Vòng 3 - Vòng chung khảo

QUESTION 1 BIT HARD: The degree measures of the interior angles of a quadrilateral form a geometric sequence whose terms have integer values and are all integer multiples of the first term. What is the largest possible degree measure of an angle in this quadrilateral?

QUESTION 2 BIT HARD: To win a certain lottery, one must match three different numbers chosen from the integers 1 through 25, in any order. Liam buys 500 tickets, each with a unique combination of numbers. What is his probability of winning? Express your answer as a decimal to the nearest hundredth.

QUESTION 3 HARD: There exist two non-congruent right triangles for which the length of the shorter leg in each triangle is 9 units and all sides have integer lengths. What is the sum of the lengths of the longer legs of these two triangles?

QUESTION 4 HARD: What is the perimeter of a right triangle with an area of 10 cm² and a hypotenuse of length 10 cm? Express your answer in simplest radical form.

QUESTION 5 HARD: As a used-car salesperson, Noah has a monthly sales quota, which is the minimum number of cars he must sell each month. Noah had not sold any cars in June, as of the 24th of the month. However, on June 25th, Noah sold half of the number of cars in his monthly quota, plus one more car. On June 26th, he sold half of the remaining number of cars he needed to sell, plus one more car. The same pattern continued until June 30th, when Noah sold half of the remaining cars he needed to sell, plus one more car and reached his monthly sales quota. Noah has a monthly sales quota to sell how many cars?

QUESTION 6 HARD: Edna enters a room with 1000 bottles lined up in a row left to right. One bottle contains a tasteless magic potion. All bottles to the left of the magic potion contain tasteless water. All bottles to the right of the magic potion contain a bitter poison. Edna can drink from no more than two bottles containing poison before becoming sick and being unable to drink anything else. She can take an unlimited number of drinks from any other bottle. What is the minimum number of bottles from which Edna may need to drink to ensure she can identify the bottle containing the magic potion no matter where it is in the lineup?

QUESTION 7 HARD: If four people each randomly pick an integer from 1 to 10, inclusive, what is the probability that at least two of the people pick the same integer? Express your answer to the nearest tenth. 

QUESTION 8 HARD: What is the radius of the largest circle that can be inscribed in an acute triangle with sides 13, 14 and 15 units?

QUESTION 9 HARD: What is the sum of all prime numbers p less than 60 such that there exists a right triangle whose side lengths are all integers and whose hypotenuse has length p?

QUESTION 10 HARD: The digits 2, 0, 1 and 4 are used to create every possible positive four-digit integer, with each digit used exactly once in each integer. What is the arithmetic mean of all these integers?

QUESTION 11 HARD: If the average length of the edges of a right rectangular prism is 13 inches, and the dimensions of the prism are distinct integers in geometric progression, what is the sum of the volumes of the distinct prisms that meet these criteria? 

QUESTION 12 HARD: A particular date is called a difference date if subtracting the month number from the day gives you the two-digit year. For example, June 29, 2023 and January 1, 2100 are difference dates since 29 − 6 = 23 and 1 − 1 = 00. Including these two dates, how many dates during the 21st century (January 1, 2001 to December 31, 2100) can be classified as difference dates?

QUESTION 13 SUPER HARD: Two standard six-sided dice, each with faces numbered with the positive integers 1 through 6, have the probability distribution shown for the sum of the top-facing values on the dice. Two non-standard but fair six-sided dice can be numbered differently with nonnegative integers on each face and still yield the same probability distribution. Though a number may be on both dice, a number may not appear more than once on either die. If a is the sum of the six numbers on one of these non-standard dice, and b is the sum of the six numbers on the other die, what is the value of the product a × b?

QUESTION 14 SUPER HARD: We define a Heronian triangle to be a triangle with three integer side lengths and integer area. What is the least possible positive area of a Heronian triangle whose longest side has a length of 17 units? 

QUESTION 15 SUPER HARD: A rectangle measures 2 × 2√3 units. Two arcs are drawn with their centers at the midpoints of the shorter sides, as shown. What is the area of the shaded region? Express your answer as a decimal to the nearest hundredth.

https://drive.google.com/open?id=12uf97E5yqVz_pk7Tu6xYOyJlIWYqZEk5