Found problems: 85335
Let the sequence $ a(n), n \equal{} 1,2,3, \ldots$ be generated as follows with $ a(1) \equal{} 0,$ and for $ n > 1:$
\[ a(n) \equal{} a\left( \left \lfloor \frac{n}{2} \right \rfloor \right) \plus{} (\minus{}1)^{\frac{n(n\plus{}1)}{2}}.\]
1.) Determine the maximum and minimum value of $ a(n)$ over $ n \leq 1996$ and find all $ n \leq 1996$ for which these extreme values are attained.
2.) How many terms $ a(n), n \leq 1996,$ are equal to 0?
Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.
A right triangle has perimeter $ 32$ and area $ 20$. What is the length of its hypotenuse?
$ \textbf{(A)}\ \frac{57}{4} \qquad
\textbf{(B)}\ \frac{59}{4} \qquad
\textbf{(C)}\ \frac{61}{4} \qquad
\textbf{(D)}\ \frac{63}{4} \qquad
\textbf{(E)}\ \frac{65}{4}$
Estimate the number of $1$s in the hexadecimal representation of $2020!$.
If $E$ is your estimate and $A$ is the correct answer, you will receive $\max (25 - 0.5|A - E|, 0)$ points, rounded to the nearest integer.
Show that for $n > 1$ we can find a polynomial $P(a, b, c)$ with integer coefficients such that
$$P(x^{n},x^{n+1},x+x^{n+2})=x.$$
Let $a,b,c,d$ be positive integers such that the number of pairs $(x,y) \in (0,1)^2$ such that both $ax+by$ and $cx+dy$ are integers is equal with 2004. If $\gcd (a,c)=6$ find $\gcd (b,d)$.
(a) Let $f(x)=x-xe^{-\frac1x}, \ \ x>0$. Show that $f(x)$ is an increasing function on $(0,\infty)$, and $\lim_{x\to\infty} f(x)=1$.
(b) Using part (a) or otherwise, draw graphs of $y=x-1, y=x, y=x+1$, and $y=xe^{-\frac{1}{|x|}}$ for $-\infty<x<\infty$ using the same $X$ and $Y$ axes.
A sheet of paper has the shape of an isosceles trapezoid $C_1AB_2C_2$ with the shorter base $B_2C_2$. The foot of the perpendicular from the midpoint $D$ of $C_1C_2$ to $AC_1$ is denoted by $B_1$. Suppose that upon folding the paper along $DB_1, AD$ and $AC_1$ points $C_1,C_2$ become a single point $C$ and points $B_1,B_2$ become a point $B$. The area of the tetrahedron $ABCD$ is $64$ cm$^3$ . Find the sides of the initial trapezoid.
Caroline wants to plant 10 trees in her orchard. Planting $n$ apple trees requires $n^2$ square meters, planting $n$ apricot trees requires $5n$ square meters, and planting $n$ plum trees requires $n^3$ square meters. If she is committed to growing only apple, apricot, and plum trees, what is the least amount of space in square meters, that her garden will take up?
Find the units digit of the sum \[(1!)^2+(2!)^2+(3!)^2+(4!)^2+\cdots+(2007!)^2.\]
$\textbf{(A) }0\hspace{14em}\textbf{(B) }1\hspace{14em}\textbf{(C) }3$
$\textbf{(D) }5\hspace{14em}\textbf{(E) }7\hspace{14em}\textbf{(F) }9$
A spider sits on the circumference of a circle and wants to weave a web by making several passes through the circle's interior. On each pass, the spider starts at some location on the circumference, picks a destination uniformly at random from the circumference, and travels to that destination in a straight line, laying down a strand of silk along the line segment they traverse. After the spider does $2022$ of these passes (with each non-initial pass starting where the previous one ended), what is the expected number of points in the circle's interior where two or more non-parallel silk strands intersect?
In the Martian language every finite sequence of letters of the Latin alphabet letters is a word. The publisher “Martian Words” makes a collection of all words in many volumes. In the first volume there are only one-letter words, in the second, two-letter words, etc., and the numeration of the words in each of the volumes continues the numeration of the previous volume. Find the word whose numeration is equal to the sum of numerations of the words Prague, Olympiad, Mathematics.
In the middle of the number 1996 (i.e. between $19$ and $96$) you need to insert several digits so that the resulting number is divisible by $1997$. In this case, you need to get by with the least number of inserted digits.
