Math Problem Marathon
by
Milki
August 5, 2015 at 12:08 AM UTC
I was inspired by Gil's post on math earlier.
There's only one rule: have fun.
Of course, the point is to solve and give problems every time, but don't worry too much about that.
Please give sources, or if it is original, simply state it is original.
Whatever problem you would like to submit is up to you. Whether or not it's a proof-based problem, Algebra, Combinatorics, Geometry, or Number Theory, all is accepted, no matter what difficulty!
Don't look up problems, please (unless you really, really want to); it ruins some of the fun of solving a problem!
Format: Solution/Answer for Problem (with problem number), if applicable
New Problem (with problem number), if applicable
Problems:
What is the largest number that cannot be expressed as the sum of a multiple of 6 and a multiple of 7? (Original)
Bob and Joe are running buddies. Bob runs at 10 m/s, and Joe runs at 12 m/s, and whenever they meet each other on the track, they high-five. They don't like turning much, so their track has the minimum perimeter with the sides being integers in terms of meters. Given that the track has width 0 and that they first high-five 10 seconds after they start running, what is the perimeter of the track? (Original)
Cosmin is at (0,0), and he would like to visit Combo 3 at (3,3), Geo 3 at (-3,-3), Alg 3.5 at (-3,3), and NT 3 at (3,-3). He can only walk on lines parallel to the x and y axes, and before he starts, he decides which class to go to first. Given that he takes the shortest path possible between two points after he has decided an order of classes to visit, and N is the number of paths he can take, find the remainder when N is divided by 1000. (Original)
There are n sweets in a bag. 6 of the sweets are orange. The rest of the sweets are yellow.
Hannah takes a random sweet from the bag. She eats the sweet.
Hannah then takes at random another sweet from the bag. She eats the sweet.
The probability that Hannah eats two orange sweets is 1/3.
Show that n² – n – 90 = 0.
Edexcel maths paper 2015 GCSE
If Hannah keeps eating those sweets at that rate, she'd probably get a sugar rush.. Then she'd be all hyper.. The sweet eating might have to be dialed back a few notches.
There are n sweets in a bag. 6 of the sweets are orange. The rest of the sweets are yellow.
Hannah takes a random sweet from the bag. She eats the sweet.
Hannah then takes at random another sweet from the bag. She eats the sweet.
The probability that Hannah eats two orange sweets is 1/3.
Show that n² – n – 90 = 0.
Edexcel maths paper 2015 GCSE
Solution for (what I would think is #4): We have that (6/n)*(5/(n-1))=1/3.
Therefore, n(n-1)=90, and so n^2-n-90=n(n-1)-90=0.
Problem 5. The sequence 1,2,1,2,2,1,2,2,2,1,2,2,2,2,1,... is formed by a 1 followed by a block of 2, where the nth block of twos contains n twos. What is the sum of the first 1234 terms of the sequence? (Reworded 1996 AHSME #24)
Solution for (what I would think is #4): We have that (6/n)*(5/(n-1))=1/3.
Therefore, n(n-1)=90, and so n^2-n-90=n(n-1)-90=0.
Problem 5. The sequence 1,2,1,2,2,1,2,2,2,1,2,2,2,2,1,... is formed by a 1 followed by a block of 2, where the nth block of twos contains n twos. What is the sum of the first 1234 terms of the sequence? (Reworded 1996 AHSME #24)
The probability that Hannah takes an orange sweet at first is 6/n.
The probability that Hannah takes an orange sweet after taking the first orange sweet is 5/(n-1) since there are 5 orange sweets left, and n-1 total sweets left.
So, we get that (6/n)*(5/(n-1))=1/3, or n(n-1)=3*30=90.
There are 5 houses (along the street) in 5 different colors: blue, green, red, white and yellow.
2.
In each house lives a person of a different nationality: Brit, Dane, German, Norwegian and Swede.
3.
These 5 owners drink a certain beverage: beer, coffee, milk, tea and water, smoke a certain brand of cigar: Blue Master, Dunhill, Pall Mall, Prince and blend, and keep a certain pet: cat, bird, dog, fish and horse.
4.
No owners have the same pet, smoke the same brand of cigar, or drink the same beverage.
Hints:
1.
The Brit lives in a red house.
2.
The Swede keeps dogs as pets.
3.
The Dane drinks tea.
4.
The green house is on the left of the white house (next to it).
5.
The green house owner drinks coffee.
6.
The person who smokes Pall Mall rears birds.
7.
The owner of the yellow house smokes Dunhill.
8.
The man living in the house right in the center drinks milk.
9.
The Norwegian lives in the first house.
10.
The man who smokes blend lives next to the one who keeps cats.
11.
The man who keeps horses lives next to the man who smokes Dunhill.
12.
The owner who smokes Blue Master drinks beer.
13.
The German smokes Prince.
14.
The Norwegian lives next to the blue house.
15.
The man who smokes blend has a neighbor who drinks water.
bob has 9 fruit. 2 are apples, 4 are oranges. how many bananas does he have?
sorry for the difficulty
Excellence, I'm going to say the answer is: It's the end of the world and nothing matters anymore. Except my temp ban. 626046. I didn't know. Please understand. I love Avicus. I don't want a temp ban to stand in the way. I love Nebula, CTH, and CTW. I want to organize a scrimmage for my team. Otherwise, I can go to Cube Craft or Happy Hunger Games.
Excellence, I'm going to say the answer is: It's the end of the world and nothing matters anymore. Except my temp ban. 626046. I didn't know. Please understand. I love Avicus. I don't want a temp ban to stand in the way. I love Nebula, CTH, and CTW. I want to organize a scrimmage for my team. Otherwise, I can go to Cube Craft or Happy Hunger Games.
Please appeal your infraction under the Appeal an Infraction tab. This is not the place for it.
Excellence, I'm going to say the answer is: It's the end of the world and nothing matters anymore. Except my temp ban. 626046. I didn't know. Please understand. I love Avicus. I don't want a temp ban to stand in the way. I love Nebula, CTH, and CTW. I want to organize a scrimmage for my team. Otherwise, I can go to Cube Craft or Happy Hunger Games.
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