misc.
How is Euler pronounced?
Oiler, not yuler
A card is drawn at random from a card deck. What is the probability that the card is a king?
A deck has 52 cards.
13 of each type (red hearts, black clubs, red diamonds, and black spades)
1/Ace, 2-10, Jack, Queen, King
Thus, there are 4 King cards, 1 for each card type.
Probability: 4/52 = 1/13
Prove that the area of any triangle ABC is equal to 0.5bcsinA.
Using 0.5 x base x height, where the base is one of A's sides and the height is the line perpendicular to A that touches a, and the definition of sine (where the height is equal to b x sin C), we get our desired result.
How many prime numbers are there below 100?
Note: Bonus Qs must be answered at the same time as the original question.
Bonus Q1 (200 pts): How many prime numbers are there below 1000?
Bonus Q2 (300 pts): State all the primes below 100
25
BQ1: 168
BQ2: Search it up lol
A looks at B, who looks at C. A has a 4.0 GPA, while C does not. Is a 4.0 GPA student looking at a non-4.0 GPA student?
Yes, because either A (a 4.0 GPA student) looks at B or B (if a 4.0 GPA student) looks at C, who is non-4.0.
How did Évariste Galois die?
Fought in a duel and died from the wounds.
How many combinations are there to a standard luggage lock?
Bonus: Give an estimation for the minimum amount of time it would take to find the correct configuration no matter the starting configuration.
-> We will award points based on the attention to detail used to make the estimation.
10^3 = 1000
What was Archimedes' famous method to approximate Pi?
A. Find a square of the same area of a circle and deduce an approximation for Pi
B. Calculate the perimeters of n-sided polygons inscribed and circumscribed in a circle and make n tend to infinity
C. Calculate the diameter of a circle fixing it's circumference at 1
D. Bake pies and calculate it's circumference
B. Calculate the perimeters of n-sided polygons inscribed and circumscribed in a circle and make n tend to infinity
Prove that every prime greater than 3 can be written in the form 6n ± 1.
A new government decides to transition into democracy, and this year, there are two candidates, A and B. You are tasked with surveying the citizens to predict the winner, and you will do so by going door to door and asking each voter who they prefer. However...
Because A is an ex-serial killer, no one is willing to admit that they support him, even if they do (e.g., if you ask a pro-A voter, they may lie and claim B).
You check your pockets, but all you got is a coin. What should you do to conduct the most accurate poll?
Use the coin to add randomness, as it gives the supporters of A a level of deniability (if they say A, it may be because of the coin, or because they truly support him, but done in a way where you know how skewed it has become).
Ask everyone to flip your coin in private. HEADS: State A, whether or not they actually support him. TAILS: State their true preference.
How many efficient solutions are there for the standard Rubix Cube?
-> group with the closest number will get the point
(Group who chose the question will get 2 guesses, everyone else will get 1)
Bonus 300 if the winning group can guess the minimum number of moves that can solve any configuration.
43,252,003,274,489,856,000 (over 43 quintillion) possible configurations
Bonus: 20 moves
Let set A be a 90-element subset of {1, 2, 3, ..., 100} and let S be the sum of the elements of A. Find the number of possible values of S.
901
arcsin(1/3) + arccos(1/3) + arctan(1/3) + arccot (1/3) = ?
pi radians/180 degrees
Derive the infinite geometric series formula a/(1-r)
Officers will check if your method is valid.
Example solution will be put up in a moment.
Which answer in this list is the correct answer to this question, and why?
1. All of the below
2. None of the below
3. All of the above
4. One of the above
5. None of the above
6. None of the above
If 1 is true, then 5 is true, which means that 1 is false. This is a contradiction, so 1 cannot be true. If 3 is true, then 1 must be true, which it isn't, so 3 cannot be true.
If 2 is true, then 4 must be false; but if 4 is false, then NONE of the above are true, which is contradictory, so 2 cannot be true. If 4 is true, then 1, 2, or 3 is true, but we've concluded that they're all false, so 4 cannot be true.
If 6 is false, then 5 has to be false, implying that one of the others are true, which since we've proven they're not, 6 cannot be true.
Therefore, answer 5 is the correct answer.
Why is a mug a donut?
In topology, we only care about shape in terms of holes, not exact form.
If you can stretch, squish, or bend one shape into another without cutting or gluing, they are the same in topology.
You have three water jugs that measure 3 L, 7 L, and 10 L. Find the minimum number of steps to precisely measure out 5L using ONLY the three jugs.
9 steps.
Fill 10L jug: (0,0,10)
Pour 10 into 7: (0,7,3)
Pour 7 into 3: (3,4,3)
Pour 3 into 10: (0,4,6)
Pour 7 into 3: (3,1,6)
Pour 3 into 10: (0,1,9)
Pour 7 into 3: (1,0,9)
Pour 10 into 7: (1,7,2)
Pour 7 into 3: (3,5,2)
Recite the first 20 digits of Pi
3.14159265358979323846
Prove that if x is a natural number in the form 4666...69, then x is divisible by 7.
