Question 1

Each birthday you extinguish candles equal to your age. After a lifetime total of 666 candles, how old were you at the last celebration?

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Question 2

On average, how many times must a 6-sided die be rolled until a 6 turns up twice in a row?

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Question 3

You have 5 indistinguishable snickerdoodle cookies and 7 indistinguishable chocolate chip cookies. How many ways can you arrange 6 of these cookies in a straight line?

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Question 4

You baked 5 snickerdoodle cookies (S) and 6 chocolate cookies (C). You will place exactly 6 cookies in a straight line. Cookies of the same type are indistinguishable. How many distinct lines can you make?

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Question 5

How many positive integers less than or equal to 10000 contain the digit 0 at least once?

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Question 6

A solid cube with dimensions 8x8x8 is composed of small 1x1x1 cubes. How many of these small cubes are exposed to the surface?

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Question 7

How many digits are in $ 99^{99} $?

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Question 8

How many four-digit integers exist such that each digit is unique? Assume the integers range from 1000 to 9999.

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Question 9

Nineteen clubs enter a league where every pair of clubs meets twice: once at each home ground.
How many distinct matches will the season contain?

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Question 10

Let $S_n = (2n)^2 - (2n-1)^2 + (2n-2)^2 - (2n-3)^2 + \dots + 2^2 - 1^2$ . Report the value of $S_{50}$ .

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Question 11

A bath can hold 572 L. The cold tap pours 16 L/min, the hot tap 11 L/min, while the unplugged drain removes 5 L/min.
If you open both taps but forget the plug, how long until the tub is finally full?

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Question 12

Evaluate the expression $\sqrt{12+\sqrt{12+\sqrt{12+\dots}}}$

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Question 13

Let $1 \leq r \leq 90$ be an odd integer. Evaluate $\displaystyle \sum_{k=0}^r \sin^2\left(\frac{k\pi}{2r}\right)$ . Report the answer when $r = 45$ .

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Question 14

Let $1 \leq r \leq 90$ be an even integer. Evaluate $\displaystyle \sum_{k=0}^r \sin^2\left(\frac{k\pi}{2r}\right)$ . Report the answer when $r = 30$ .

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Question 15

Let $r  
eq 1$ and $n \geq 1$ be fixed. Compute the value of $\displaystyle \sum_{k=1}^n kr^k$ . Evaluate when $r = 2$ and $n = 8$ .

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Question 16

Does there exist a non-negative integer $k  
eq 0,1$ such that $\binom{3500}{k} = \binom{3500}{k^2}$ ? If so, report the value of $k$ . Otherwise, report $-1$ .

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Question 17

Does there exist a non-negative integer $k  
eq 0,1$ such that $\binom{2070}{k} = \binom{2070}{k^2}$ ? If so, report the value of $k$ . Otherwise, report $-1$ .

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Question 18

Fix non-negative integers $0 \leq r \leq n$ . Evaluate the following sum: $$\sum_{i=r}^n \binom{i}{r}$$ Report the answer when $n =12$ and $r = 2$ .

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Question 19

You are asked to compute, without a calculator, the sum of all even integers from $1$ to $1000$. Find $2+4+\cdots+1000$ as quickly as possible.

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Question 20

Find the unique real value $x$ such that exactly one of $x - \sqrt{2}, x + 1/x, x-1/x,$ and $x^2 + 2\sqrt{2}$ is not an integer. Report the value of $x$ to the nearest thousandth.

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Question 21

Two trains are traveling on the same track towards each other. The first train moves at a speed of 50 km/h, and the second train moves at a speed of 70 km/h. When the distance between the two trains is 100 km, a bee starts flying back and forth between them. The bee flies from the first train to the second train, and upon reaching the second train, it immediately turns around and flies back to the first train. This pattern continues until the trains collide. Given that the bee's speed is 80 km/h, can you determine the total distance traveled by the bee before the trains collide?

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Question 22

There are several ducks and rabbits in a cage. In total, we observe $72$ heads and $200$ feet inside the cage. What is the product of the number of ducks and the number of rabbits in the cage?

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Question 23

At exactly 3:15 (quarter past three), what is the smaller angle between the hour hand and the minute hand of an analog clock?

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Question 24

You're given a large dataset containing daily returns of various stocks over several years. You suspect some noise in the data due to microstructural effects. How might you use Singular Value Decomposition (SVD) to isolate the main components driving these returns, separating them from noise?

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Question 25

You are on a straight bike road, from your house to the supermarket. You also take the same road back to your house.
- On Monday, there was no wind, and you had a constant speed of 20 km/h.
- On Tuesday, there is a 5 km/h wind from the supermarket to your home. Assume you're putting the same amount of effort to bike.
On Monday, the total bike ride took you 30 minutes. How many minutes will it take you on Tuesday?

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Question 26

Every year I receive a birthday cake with a number of candles equal to my age. If I have blown out 666 candles in my life (assuming I was already able to blow out a candle at the age of 1), how old am I?

