There are four contestants and 25 questions in a television quiz show. Each question is attempted by only one contestant, and several points, from 0 to 150, are awarded according to how it is answered. The results for the 25 questions are shown in the table below, without indicating which contestants receive the points.
The first contestant gets twice as many points as the third while the second gets three times as many points as the third. The fourth contestant attempts only one question. How many points does he get on that question?
In the diagram, $ABCD$ and $EFGH$ are both unit squares, with a common centre at $O$. If the length of $PW$ is equal to $\dfrac{45}{101}$ units, find the area, in square units, of the octagon $PQRSTUVW$.
Four positive integers are arranged to form a $2\times2$ table. The product of the two numbers in each row and in each column is computed. These four products are added to the sum of the four numbers in the table, and the result is 2018. What is the sum of the four numbers in the table?
In quadrilateral $ABCD$, $\angle ABC=\angle BAD=90^\circ$. Point $E$ is on $AB$ such that $BE\times(BC-AD)=AE\times BC$. If $BC-AD=1$ cm and $\angle ADB=2\angle BCE$, find the length of $BD$ in cm.
A magic crystal is in the shape of a shaded square. On the first day, it grows a white head in the shape of an isosceles right triangle whose hypotenuse is a side of the square, and two shaded ears in the shape of squares each with a leg of the white triangle as a side.
The diagram above shows what it looks like after the first day. On the second day, each ear grows a head and two ears like in the first day. The diagram below shows what it looks like after the second and third day. Note that the new head always grows on the side of the ear opposite to the head of the day before.
The crystal explodes whenever two of its ears touch. After how many days will this happen?
Four different three-digit numbers have the same hundreds digit. Their sum is divisible by three of them. Find the remainder when the sum is divided by the fourth number.
Use each of the two pieces below exactly once to form a shape which can be divided into two parts by a line such that the two parts are mirror images of each other across the line. The pieces may be rotated and/or reflected multiple times, but they may not overlap. Find two solutions.
At noon, Donny and Ronny started from Burgas towards Varna while Lonny started from Varna towards Burgas. The three boys are riding their bicycles at different constant speeds. At 13:00, Ronny was midway between Donny and Lonny, and at 13:20, Lonny was midway between Donny and Ronny. At what time will Donny be midway between Ronny and Lonny?
A town has 25 building blocks in a $5\times5$ arrangement. Each block has a height from 1 to 5. There is a block of each height in every row and every column. Twenty observers are standing outside looking in. Each observer records the number of blocks he can see, which are blocks not hidden behind a taller block. Determine the height of each block.
The greatest common divisor of four positive integers, not necessarily distinct, is equal to 1. Their least common multiple is equal to their sum. Find the number of possible values of the sum of these four numbers.
