Practice (9)
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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 $ABCD$.
Totoro wants to cut a 3 meter long bar of mixed metals into two parts with equal monetary value. The left meter is bronze, worth 10 zoty per meter, the middle meter is silver, worth 25 zoty per meter, and the right meter is gold, worth 40 zoty per meter. How far, in meters, from the left should Totoro make the cut?
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 $|x_1 - x_2| + |x_2 - x_3| + |x_3 - x_4| + |x_4 - x_5|$.
In a $3 \times 4$ grid of 12 squares, find the number of paths from the top left corner to the bottom right corner that satisfy the following two properties:
- The path passes through each square exactly once.
- Consecutive squares share a side.
Two paths are considered distinct if and only if the order in which the twelve squares are visited is different. For instance, in the diagram below, the two paths drawn are considered the same.
Scott, Demi, and Alex are writing a computer program that is 25 lines long. Since they are working together on one computer, only one person may type at a time. To encourage collaboration, no person can type two lines in a row, and everyone must type something. If Scott takes 10 seconds to type one line, Demi takes 15 seconds, and Alex takes 20 seconds, at least how long, in seconds, will it take them to finish the program?
A hand of four cards of the form $(c, c, c + 1, c + 1)$ is called a $tractor$. Vinjai has a deck consisting of four of each of the numbers $7$, $8$, $9$ and $10$. If Vinjai shuffles and draws four cards from his deck, compute the probability that they form a tractor.
The parabola $y = 2x^2$ is the wall of a fortress. Totoro is located at (0, 4) and fires a cannonball in a straight line at the closest point on the wall. Compute the $y$-coordinate of the point on the wall that the cannonball hits.
How many ways are there to color the squares of a 10 by 10 grid with black and white such that in each row and each column there are exactly two black squares and between the two black squares in a given row or column there are exactly 4 white squares? Two configurations that are the same under rotations or reflections are considered different.
In rectangle $ABCD$, points $E$ and $F$ are on sides $AB$ and $CD$, respectively, such that $AE = CF > AD$ and $\angle CED = 90^{\circ}$. Lines $AF$, $BF$, $CE$ and $DE$ enclose a rectangle whose area is 24% of the area of $ABCD$. Compute $\frac{BF}{CE}$ .
Link cuts trees in order to complete a quest. He must cut 3 Fenwick trees, 3 Splay trees and 3 KD trees. If he must also cut 3 trees of the same type in a row at some point during his quest, in how many ways can he cut the trees and complete the quest? (Trees of the same type are indistinguishable.)
Find all ordered pairs $(a, b)$ of positive integers such that $\sqrt{64a + b^2} + 8 = 8\sqrt{a} + b$.
Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle BCD = 108^{\circ}$, $\angle CDE = 168^{\circ}$ and $AB =BC = CD = DE$. Find the measure of $\angle AEB$.
Alec rated the movie Frozen 1 out of 5 stars. At least how many ratings of 5 out of 5 stars does Eric need to collect to make the average rating for Frozen greater than or equal to 4 out of 5 stars?
Bessie shuffles a standard 52-card deck and draws five cards without replacement. She notices that all five of the cards she drew are red. If she draws one more card from the remaining cards in the deck, what is the probability that she draws another red card?
Find the value of 121 -1020304030201
Find the smallest positive integer $c$ for which there exist positive integers $a$ and $b$ such that $a \neq b$ and $a^2 + b^2 = c$.
A semicircle with diameter $AB$ is constructed on the outside of rectangle $ABCD$ and has an arc length equal to the length of $BC$. Compute the ratio of the area of the rectangle to the area of the semicircle.
There are $10$ monsters, each with 6 units of health. On turn $n$, you can attack one monster, reducing its health by $n$ units. If a monster's health drops to $0$ or below, the monster dies. What is the minimum number of turns necessary to kill all of the monsters?