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Question 27

There is a wide river, with a boat on each side. Both boats move with a constant speed towards the opposite side. Note: they don’t necessarily have the same constant speed. When a boat reaches the other side, it immediately turns around and heads back to its original side.
The first time the boats cross each other, they are 750 meters away from one shore. After they reach the opposite sides and turn back, the second time they cross, it's 250 meters away from the opposite shore.
Given this information, how wide is the river?

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Question 28

On a chessboard with a length of four and a width of five, a pawn starts at the top-left corner. It wants to get to the bottom-right corner, however, it's only allowed to move 1 cell to the right or 1 cell down. How many unique paths can the pawn take?

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Question 29

Imagine young Gauss is challenged to add every integer from 1 up to 20. Without jotting down partial sums, how can he find the total almost instantly?

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Question 30

You have 11 blocks, where nine of them are labeled 1 through 9 and the remaining two are indistinguishably labeled with 10. Compute the number of ways you can pick a set of three blocks such that at least one block is even.

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Question 31

You baked 6 indistinguishable coconut cookies and 8 indistinguishable chocolate cookies. In how many ways can you select and arrange 7 of these cookies in a straight line?

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Question 32

How many steps have I taken if I walk 343 steps plus half of the total number of steps?

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Question 33

What is the purpose of Cramer's rule?

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Question 34

Imagine you are in the mood for a coffee. You eagerly walk to your favorite coffee shop at a brisk pace of 6 km/h. Once you have your coffee in hand, you leisurely make your way back home at a relaxed speed of 4 km/h. Considering the entire trip, what would be your average speed?

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Question 35

You are building a risk model to incorporate the impact of a set of macroeconomic factors on a portfolio's returns. You are given the correlation matrix of these factors. How can you use the eigenvalues of this matrix to determine the stability and robustness of the model?

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Question 36

A bacteria population doubles in size every $10$ minutes. If at $t_1 = 55$ minutes the population consists of $200$ bacteria, how many bacteria will there be at $t_2 = 95$ minutes?

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Question 37

What is the significance of eigenvalues and eigenvectors in linear transformations, and how might they be relevant in a quantitative trading context?

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Question 38

Find the eigenvalues of a 3x3 matrix
A
, where
$$\begin{equation} A = \begin{bmatrix} 2&1&-2\1&0&0\0&1&0 \end{bmatrix} \end{equation}$$

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Question 39

There are two escalators, one moving up and the other moving down at the same constant speed. Two individuals start at the top of each escalator, both walking down at a pace of 1 step per second.
- The person walking down the downward-moving escalator reaches the bottom after taking 20 steps.
- The person walking down the upward-moving escalator takes 60 seconds to reach the bottom.
How many steps would each escalator have if they were stationary (i.e., paused)?

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Question 40

The cold tap in my bathtub pours water at a rate of 14 litres per minute and the hot tap pours at a rate of 9 litres per minute. The plug hole drains water out of the 572 litre bath at a rate of 12 litres per minute. If both taps are turned on but I forgot to put the plug in, how many minutes does it take for the bathtub to be completely full?

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Question 41

You finish a stroll and report: “I walked 512 steps plus half of my total steps.” How many steps did you actually take?

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Question 42

You have a cube shaped room with dimensions 10x10x10. There is fly and an ant at the center of the floor. If they both travel at the same speed, how much faster is the fly than the ant to reach a corner of the ceiling? Provide your answer as a percentage.

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Question 43

Elaborate on the differences between the Frobenius norm and the spectral norm of a matrix. How are they computed?

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Question 44

You are provided with a set of factors believed to drive asset returns in a portfolio. These factors are correlated. How might you use linear algebra techniques to derive a set of orthogonal (uncorrelated) factors?

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Question 45

In a football league, there are 20 teams that play each other twice each season. How many games are played each season in total?

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Question 46

Ten identical cars sit at mile 0, each with a tank capacity of 1 gallon, initially filled, and a fuel economy of 1 mile per gallon. Cars may drive, stop, and siphon fuel between each other without loss, but only while they occupy the same location.
With clever refuelling, what is the furthest mile marker that any single car can reach before its tank finally runs dry?

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Question 47

Some friends are walking to the beach. They call a taxi to return home, instead of walking back.
- The friends are walking at 4 miles per hour.
- The taxi travels on average at 12 miles per hour.
How far from the beach is their home if the total traveling time is 8 hours?

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Question 48

What is the purpose of the Gram-Schmidt process?

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Question 49

Suppose
$P = span\{	extbf{v}_1, 	extbf{v}_2,...,	extbf{v}_n\}$
is a subspace in
$\mathbb{R}^n$
. If
$\{	extbf{v}_1, 	extbf{v}_2,...,	extbf{v}_n\}$
is a dependent set, can the Gram-Schmidt process still find an orthogonal basis for
$P$
?

