Practice (90/1000)
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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.
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 $\angle AEB$.
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?
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?
Chad has $100$ cookies that he wants to distribute among four friends. Two of them, Jeff and Qiao, are rivals; neither wants the other to receive more cookies than they do. The other two, Jim and Townley, don't care about how many cookies they receive. In how many ways can Chad distribute all $100$ cookies to his four friends so that everyone is satisfied? (Some of his four friends may receive zero cookies.)
Compute the smallest positive integer with at least four two-digit positive divisors.
Let $ABCD$ be a trapezoid such that $AB$ is parallel to $CD$, $BC$ = 10 and $AD$ = 18. Given that the two circles with diameters $BC$ and $AD$ are tangent, find the perimeter of $ABCD$.
How many length ten strings consisting of only $A$s and $B$s contain neither "$BAB$" nor "$BBB$" as a substring?
Let $A$ be the answer to problem 13, and let $C$ be the answer to problem 15. In the interior of angle $NOM$ = $45^{\circ}$, there is a point $P$ such that $\angle MOP = A^{\circ}$ and $OP = C$. Let $X$ and $Y$ be the reflections of $P$ over $MO$ and $NO$, respectively. Find $(XY )^2$.
Let $ABCD$ be a trapezoid such that $AB$ is parallel to $CD$, $AB$ = 4, $CD$ = 8, $BC$ = 5, and $AD$ = 6. Given that point $E$ is on segment $CD$ and that $AE$ is parallel to $BC$, find the ratio between the area of trapezoid $ABCD$ and the area of triangle $ABE$.
Find the maximum possible value of the greatest common divisor of $MOO$ and $MOOSE$, given that $S$, $O$, $M$, and $E$ are some nonzero digits. (The digits $S$, $O$, $M$, and $E$ are not necessarily pairwise distinct.)
$\textbf{Lying Politicians}$
Suppose $125$ politicians sit around a conference table. Each politician either always tells the truth or always lies. (Statements of a liar are never completely true, but can be partially true.) Each politician now claims that the two people beside him or her are both liars. What are the maximum possible number and minimum possible number of liars?
Let line segment $AB$ have length 25 and let points $C$ and $D$ lie on the same side of line $AB$ such that $AC$ = 15, $AD$ = 24, $BC$ = 20, and $BD$ = 7. Given that rays $AC$ and $BD$ intersect at point $E$, compute $EA + EB$.
A $3 \times 3$ grid is filled with positive integers and has the property that each integer divides both the integer directly above it and directly to the right of it. Given that the number in the top-right corner is 30, how many distinct grids are possible?
Define a sequence of positive integers $s_1, s_2, . . . , s_{10}$ to be $terrible$ if the following conditions are satisfied for any pair of positive integers $i$ and $j$ satisfying $1 \le i < j \le 10$:
- $s_i > s_j$
- $j - i + 1$ divides the quantity $s_i + s_{i+1} + \cdots + s_j$
Determine the minimum possible value of $s_1 + s_2 + \cdots + s_{10}$ over all terrible sequences.
The four points $(x, y)$ that satisfy $x = y^2 - 37$ and $y = x^2 -37$ form a convex quadrilateral in the coordinate plane. Given that the diagonals of this quadrilateral intersect at point $P$, find the coordinates of $P$ as an ordered pair.
Consider a non-empty set of segments of length 1 in the plane which do not intersect except at their endpoints. (In other words, if point $P$ lies on distinct segments $a$ and $b$, then $P$ is an endpoint of both $a$ and $b$.) This set is called $3-amazing$ if each endpoint of a segment is the endpoint of exactly
three segments in the set. Find the smallest possible size of a 3-amazing set of segments.
Let $z_1$, $z_2 \in \mathbb{C}$. Prove that the number $E= z_1\cdot z_2 + \overline{z_1}\cdot \overline{z_2}$ is a real number.