Which of the following, if any, are false?
1. In this list, at least 1 statement is false.
2. In this list, at least 2 statements are false.
3. In this list, at least 3 statements are false.
4. In this list, at least 4 statements are false.
5. In this list, at least 5 statements are false.
6. In this list, at least 6 statements are false.
7. In this list, at least 7 statements are false.
8. In this list, at least 8 statements are false.
9. In this list, at least 9 statements are false.
10. In this list, at least 10 statements are false.
If 10 is true, all statements, including 10, are false, which is contradictory. Therefore, statement 10 is false. As at least one statement is false, statement 1 is true.
Similarly, if 9 is true, then 2-9 must be false (as one is true), which is self-contradictory, so statement 2 is true. Repeat this reasoning for statements 3-8 to find that the first 5 statements are true, and the last 5 are false.
Let {c1, ..., ck} be a non-empty and finite set of colours. A partially coloured directed graph is a structure <N, R, C> where
• N is a non-empty set of nodes
• R is a binary relation on N
• C associates colours to nodes (not all the nodes are necessarily coloured, and each node has at most one colour)
Provide a first order language and a set of axioms that formalize partially coloured graphs. Show that every model of this theory corresponds to a partially coloured graph, and vice-versa. For each of the following properties, write a formula which is true in all and only the graphs that satisfies the property:
1. connected nodes do not have the same colour
2. the graph contains only 2 yellow nodes
3. starting from a red node one can reach in at most 4 steps a green node
4. for each colour there is at least a node with this colour
5. the graph is composed of |C| disjoint non-empty subgraphs, one for each colour
Language:
• a binary predicate edge, where edge(n, m) means that node n is connected to node m
• a binary predicate color, where color(n, x) means that node n has color x
• the following constants: yellow, green, red (or any three distinct colours)
Axioms:
https://docs.google.com/document/d/1I-92mcOrca3UJ2QsYAFbgyki1kn_iHaQgV86H_oSo40/edit?usp=sharing
The King of a small country invites 1000 senators to his annual party. As a tradition, each senator brings the King a bottle of wine. Soon after, the Queen discovers that one of the senators is trying to assassinate the King by giving him a bottle of poisoned wine. Unfortunately, they do not know which senator, nor which bottle of wine is poisoned, and the poison is completely indiscernible. However, the King has 10 prisoners he plans to execute. He decides to use them as taste testers to determine which bottle of wine contains the poison. The poison when taken has no effect on the prisoner until exactly 24 hours later when the infected prisoner suddenly dies. The King needs to determine which bottle of wine is poisoned by tomorrow so that the festivities can continue as planned. Hence he only has time for one round of testing. How can the King administer the wine to the prisoners to ensure that 24 hours from now he is guaranteed to have found the poisoned wine bottle?
We have 1000 bottles of wine, one of which is poisoned and somehow we need to test all of the wine bottles using only 10 prisoners as taste testers. However we decide to administer the wine to the prisoners, we need to use the prisoners deaths as a code to trace back to the poisoned wine bottle.
Since we have only 24 hours to test the wine, we know that there is not enough time nor enough prisoners to test the wine one-by-one.
To identify the poisoned bottle we must use the binary number system. Since log2 1000≈10, we need 10 bits to uniquely represent each bottle from 1 to 1000 in binary.
Given a right-angled triangle where the side lengths of the triangle are integers, determine all the possible side lengths of the triangle if the product of the legs* of the right triangle equals to three times the perimeter of the triangle.
*the sides that form the right angle of the triangle
(7, 24, 25), (8, 15, 17), and (9, 12, 15)
Prove by contradiction that there are infinitely many primes.
Assume by way of contradiction that there are only a finite number of primes: p1 < p2 < ··· < pm. Consider the number P = p1 x p2 x ··· x pm + 1.
If P is a prime, then P > pm, contradicting the maximality of pm. Hence P is composite, and consequently, it has a prime divisor p > 1, which is one of the primes p1, p2, ..., pm. Let that prime be pk. It follows that pk divides P.
This, together with the fact that pk divides P - 1, implies pk divides 1, a contradiction.
You (A) and 4 other pirates (B, C, D, E) have just struck gold, finding a chest with 100 gold bars. Starting with the captain (you), one pirate proposes a distribution, and all pirates vote on it. If at least half vote in favor, the proposal is accepted; otherwise, the proposer is thrown overboard, and the next in line makes the offer.
Everyone is perfectly rational, prioritizing both survival and the most amount of money, considering all outcomes.
What should your proposal be to guarantee survival AND the most amount of gold bars?
98, 0, 1, 0, 1
(correct answer must include the correct logical explanation)