Practice (TheColoringMethod)

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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?

It is known that 2 students make up 5% of a class, when rounded to the nearest percent. Determine the number of possible class sizes.

At 17:10, Totoro hopped onto a train traveling from Tianjin to Urumuqi. At 14:10 that same day, a train departed Urumuqi for Tianjin, traveling at the same speed as the 17:10 train. If the duration of a one-way trip is 13 hours, then how many hours after the two trains pass each other would Totoro reach Urumuqi?

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 $B$ be the answer to problem 14, and let $C$ be the answer to problem 15. A quadratic function $f(x)$ has two real roots that sum to 210 + 4. After translating the graph of $f(x)$ left by $B$ units and down by $C$ units, the new quadratic function also has two real roots. Find the sum of the two real roots of the new quadratic function.

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?


Define a lucky number as a number that only contains 4s and 7s in its decimal representation. Find the sum of all three-digit lucky numbers.

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.