Compared to a standard clock, an old clock runs 9 seconds faster every hour, while an old watch runs 6 seconds slower every hour. At 10:30:00 AM today, they show the same correct time, as shown in the diagram below.
After today, in how many days will this old clock and old watch show the same correct time again at 10:30:00 AM?
Inside a rectangle, a certain point has been chosen. By connecting this point with the midpoints of the four sides of the rectangle, the rectangle is split into four disjoint regions. It is known that the areas of three of these regions are $95\text{ cm}^2$, $86\text{ cm}^2$ and $77\text{ cm}^2$ respectively, as shown in the diagram below.
What is the area, in $\text{cm}^2$, of the shaded region? Note: The diagram is NOT drawn to scale.
Ann bought a total of 17 blue and red pens for 222 dollars. Each pen costs an integer number of dollars, and a red pen costs 3 dollars more than a blue pen. How many red pens did Ann buy?
A game board is shown in the diagram below. 15 balls are placed at the points indicated by the letters and will be shot in the directions indicated by the arrows. The black squares are obstacles such that when each ball hits the obstacle, it will be reflected by its path, where the angle of incidence is equal to the angle of reflection. For example, when the ball located at point X is shot, it would end up in the goal on the left (as shown by the dotted lines); while when the ball located at point Y is shot, it will eventually go outside (as shown by the dotted lines). The balls cannot be reflected from any border of the game board.
If the balls located at points A, B, C, D, E, F, G, H, I, J, K, L and M are shot one after another in the directions indicated by their corresponding arrows, how many balls will eventually end up in one of the goals, without counting the example (the ball starting from X)?
A string, which has an integer length, in cm, is to be cut into 4 pieces. It is known that the lengths of all 4 pieces are all different integers. If the length of each piece is $\frac12$ times the length of the previous piece plus 1 cm, except for the longest piece, what is the shortest possible length, in cm, of the string?
In triangle ABC, $\angle ACB=30^\circ$ and $\angle BAC=90^\circ$, where D is a point on BC. Points X and Y are on AD such that the areas of triangles AXB, AXC, XBY, XCY and YBC are all equal, as shown in the diagram below.
Select nine different digits from 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, and then use these digits to create three positive three-digit numbers, namely A, B and C, where $A>B>C$. What is the smallest possible sum of the pairwise positive differences (subtracting the smaller number from the larger number, that is $A-B$, $A-C$ and $B-C$) between these three three-digit numbers?
Alex and Boris live in the same village. One day, both of them left their village at the same time to travel to the city, where it is known that they have different constant speeds. At a certain moment in the journey, we know that if Alex had gone three times as far as he had so far, he would have half as far to go, and if Boris had gone half as far as he had so far, he would have three times as far to go. How many times is Boris faster than Alex?
The area of a rectangular paper ABCD is $120\text{ cm}^2$. Now, fold this paper along line EF so that C coincides with A, where E is on AB and F is on CD, as shown in the diagram below. If $FD=\frac13CD$, what is the area, in $\text{cm}^2$, of the shaded region?
Amy selects one number from 1, 2, 3, …, 99 and 100. After that, Brandon selects a number from the remaining ninety-nine numbers such that the last digit of the product of their numbers is 4. How many ways are there for Amy and Brandon to select their numbers?
Excluding the cue ball (white ball), a complete set of snooker balls consist of 21 balls, of which there are 15 red balls, which are worth 1 point each, and the 6 other non-red balls which are worth 2, 3, 4, 5, 6 and 7 points respectively. Two balls are randomly selected from these 21 balls at the same time. Let $p$ be the probability that the total value of the two balls selected is at most 5 points. What is the value of $210p$?
If a positive integer, composed of at least three digits, has all its digits written in strictly decreasing order from left to right, then this positive integer is called a good integer. For example, 876 and 97531 are both good integers, while each of 87, 5331 and 97513 are not. How many good integers are there in total?
Let $a$, $b$ and $c$ be digits so that the decimal number $\overline{2a41b3c125}$ is a multiple of 7, 9 and 11. List down all the possible numerical values of $\overline{2a41b3c125}$. Give a detailed solution.
In one move, a bunny can either jump forward 2 or 3 steps or jump backward 1 step, as shown in the diagrams below.
Starting from cell number 2, in how many different ways can the bunny reach cell number 8 in less than or equal to 5 moves? Note that the different order of jumping forward 2 or 3 steps and backward 1 step are viewed as different ways. Also, the bunny is only allowed to jump on any cell from 0 to 10.
