrac{c}{n}+\dfrac{d}{n^2}+O\left(\dfrac{1}{n^3}\right),\] where the $O\left(\dfrac{1}{n^3}\right)$ means terms of order $\dfrac{1}{n^3}$ or lower. Compute the ordered pair $(c,d)$.
For positive reals $a,b$ and $c$ with $ab+bc+ca=1$, show that
\[\sqrt{3}({\sqrt{a}+\sqrt{b}+\sqrt{c})\le \frac{a\sqrt{a}}{bc}+\frac{b\sqrt{b}}{ca}+\frac{c\sqrt{c}}{ab}.}\]
[i]Proposed by Morteza Saghafian[/i]
Let $ G$ be an infinite group generated by nilpotent normal subgroups. Prove that every maximal Abelian normal subgroup of $ G$ is infinite. (We call an Abelian normal subgroup maximal if it is not contained in another Abelian normal subgroup.)
[i]P. Erdos[/i]
There are $n\ge3$ integers around a circle. We know that for each of these numbers the ratio between the sum of its two neighbors and the number is a positive integer. Prove that the sum of the $n$ ratios is not greater than $3n$.
The polynomial $W(x)=x^2+ax+b$ with integer coefficients has the following property: for every prime number $p$ there is an integer $k$ such that both $W(k)$ and $W(k+1)$ are divisible by $p$. Show that there is an integer $m$ such that $W(m)=W(m+1)=0$.
Determine the digits $0\leqslant c\leqslant 9$ such that for any positive integer $k{}$ there exists a positive integer $n$ such that the last $k{}$ digits of $n^9$ are equal to $c{}.$
What is the last digit of $4^{3^{2019}}$?
[b]A.[/b] $0$ [b]B.[/b] $2$ [b]C.[/b] $4$ [b]D.[/b] $6$ [b]E.[/b] $8$
Let us consider mathematical crossword which we fill with numbers $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ such that:
1) All digits occur exactly twice
2) $10$ horizontally divides $4$ vertically
3) $4 \cdot$ ($4$ horizontally - $4$ vertically +$5$) equals $1$ vertically
4) $36$ divides $1$ horizontally and $5$ vertically
5) $9$ vertically divides $5$ vertically
In how many ways we can solve this mathematical crossword?
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvOC85LzgyNjUzYmNkNTVmNDE1YTg4OWVkNzAzYzE1M2JkZWE0MThiYWY1LnBuZw==&rn=Y3Jvc3N3b3JkLnBuZw==[/img]
Calculate the sum of vector products $[[x, y], z] + [[y, z], x] + [[z, x], y]$
Let $M$ be the set of all integers in the form of $a^2+13b^2$, where $a$ and $b$ are distinct itnegers.
i) Prove that the product of any two elements of $M$ is also an element of $M$.
ii) Determine, reasonably, if there exist infinite pairs of integers $(x,y)$ so that $x+y\not\in M$ but $x^{13}+y^{13}\in M$.
Let $ n > 3$ be a fixed integer and $ x_1,x_2,\ldots, x_n$ be positive real numbers. Find, in terms of $ n$, all possible real values of
\[ {x_1\over x_n\plus{}x_1\plus{}x_2} \plus{} {x_2\over x_1\plus{}x_2\plus{}x_3} \plus{} {x_3\over x_2\plus{}x_3\plus{}x_4} \plus{} \cdots \plus{} {x_{n\minus{}1}\over x_{n\minus{}2}\plus{}x_{n\minus{}1}\plus{}x_n} \plus{} {x_n\over x_{n\minus{}1}\plus{}x_n\plus{}x_1}\]
Prove that total number of ones which is showed in all nonrestricted partitions of natural number $n$ is equal to sum of numbers of distinct elements in that partitions.
Prove that $\frac{5^{125}-1}{5^{25}-1}$ is a composite number.
Let $ABC$ and $PQR$ be two triangles such that
[list]
[b](a)[/b] $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$.
[b](b)[/b] $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$
[/list]
Prove that $AB+AC=PQ+PR$.
Let $ABC$ be an acute triangle, and let $M$ and $N$ be two points on the line $AC$ such that the vectors $MN$ and $AC$ are identical. Let $X$ be the orthogonal projection of $M$ on $BC$, and let $Y$ be the orthogonal projection of $N$ on $AB$. Finally, let $H$ be the orthocenter of triangle $ABC$.
Show that the points $B$, $X$, $H$, $Y$ lie on one circle.
Let $\triangle{ABC}$ be an isosceles right triangle with $AB=AC=10.$ Let $M$ be the midpoint of $BC$ and $N$ the midpoint of $BM.$ Let $AN$ hit the circumcircle of $\triangle{ABC}$ again at $T.$ Compute the area of $\triangle{TBC}.$
The projections of the body on two planes are circles. Prove that they have the same radius.