In the diagram below, both diagonals of the rectangle $ABCD$ are divided into $12$ equal segments. If the total area of all the white regions inside the rectangle $ABCD$ is $10$ cm², then what is the total area, in cm², of the shaded regions?
Ten friends were sitting at a round table. Following the clockwise direction, each of them was asked: “What is the sum of the ages of your two neighbours?” The answers (given in the same order) were: $20,22,24,26,28,30,32,34,36$ and $38$. How old is the person who answered $26$?
Alvin and Bob take turns taking marbles from a bag. Alvin starts by taking one marble, then Bob takes two marbles, then Alvin takes three marbles, then Bob takes four marbles, and so on. If, during a turn, one of them doesn’t find enough marbles in the bag, then he takes all the marbles that are left. In the end, Alvin has $101$ marbles. How many marbles were in the bag at the beginning?
In a mathematics class, students’ grades are based solely on exams, all of which carry the same weight. When Eva scored $98$ on a certain exam, the average of her exam scores up to that moment increased by $1$ point. However, when she scored $70$ on the next exam, her average dropped by $2$ points. Including these two exams, how many exams had she taken?
What is the sum of the three largest four-digit numbers in the form $\overline{VIMC}$ such that $V,I,M$ and $C$ are all distinct, $V$ is even, and $\overline{VI},\overline{IM}$ and $\overline{MC}$ are all two-digit prime numbers?
Five squares are drawn next to each other, as shown in the diagram below. If the area of each smallest square is $30$ cm², what is the area, in cm², of the shaded triangle?
In addition, we also know the following:
- The maximum number of marbles that a student has is $15$.
- The students with $3$ or more marbles have an average of $6$ marbles.
- The students with $12$ or fewer marbles have an average of $5$ marbles.
What is the total number of marbles combined across all the students?
In the diagram below, $ABC$ is a right triangle with $\angle A=90^\circ$, $AB=15$ cm and $AC=36$ cm. Let $D$ and $E$ be points on $BC$ such that $BD=DE=EC$. What is the value, in cm², of $AD^2+AE^2$?
Five children, Alice, Bob, Carter, David and Emily, and three adults, Frank, Gary and Henry, are to be seated in a row with eight seats so that:
- No two adults sit in adjacent seats;
- Bob sits to the left of Alice and to the right of Carter;
- David sits to the right of Emily.
In how many ways can these eight people be seated?
In how many different ways can the shape below be divided along the grid lines into two identical whole parts (each part has seven unit squares, connected by sides)? (Note: Two parts are identical if they can be obtained from each other by a finite number of reflections and/or rotations.)
A 7-digit number $\overline{abcdefg}$ is said to be V-shaped if it has the following properties:
- No two digits among $a,b,c,d,e,f$ and $g$ have the same value;
- $a>b>c>d$;
- $d<e<f<g$.
A three-digit number $\overline{abc}$ is said to be remarkable if $\overline{abc}$ is less than $500$ and the last three digits of $\overline{abc}^{\,3}$ are $\overline{abc}$. What is the sum of the largest and smallest possible remarkable numbers?
Below, the left diagram shows a chain that has two connected rings and the right diagram shows a chain that has seven connected rings.
Mr. Metal owns $10$ chains. Two of them are identical, each consisting of a single ring. The remaining eight chains have $2,3,5,8,13,21,34$ and $55$ connected rings, respectively, as shown below.
Mr. Metal wants to make two or more chains, all of the same length, with no loops or any opened ring in it. All rings must be used. What is the shortest amount of time, in minutes, to make them if it takes exactly $1$ minute to cut a ring (which disconnects it from other rings) and exactly $3$ minutes to connect rings by welding one open ring (after connecting it to zero or more other rings)?
Examples of Cutting Process A and B: Cut; takes exactly one minute. Examples of Connecting Process A and B: Connect; takes exactly three minutes.
The diagram below shows a $4\times4$ board made of $16$ unit squares, which contains one star in each corner cell. The board is to be partitioned - by cutting along the grid lines - into non-overlapping regions, where each region is connected by sides. A partition is said to be valid if the regions all have the same area and contain exactly one star each. Draw all the valid partitions of the board.
(Note: Two partitions are considered the same if they can be obtained from each other by a finite number of reflections and/or rotations.)
Answer:
(Note: You may not need to fill out all the provided answer diagrams.)
In the diagram below, $ABCD$ is a quadrilateral of area $675$ cm², where $AB$ and $CD$ are parallel and $AB=20$ cm, $CD=25$ cm. Points $M,R$ and $T$ are on segments $BD,AB$ and $AC$, respectively, such that $\frac{MD}{MB}=\frac16$ and $\frac{TC}{TA}=\frac14$.
If the area of the triangle $MRT$ is $216$ cm², what is the length, in cm, of $AR$?
