COM-I Homework 3

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COM-I Homework 3

Deadline: 8 August (Tuesday) 11:59 PM

ဒီအိမ်စာမှာ စုစုပေါင်း 22 points ရှိပါတယ်။ ဒါပေမယ့် အနည်းဆုံး 9 points ရတာနဲ့တင် ဒီအိမ်စာကို အမှတ်ပြည့်ပေးမယ်။ ဒီလိုဆိုပေမယ့်လည်း ဖြစ်နိုင်ရင် ပုစ္ဆာစေ့အကုန်တွက်ကြည့်ကြပါ။ များများတွက်မှ trick တွေနဲ့များများရင်းနှီးလာမှာပါ။ ပုစ္ဆာတွေက ဂဏန်းကိုင် heavy ဖြစ်တာမို့လို့ calculator သုံးကြပါ။ ကိုယ့်အမှတ်ကိုယ်မကြိုက်ရင် deadline မကျော်ခင် ကျေနပ်တဲ့အထိအကြိမ်ကြိမ်ပြန်ဖြေပြီး submit လို့ရပါတယ်။ မှန်ပြီးသားအပုဒ်တွေကို အဖြေထပ်ဖြည့်စရာမလိုပါဘူး။ တစ်ခါမှန်တာနဲ့ အမှန်ပေးပါမယ်။

စတား (*) ပြထားတဲ့ပုစ္ဆာတွေက နည်းနည်းပိုခက်ပါတယ်။

Homework ကို ကိုယ့်သူငယ်ချင်းတွေနဲ့ ပေါင်းတွက်၊ ပေါင်းစဉ်းစားလို့ရပါတယ်။ အဖြေတွေကို တိုက်ရိုက်ကူးမချရင် ပြီးတာပါပဲ။ အစ်ကိုနဲ့ အစ်ကိုနိုင်ဇော်လူကိုလည်း မေးခွန်းနားမလည်တာတို့၊ Hint လိုချင်တာတို့ဆိုရင် အချိန်မရွေးပြောလို့ရပါတယ်။

Name *

1 point

Question 1: Tennis Team of the Class

A class has 12 boys. Coach Thiha wants to create a 5-person boys-tennis team representing the class.

Q1a. In how many ways can he create the team if there are no restrictions?

1 point

Q1b. Five boys in the class are not interested in tennis at all. In how many ways can he create the team if he wants to exclude them from the team?

1 point

Q1c. Two of the boys in the class turns out to be really good at tennis. In how many ways can he create the team if he always want to include them in the team?

1 point

Question 2: E-Sports Matchup

There are two esports squads where squad Spectra has 6 players and squad NeonBurma has 8 players. They want to make a 3 vs 3 match such that 3 Spectra players are on one side and 3 NeonBurma players on the other.

(Note: the two sides are considered the same i.e. NeonBurma vs Spectra is the same is Spectra vs NeonBurma)

Q2a. How many different matches are possible if there are no restrictions?

1 point

Q2b. Each squad has a leader. In how many different matches are possible if leaders from each team must be included in the match?

1 point

Q2c*. Suppose that Spectra has 4 young players and 2 veterans while NeonBurma has 5 young players and 3 veterans. Each team wants to include at least one young player and at least one veteran in the match. How many matches are possible?

1 point

Question 3: Arranging OLYMPIAD Again

Ma Dar Dar wants to create a 5-letter word using the letters of the word OLYMPIAD each no more than once.

Q3a. How many words can she create if she wants to use exactly 3 consonants?

1 point

Q3b. How many words can she create if she wants to use at least 3 consonants?

1 point

Q3c*. She wants her word to have the following property:

"when the consonants are erased from the word, the remaining words spell IOA."

How many words can she create?

1 point

Question 4: Drawing Pictures from Points

Wutt Yee drew 12 points on the paper. She wants to draw several figures using those points.

Q4a. Suppose no three points are collinear. How many lines can she draw by joining the chosen points?

1 point

Q4b. Eleven of the points are on a line and one point is not on that line. How many (non-degenerate) triangles can she draw?

1 point

Q4c. She puts those points on a square, 3 points on each side. How many (non-degenerate) triangles can she draw using the points as the vertices?

1 point

Q4d*. The points are in the same configuration as in Q4c. How many (non-degenrate) pentagons can she draw using the points as vertices?

1 point

Question 5: Overcounting-related

Q5a. The logo for 60th International Mathematical Olympiad is a star shaped 3D-solid whose surface is made up with 60 isosceles triangles as shown. How many edges does this figure have?

1 point

Q5b. In the following picture, how many segments are there whose endpoints are blue points?

1 point

Q5c. Let N be a positve integer. A bakery sells 20 types of cookies. One day, a group consisting N children arrived at the bakery and each child ordered 4 different types of cookies. If each type of cookie is bought by
exactly 5 children, what is the value of N.

(Note: Write down the value of N only. For example, if the answer is N = 10, only write down 10)

1 point

Q5d. How many ways are there to seat six married couples around a circular bench so that everyone sits together with their partner?

1 point

Q5e*. Suppose there a red circular bench and blue circular bench in Q5d instead of just one bench. In how many ways can we place the couples if three couples are to sit on each bench, and each person is to sit together with their partner?

1 point

Challenge Problem

Note: This week's challenge is a bit easier than previous weeks.

How many rectangles are there in the following picture?

3 points

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