Given a regular hexagon, we construct all possible triangles from its vertices. We find that:
- There are $L$ possible acute triangles.
- There are $M$ possible right triangles.
- There are $N$ possible obtuse triangles.
Triangles are considered distinct if they have at least one different vertex, even if the triangles are congruent. What is the value of $L\times M\times N$?
Alex, having only one 20-dollar bill, went to buy lunch. While giving him back change, the cashier got distracted and mixed up the dollars and cents—paying dollars instead of cents and cents instead of dollars. Alex put the change in his wallet without looking at it. After a while he decided to buy a slice of bread as an addition to his lunch. He paid 15 cents for it and then noticed that the amount he had with him now was double of what he should have had after paying for the lunch and the slice of bread (if the cashier had not made a mistake). How many dollars and how many cents is the correct change that Alex should have received after buying lunch? (Note: 1 dollar = 100 cents)
Two couriers, Alice and Bob, travel at different constant speeds. Alice departs from village $A$ towards village $B$ and Bob departs from village $B$ towards village $A$ at the same time. When they meet at point $C$, both turn around and return to their starting points. After some time, Alice realizes that she has forgotten to give Bob a package, so she turns around again and catches up with Bob halfway between $C$ and $B$. After delivering the package, Alice turns around to go back to $A$ and Bob continues to $B$. If the distance between $A$ and $B$ is 24 km, when Bob arrives at $B$, what is the distance, in km, of Alice from $A$?
There are 8 circles on a rectangular board as shown in the diagram below. The distances between any pair of adjacent circles, horizontally or vertically, are all equal. Boris wants to colour some circles in red. How many ways can Boris colour some of the circles so that the resulting board has no line of symmetry?
The number 3 has the following properties: it is one less than a square number, and five times of it is also one less than a square number.
$$3=4-1=2^2-1$$
$$5\times3=16-1=4^2-1$$
24 is another number with the same properties:
$$24=25-1=5^2-1$$
$$5\times24=121-1=11^2-1$$
What is the smallest number with the same properties that is greater than 25?
An $a\times b\times c$ rectangular block with $a\le b\le c\le20$ is made from unit cubes. When it is viewed from one vertex, such that the three faces intersecting at this vertex can be entirely seen, exactly 487 cubes are visible. What is the number of the unit cubes not visible? List down all possible answers.
For example, the diagram on the right is a $3\times4\times5$ rectangular block. When it is viewed from vertex $A$, 36 cubes are visible.
In triangle $ABC$, $\angle BAC=120^\circ$ and $\angle ABC=40^\circ$. Let $N$ be a point on $AC$ such that $AN=AB$. Let $M$ be a point on the line $BA$ such that $\angle ANM=40^\circ$, as shown in the diagram below. What is the measure, in degrees, of $\angle BMC$?
A wizard paints in grey four squares of an infinite grid table, as shown in the diagram below on the left. Then he casts a spell so that, at every second, each square that has at least one common side with those grey squares becomes also grey. For example, after 1 second, there will be 18 grey squares as shown in the diagram below on the right. What is the number of grey squares after 60 seconds?
The rhombus shape below is formed by using 25 dots arranged in a triangular grid. The distances between any two adjacent dots are equal. What is the number of equilateral triangles that can be drawn with 3 dots from these 25 dots as vertices?
