The “IMSO Robot” is controlled by a gamepad with buttons ↑ ↓ → and ←. When ↑ is pressed 1 time, the “IMSO Robot” will go 1 unit to the north, ↓ is pressed 1 time, the “IMSO Robot” will go 1 unit to the south, → is pressed 1 time, the “IMSO Robot” will go 1 unit to the east, ← is pressed 1 time, the “IMSO Robot” will go 1 unit to the west.
Anna controls the “IMSO Robot” from A to B by pressing the buttons as follows:
Annie is 22 years younger than Betty and Cindy is 6 years younger than Betty. The age difference between Diana and Elsa is 29 years. Diana is twice as old as Annie, and the sum of their ages is 54. Elsa is not older than Cindy. What is the sum of the ages of Betty, Cindy, and Elsa?
In the diagram, the parallelogram ABCD is divided into 16 identical small parallelograms. The area of the shaded quadrilateral is $68\ \mathrm{cm}^2$. What is the area, in $\mathrm{cm}^2$, of the parallelogram ABCD?
Yuan bought mangoes and melons at the market. The price of a mango is $\frac37$ of the price of a melon. At first, Yuan used $\frac38$ of his money to buy 6 melons and 2 mangoes. The following week, he used the rest of his money to buy 2 melons and some mangoes. How many mangoes did Yuan buy in the second week?
In a TV show, four women (Ana, Bianca, Celine and Diana) and four men (Andrew, Ben, Chris and Daniel) are to be divided into four groups. Each group consists of a man and a woman. If Ana cannot be grouped with Ben, in how many different ways can we group them?
Let M be a two-digit positive integer such that the greatest common divisor of $M+9$, $M+35$, and $M+74$ is greater than 1. What is the sum of all possible values of M?
A team participating in the IMSO 2024 competition consists of six male and six female students. They will be staying in a dormitory consisting of six rooms in a row. Each room must be occupied by exactly two students of the same gender. Moreover, any two adjacent rooms must be occupied by students of different genders. In how many ways can they make such arrangements?
Inspector Daniel is investigating a crime. He apprehends three suspects: Anne, Bill, and Caroline. He knows that no one else could have participated in the crime. During the interrogation, each of the suspects make two statements:
- Anne: I didn’t do it. Bill did it.
- Bill: I didn’t do it. Caroline did it.
- Caroline: I didn’t do it. Anne did it alone.
Daniel also discovered that one of the suspects always tells the truth and the other two always give false statements. Who is the truth-teller?
Let n, m and k be positive integers such that $n+n\times m+n\times m\times k=13$. What is the sum of all possible value of $k+k\times m+k\times m\times n$?
Xiaoming wants to buy school supplies. The total number of pens, pencils and erasers that he wants to buy is 100. The price of one pen is 6 yuan, one pencil is 4 yuan and one eraser is 1 yuan. He spends exactly 200 yuan and buys at least one of each type. How many different combinations of the number of pens, pencils and erasers can he buy?
Wei calculated the square and the cube of a positive integer. When she looked at the results, she noticed that all the digits from 1 to 8 had been used, each exactly once. What is the value of this positive integer?
In the diagram, the ratio of the width and the length of the rectangle is 1 : 4. The ratio of the radius of the semicircle to the radius of the biggest circle to the radius of the grey circle to the radius of the two smallest circles is 8 : 4 : 2 : 1. If the area of the rectangle is $1024\ \mathrm{cm}^2$, What is the total area, in $\mathrm{cm}^2$, of the shaded regions? (Use $\pi=3.14$.)
Huang took a test and answered all the questions. The test consists of questions from three subjects: Algebra, Geometry and Combinatorics. It turned out that he correctly answered:
- 67.5% of the Algebra questions,
- 86% of the Geometry questions,
- 80% of the Combinatorics questions,
- 70% of the questions on the subjects of Algebra and Combinatorics and
- 85% of the questions on the subjects of Geometry and Combinatorics.
What is the percentage of correct answers from all the questions?