Suppose $S$ is a subset of $\{1, 2, 3, 4, 5, 6, 7\}$. How many different possible values are there for the product of the elements in $S$?
Let $ABC$ be a triangle with incenter $I$. Let $M_b$ and $M_a$ be the midpoints of $AC$ and $BC$, respectively. Let $B'$ be the point of intersection of lines $M_bI$ and $BC$, and let $A'$ be the point of intersection of lines $M_aI$ and $AC$.
If triangles $ABC$ and $A'B'C$ have the same area, what are the possible values of $\angle ACB$?
An ice-cream novelty item consists of a cup in the shape of a $4$-inch-tall frustum of a right circular cone, with a $2$-inch-diameter base at the bottom and a $4$-inch-diameter base at the top, packed solid with ice cream, together with a solid cone of ice cream of height $4$ inches, whose base, at the bottom, is the top base of the frustum. What is the total volume of the ice cream, in cubic inches?
$\textbf{(A)}\ 8\pi\qquad\textbf{(B)}\ \frac{28\pi}{3}\qquad\textbf{(C)}\ 12\pi\qquad\textbf{(D)}\ 14\pi\qquad\textbf{(E)}\ \frac{44\pi}{3}$
In an $n\times{n}$ grid, there are some cats living in each cell (the number of cats in a cell must be a non-negative integer). Every midnight, the manager chooses one cell:
(a) The number of cats living in the chosen cell must be greater than or equal to the number of neighboring cells of the chosen cell.
(b) For every neighboring cell of the chosen cell, the manager moves one cat from the chosen cell to the neighboring cell.
(Two cells are called "neighboring" if they share a common side, e.g. there are only $2$ neighboring cells for a cell in the corner of the grid)
Find the minimum number of cats living in the whole grid, such that the manager is able to do infinitely many times of this process.
Let $x_1,x_2,\ldots, x_{10}$ be non-negative real numbers such that $\frac{x_1}{1}+ \frac{x_2}{2} +\cdots+ \frac{x_{10}}{10}$ $\leq9$. Find the maximum possible value of $\frac{{x_1}^2}{1}+\frac{{x_2}^2}{2}+\cdots+\frac{{x_{10}}^2}{10}$.
Al and Bob are joined by Carl and D’Angelo, and they decide to play a team game of Rock Paper Scissors. A game is called [i]perfect[/i] if some two of the four play the same thing, and the other two also play the same thing, but something different. For example, an example of a perfect game would be Al and Bob playing rock, and Carl and D’Angelo playing scissors, but if all four play paper, we do not have a perfect game. What is the probability of a perfect game?
Let $ABC$ be a triangle with $\angle BAC=120^o$, which the median $AD$ is perpendicular to side $AB$ and intersects the circumscribed circle of triangle $ABC$ at point $E$. Lines $BA$ and $EC$ intersect at $Z$. Prove that
a) $ZD \perp BE$
b) $ZD=BC$
Through the use of theorems on logarithms
\[ \log{\frac{a}{b}} \plus{} \log{\frac{b}{c}} \plus{} \log{\frac{c}{d}} \minus{} \log{\frac{ay}{dx}}\]
can be reduced to:
$ \textbf{(A)}\ \log{\frac{y}{x}}\qquad
\textbf{(B)}\ \log{\frac{x}{y}}\qquad
\textbf{(C)}\ 1\qquad
\textbf{(D)}\ 0\qquad
\textbf{(E)}\ \log{\frac{a^2y}{d^2x}}$
The midpoints of four sides of a cyclic pentagon were marked, after this the pentagon was erased. Restore it.
In cyclic quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $P$. Let $E$ and $F$ be the respective feet of the perpendiculars from $P$ to lines $AB$ and $CD$. Segments $BF$ and $CE$ meet at $Q$. Prove that lines $PQ$ and $EF$ are perpendicular to each other.
Let be a pyramid having a square as its base and the projection of the top vertex to the base is the center of the square. Prove that two opposite faces are perpendicular if and only if the angle between two adjacent faces is $ 120^{\circ } . $
A box contains 2 pennies, 4 nickels, and 6 dimes. Six coins are drawn without replacement, with each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least 50 cents?
$\text{(A)} \ \frac{37}{924} \qquad \text{(B)} \ \frac{91}{924} \qquad \text{(C)} \ \frac{127}{924} \qquad \text{(D)} \ \frac{132}{924} \qquad \text{(E)} \ \text{none of these}$