The cafeteria puts 289 pieces of bread out for the students every day. During a week, the students eat a different number of pieces each day. On some days, some pieces are left over. On other days, the cafeteria puts out more pieces until the students are full. The number of pieces left over is recorded as a positive integer. The number of extra pieces put out is recorded as a negative integer. The product of these seven numbers is $-252$. What is the total number of pieces of bread eaten during this week?
A circle of radius 12 cm touches all four sides of a quadrilateral $ABCD$ with $AB$ parallel to $DC$. If $BC=25$ cm and the area of $ABCD$ is 648 $\mathrm{cm}^2$, determine the length, in cm, of $DA$.
$D$ is a point on side $AB$ and $E$ is a point on side $AC$ of triangle $ABC$. $P$ is the point of intersection of $BE$ and $CD$. The area of triangle $ABC$ is 12 $\mathrm{cm}^2$. Triangle $BPD$, triangle $CPE$ and the quadrilateral $ADPE$ all have the same area. What is the area, in $\mathrm{cm}^2$, of $ADPE$?
Find the sum of the digits of the product of a number consisting of 2016 digits all of which are 6s, and another number consisting of 2016 digits all of which are 9s.
Let $n$ be a positive integer. Each of Tom and Jerry has some coins. If Tom gives $n$ coins to Jerry, then Jerry will have 2 times as many coins as Tom. If instead Jerry gives 2 coins to Tom, then Tom will have $n$ times as many coins as Jerry. Find the sum of all possible values of $n$.
In triangle $ABC$, $BC=13$ cm, $CA=14$ cm and $BA=15$ cm. $D$ and $E$ are points on sides $BC$ and $AC$, respectively, such that $DE$ is parallel to $AB$. If triangle $EDC$ has the same perimeter as the quadrilateral $ABDE$, determine $\frac{BD}{DC}$.
Three parallel lines $L_1$, $L_2$ and $L_3$ are such that $L_1$ is 1 cm above $L_2$ and $L_3$ is 2 cm below $L_2$. A right isosceles triangle has one of its vertices on each line. What is the sum, in $\mathrm{cm}^2$, of all possible values for the area of this triangle?
When a person with IQ 104 moved from village A to village B, the average IQ of both villages increased by 1. The sum of the population of the two villages is a prime number and the sum of the IQ of all people in both villages is 6102. Find the sum of the IQ of the people of village B including the new arrival.
Alice is at the origin $(0,0)$ of the coordinate plane. She moves according to the roll of a standard cubical die. If the die rolls is 1, she moves 1 space to the right. If the die rolls is 2 or 3, she moves 1 space to the left. If the die rolls is 4, 5 or 6, she moves 1 space up. What is the probability that after four moves, Alice lands on the point $(1,1)$ for the first time?
The diagram below shows a square $ABCD$ of side length 234 cm. The rectangles $CDPQ$ and $MNST$ are congruent. Find the length, in cm, of $CM$.
Figure note: The complete original diagram is taken from the official Traditional Chinese edition, because the supplied English PDF clips and omits its lower portion. The English question wording above is retained.
Answer entry: Give the exact expression; write sqrt(3) for $\sqrt3$.
$R$ is a point on a segment $CQ$ with $CR=4$ cm. A line perpendicular to $CQ$ intersect the circles with diameters $CR$ and $CQ$ at $A$ and $B$ respectively, with $A$ and $B$ on opposite sides of $CQ$. If the circumradius of triangle $ABC$ is $\sqrt6$ cm, find the length, in cm, of $CQ$.
Figure note: The complete original diagram is taken from the official Traditional Chinese edition; the supplied English PDF truncates the lower drawing and omits some labels. The English wording and the square-root radius are retained.
Submission guidance: Include a detailed explanation, as required for Section B.
