Practice (90/1000)
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Let $f(x) = \sqrt{2^2-x^2}$. Find the value of $f(f(f(f(f(-1)))))$.
Find the smallest positive integer $n$ such that 20 divides $15n$ and 15 divides $20n$.
A circle is inscribed in equilateral triangle $ABC$. Let $M$ be the point where the circle touches side $AB$ and let $N$ be the second intersection of segment $CM$ and the circle. Compute the ratio $\frac{MN}{CN}$ .
Four boys and four girls line up in a random order. What is the probability that both the first and last person in line is a girl?
In square $ABCD$, $M$ is the midpoint of side $CD$. Points $N$ and $P$ are on segments $BC$ and $AB$ respectively such that $\angle AMN = \angle MNP = 90^{\circ}$. Compute the ratio $\frac{AP}{PB}$ .
Meena writes the numbers $1$, $2$, $3$, and $4$ in some order on a blackboard, such that she cannot swap two numbers and obtain the sequence $1$, $2$, $3$, $4$. How many sequences could she have written?
Find the smallest positive integer $N$ such that $2N$ is a perfect square and $3N$ is a perfect cube.
A polyhedron has $60$ vertices, $150$ edges, and $92$ faces. If all of the faces are either regular pentagons or equilateral triangles, how many of the $92$ faces are pentagons?
All positive integers relatively prime to 2015 are written in increasing order. Let the twentieth number be $p$. The value of $\frac{2015}{p}\u2212 1$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $a + b$.
Five red lines and three blue lines are drawn on a plane. Given that $x$ pairs of lines of the same color intersect and $y$ pairs of lines of different colors intersect, find the maximum possible value of $(y - x)$.
In triangle $ABC$, where $AC$ > $AB$, $M$ is the midpoint of $BC$ and $D$ is on segment $AC$ such that $DM$ is perpendicular to $BC$. Given that the areas of $MAD$ and $MBD$ are 5 and 6, respectively, compute the area of triangle $ABC$.
For how many ordered pairs $(x, y)$ of integers satisfying $0 \le x$, $y \le 10$, and $(x + y)^2 + (xy - 1)^2$ is a prime number?
A solitaire game is played with $8$ red, $9$ green, and $10$ blue cards. Totoro plays each of the cards exactly once in some order, one at a time. When he plays a card of color $c$, he gains a number of points equal to the number of cards that are not of color $c$ in his hand. Find the maximum number of points that he can obtain by the end of the game.
Find the least composite positive integer that is not divisible by any of 3, 4, and 5.
Five checkers are on the squares of an $8 \times 8$ checkerboard such that no two checkers are in the same row or the same column. How many squares on the checkerboard share neither a row nor a column with any of the five checkers?
Equilateral triangles $ABE$, $BCF$, $CDG$ and $DAH$ are constructed outside the unit square $ABCD$. Eliza wants to stand inside octagon $AEBFCGDH$ so that she can see every point in the octagon without being blocked by an edge. What is the area of the region in which she can stand?
Let $S$ be the string $0101010101010$. Determine the number of substrings containing an odd number of $1$'s. (A substring is defined by a pair of (not necessarily distinct) characters of the string and represents the characters between, inclusively, the two elements of the string.)
Let the positive divisors of $n$ be $d_1, d_2, \dots$ in increasing order. If $d_6 = 35$, determine the minimum possible value of $n$.
The unit squares on the coordinate plane that have four lattice point vertices are colored black orwhite, as on a chessboard, shown on the diagram below. For an ordered pair $(m, n)$, let $OXZY$ be the rectangle with vertices $O = (0, 0)$, $X = (m, 0)$, $Z = (m, n)$ and $Y = (0, n)$. How many ordered pairs $(m, n)$ of nonzero integers exist such that rectangle $OXZY$ contains exactly 32 black squares?
In triangle $ABC$, $AB = 2BC$. Given that $M$ is the midpoint of $AB$ and $\angle MCA = 60^{\circ}$, compute $\frac{CM}{AC}$ .
Nicky is studying biology and has a tank of $17$ lizards. In one day, he can either remove $5$ lizards or add $2$ lizards to his tank. What is the minimum number of days necessary for Nicky to get rid of all of the lizards from his tank?
What is the maximum number of spheres with radius 1 that can fit into a sphere with radius 2?
A positive integer $x$ is $sunny$ if $3x$ has more digits than $x$. If all sunny numbers are written in increasing order, what is the 50th number written?
Quadrilateral $ABCD$ satisfies $AB = 4$, $BC = 5$, $DA = 4$, $\angle DAB = 60^{\circ}$, and $\angle ABC = 150^{\circ}$. Find the area of $ABCD$.
If the numbers $x_1, x_2, x_3, x_4,$ and $x_5$ are a permutation of the numbers 1, 2, 3, 4, and 5, compute the maximum possible value of $|x_1 - x_2| + |x_2 - x_3| + |x_3 - x_4| + |x_4 - x_5|$.