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.
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.
For a positive integer m, we define $m$ as a $\textit{factorial}$ number if and only if there exists a positive integer $k$ for which $m = k\cdot(k - 1)\cdots 2\cdot 1$. We define a positive integer $n$ as a $\textit{Thai}$ number if and only if $n$ can be written as both the sum of two factorial numbers and the product of two factorial numbers.
What is the sum of the five smallest $\textit{Thai}$ numbers?
While playing table tennis against Jordan, Chad came up with a new way of scoring. After the first
point, the score is regarded as a ratio. Whenever possible, the ratio is reduced to its simplest form. For
example, if Chad scores the first two points of the game, the score is reduced from $2:0$ to $1:0$. If later
in the game Chad has $5$ points and Jordan has $9$, and Chad scores a point, the score is automatically
reduced from $6:9$ to $2:3$. Chad's next point would tie the game at $1:1$. Like normal table tennis, a
player wins if he or she is the first to obtain $21$ points. However, he or she does not win if after his
or her receipt of the $21^{st}$ point, the score is immediately reduced. Chad and Jordan start at $0:0$ and
finish the game using this rule, after which Jordan notes a curiosity: the score was never reduced. How
many possible games could they have played? Two games are considered the same if and only if they
include the exact same sequence of scoring.