CLMC 2023 Q9 - Q14 Walkthrough

This is my recorded video, with a few polishes, explaining how you can solve this question most efficiently.

Question 9: Let ABCD be a quadrilateral with coordinates (2,2), (14,2), (16,0), and (0,0).
This quadrilateral is inscribed in a circle. What is the area of this circle?

(A) 61π (B) 64π (C ) 79π (D) 81π (E) 100π

Question 10: Consider the sequence $t_1,t_2,t_3,...,t_{15},t_{16},t_{17}$. This sequence has the following three properties.

(i) Each of the 17 integers from 1 to 17 appears exactly once in the sequence.

(ii) The sum of each pair of consecutive terms is a perfect square. (For example, $t_1 +t_2$ is a perfect square, $t_2 +t_3$ is a perfect square, and so on.)

(iii) $t_1 = 17$.

What is the value of $t_5$?

(A) 3 (B) 6 (C ) 10 (D) 13 (E) 15

Question 11: Alphonse drives to Beryl’s house, intending to arrive at a certain time. There is no traffic on the road between the two houses, and so Alphonse can drive at a constant speed.

If Alphonse drives at $x$ kilometres per hour, then he will arrive 7 minutes early.
If Alphonse drives at $y$ kilometres per hour, then he will arrive 7 minutes late.
If Alphonse drives at 72 kilometres per hour, then he will arrive exactly on time.

If $x$ and $y$ are positive integers, what is the smallest possible value of $x+ y$?

(A) 144 (B) 147 © 150 (D) 156 (E) 162 (F) 180

Question 12: Suppose we remove two diagonally-opposite corner squares from an 8 by 8 board, as shown in the diagram below.

How many squares and rectangles, of all sizes, appear on this modified board?

(A) 1127 (B) 1158 (C ) 1159 (D) 1169 (E) 1200 (F) 1296

Question 13: You are given a biased coin, where Heads comes up with probability $\frac{2}{3}$ and Tails comes up with probability $\frac{1}{3}$.

You play a game where you start with 0 points. Each time you flip Heads, you add 2 points to your score. Each time you flip Tails, you add 1 point to your score.

If you reach a total of $exactly$ n points, then you win. However, if you go over n points, then you lose.

Let $P_n$ be the probability that you win the game with a target score of n points. For example, $P_2 =\frac{7}{9}$ because you win the game by flipping H (probability $\frac{2}{3}$ ) or TT (probability $\frac{1}{9}$ ) but lose by flipping TH (probability $\frac{2}{9}$ ).

Determine the value of $P_8$, rounded to three decimal places.

(A) 0.576 (B) 0.584 (C ) 0.592 (D) 0.600 (E) 0.608 (F) 0.616

Question 14: Let $A,B,C,D,E,F$ be six points equally spread out around a circle. Draw all 15 edges connecting two of these six points. Aponi picks 8 of these 15 edges and colours them red; the remaining 7 edges are then coloured blue.

Consider all triangles that can be formed from three of these six points. For each triangle that has only red edges, Aponi scores 1 point. For each triangle that has only blue edges, Aponi scores 2 points. For all other triangles (e.g. a triangle with two red edges and one blue edge), Aponi scores 0 points.

Let $X$ be the maximum score that Aponi can obtain, and let $Y$ be the minimum score that Aponi can obtain.

What is the value of $X−Y$?

(A) 5 (B) 6 (C ) 7 (D) 8 (E) 9 (F) 10