A ball is dropped starting from the top of the drainage system and slides downward until it reaches the bottom, as shown in the diagram below. If it is known that the ball cannot go upward, how many different paths could the ball pass through the drainage? (Note: One example is shown.)
Alice buys 80 more marbles than Bob and Bob buys 67 less than Charles. Edwards buys 19 more marbles than Dennis while Charles buys 38 more than Edward. If Bob and Edward buy a total of 41 marbles between them, how many marbles are bought altogether?
In a Mathematics test given to a certain class, the average score of those who passed is 78, while the average score of those who failed is 34. If the overall average score of the class is 67, what is the percentage of students in the class who passed the test?
Two terms of an arithmetic progression are 1987 and 2022, which are not necessarily consecutive. If the common difference $d$ of the arithmetic progression is a positive integer, then what is the number of possible values $d$ could have?
In square $ABCD$, a line through $B$ intersects the extension of $CD$ at $E$, the side $AD$ at $F$ and the diagonal $AC$ at $G$, as shown in the diagram below.
If $BG=3$ cm and $GF=1$ cm, then what is the length, in cm, of $FE$?
The letters of the word “EVERYBODY” have been placed randomly in the grid below. By starting at the bottom left-hand corner “E”, we can only move up or move to the right until we reach the top right-hand corner “Y”, how many possible ways of collecting all the letters of the word in any order are there? Note that you cannot move diagonally.
In triangle $ABC$, it is known that $\angle BCA=90^\circ$ and points $E$ and $F$ lie on the hypotenuse $AB$ such that $AE=AC$ and $BF=BC$, as shown in the diagram below.
A Mathematics test was given to a group of 12 students, where each problem was solved by exactly 10 students. It is known that 11 students solved exactly 7 problems each. How many problems did the 12th student solve?
In the base ten number system, the number $379_{10}$ means $3\times10^2+7\times10+9$.
In the base $r$ number system, where $r$ is a positive integer, if the equation $1020_r-460_r=230_r$ holds, then what is the value of $r$?
Alice was told the sum of three positive integers while Bob was told their product. “If I knew”, said Alice, “that your number is greater than mine, I could find the integers”. “But I know that my number is less than yours.” replied Bob. What is the sum of these three numbers?
The diagram below shows a head of a cartoon character which is represented by a big circle, with its eyes being two circles of radii 1 cm. The two smaller circles are touching at the centre of the big circle and they are also tangent to the big circle. The shaded region represents its hat, for which the horizontal line is tangent to the smaller circles, as shown in the diagram below.
What is the area, in $\text{cm}^2$, of the shaded region?
In the Fibonacci sequence $\{F_k\}$, $F_1=F_2=1$ and $F_{k+1}=F_k+F_{k-1}$ for $k\ge2$. What is the value of $\frac1{F_1F_3}+\frac1{F_2F_4}+\frac1{F_3F_5}+\frac1{F_4F_6}+\frac1{F_5F_7}+\frac1{F_6F_8}+\frac1{F_7F_9}+\frac1{F_8F_{10}}$?
Let $P$ be the point in the interior of triangle $ABC$ such that $\angle BAP=13^\circ$, $\angle CAP=31^\circ$, $\angle ACP=44^\circ$ and $AP=BC$, as shown in the diagram below. What is the measure, in degrees, of $\angle BCP$?