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Question 50

Arya (A), Jeff (B) and Clarice (C) all have different numbers of dogs.
They had the following conversation :
1. B to C: You have the most dogs.
2. A to B: I have exactly 30% more dogs than you.
3. A to C: the number of dogs you have is the average between the number of dogs I have and the number of dogs B has.
4. C to A : you have at least 4 more dogs than me.
However, not all of these statements are True!
- When speaking to someone with fewer dogs, the speaker always tells the truth.
- Otherwise when speaking to someone with more dogs, the speaker always lies.
How many dogs does Clarice (C) have?

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Question 51

If there are 27 people in a competition and I finish with exactly the same number of people above me as below me, in what place did I finish?

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Question 52

You walk into a barn and see a collection of spiders, chickens and cows. You notice there are 520 legs in total.
- The number of chickens is twice the number of cows,
- and the number of spiders is twice the number of chickens.
Compute the number of spiders.

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Question 53

If 128 players enter the knockout stage of a tennis tournament (for single players), how many matches will be played?

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Question 54

Imagine you are working with a large matrix similarity representing pairwise correlations between thousands of assets. You're trying to find assets that behave similarly. How can you use the concept of matrix similarity and orthogonal transformations to identify these asset clusters?

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Question 55

Suppose you are given a large correlation matrix representing the correlations among returns of various assets. How might you determine if there are any redundant assets, and what are the implications for portfolio construction?

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Question 56

Ginni needs 120 minutes to inflate a massive bouncy castle by herself, while Sheryl would puff away for 150 minutes on her own. They both start inflating a castle together. After 45 minutes of teamwork, Elon jumps in and starts to help inflate as well. With the three of them going at it, the bouncy castle is fully inflated 13 minutes after Elon started helping.
How many minutes would it take Elon to inflate a castle on his own?

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Question 57

How many distinct permutations of the word INTERVIEW are there?

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Question 58

How can you determine if a matrix is invertible?

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Question 59

Linear Diophantine Equations with Constraints are a type of mathematical problem where you are given a set of linear equations with integer coefficients, and you are asked to find unique integer solutions that satisfy the linear equations. These puzzles are a unique fusion of mathematical reasoning and logic. Your task is to deduce the distinct values of these variables based on the provided sum equations. These puzzles challenge your ability to analyze patterns, solve equations, and manage constraints simultaneously. Try to solve them as quickly as possible, because during an interview you will be timed as well!
For all questions, you have the constraint that A – E need to be unique integers in the range of 1 – 5.
1. B + A = 6
2. E + B = C
3. E + C + B = 8

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Question 60

Linear Diophantine Equations with Constraints are a type of mathematical problem where you are given a set of linear equations with integer coefficients, and you are asked to find unique integer solutions that satisfy the linear equations. These puzzles are a unique fusion of mathematical reasoning and logic. Your task is to deduce the distinct values of these variables based on the provided sum equations. These puzzles challenge your ability to analyze patterns, solve equations, and manage constraints simultaneously. Try to solve them as quickly as possible, because during an interview you will be timed as well!
For all questions, you have the constraint that A – E need to be unique integers in the range of 1 – 5.
1. A + D = C
2. E is not 5
3. D + E = B + C
4. D + 1 = C

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Question 61

Linear Diophantine Equations with Constraints are a type of mathematical problem where you are given a set of linear equations with integer coefficients, and you are asked to find unique integer solutions that satisfy the linear equations. These puzzles are a challenging combination of mathematical reasoning and logic. Your task is to deduce the distinct values of these variables based on the provided sum equations. These puzzles challenge your ability to analyze patterns, solve equations, and manage constraints simultaneously. Try to solve them as quickly as possible, because during an interview you will be timed as well!
For all questions, you have the constraint that A – E need to be unique integers in the range of 1 – 5.
1. D - 2 = C
2. E + C = D
3. A + E = C

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Question 62

Linear Diophantine Equations with Constraints are a type of mathematical problem where you are given a set of linear equations with integer coefficients, and you are asked to find unique integer solutions that satisfy the linear equations. These puzzles are a challenging combination of mathematical reasoning and logic. Your task is to deduce the distinct values of these variables based on the provided sum equations. These puzzles challenge your ability to analyze patterns, solve equations, and manage constraints simultaneously. Try to solve them as quickly as possible, because during an interview you will be timed as well!
For all questions, you have the constraint that A – E need to be unique integers in the range of 1 – 5.
1. C + D = B + E
2. B + C = A + E
3. A - D = E - 2

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Question 63

A fish has a head 8 cm long. The tail is equal to the size of the head plus half the size of the body. The body is the size of the head plus the tail. What is the total length of the fish?

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Question 64

Let x be a 5-digit integer in the form 37abc, where a, b and c are unknown integers, such that 37abc, 37bca and 37cab are all divisible by 37.
How many possible values of x exist?

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Quantitative Finance

Quantitative Finance

Quantitative Finance

Quantitative Finance