# Levels and Ratings on Brilliant

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\(x\) is a real number satisfying \(|x^2-5| = 20\) . What is the sum of all possible values of \(x\)?

**Details and assumptions**

Note: \(|x|\) is the absolute value of \(x\). For example, \(|3| = 3\) and \(|-3| = 3\).

If \(x\) is an integer where \(14 < x < 43\), how many possible values are there for \(x\)?

As \(x\) ranges over all real values, what is the minimum value of \( f(x)=|x-245|+|x-576| + |x-794| \)?

**Details and assumptions**

The notation \( | \cdot | \) denotes the absolute value. The function is given by \[ |x | = \begin{cases} x & x \geq 0 \\ -x & x < 0 \\ \end{cases} \] For example, \( |3| = 3, |-2| = 2 \).

Neutrinos are interesting little fundamental particles. These particles interact very weakly with everything else, so they essentially pass right through everyday stuff. In fact there are tremendous numbers of neutrinos passing through your body at this very second! In 2011, two experiments in Italy measured the time it took for neutrinos generated at a particle accelerator at CERN in Geneva, Switzerland to get to a pair of detectors built into the Gran Sasso mountain in central Italy. One of them mistakenly measured the speed as faster than the speed of light, which caused a furor.

The straight line distance through the earth from \(x_0\), the point of production of the neutrinos at CERN, to \(x_1\), the point of measurement at Gran Sasso was roughly \(732~\mbox{km}\). Now let \(D\) be the shortest distance between \(x_0\) and \(x_1\) over the surface of the earth, which is the path we'd have to take if we wanted to get from CERN to Gran Sasso without heavy machinery.

How much longer is \(D\) than the neutrino path, i.e. what is \(D-732~\mbox{km}\) in **meters**?

**Details and assumptions**

You may assume that \(x_0\) and \(x_1\) are on the surface of the earth and that the earth is a perfect sphere of radius \(6,370~\mbox{km}\).

In chemistry and physics courses we learn that atoms are electrically neutral because the number of electrons equals the number of protons (atomic number \(Z\)) in an atom. In addition, we know that the charges of the proton and electron are equal in magnitude. As a result, macroscopic objects do not experience considerable electrostatic interactions. Have you ever considered what would happen if the magnitudes of the charges of the electron and proton were slightly different? Suppose that the charge of the proton differs in \(1\%\) from the charge of the electron. That is, \[Q(\mbox{proton})=|Q(\mbox{electron})|(1+1/100).\] Therefore the net electric charge on an atom with atomic number \(Z\) is \(Q_{Net}=Q(\mbox{proton})-|Q(\mbox{electron})|=0.01Z|Q(\mbox{electron})|\).

In this case, what would be the magnitude of the force of interaction **in Newtons** between two small 1-gram copper balls separated by one meter? There are \(N=9.5 \times 10^{21} \) atoms in one gram of copper and its atomic number is \(Z=29\). Hint: the answer's big... real big.

**Details and assumptions**

- \(|Q(\text{electron})|=e=1.6 \times 10^{-19}~\mbox{C}\)
- \(k=9\times 10^{9}~\mbox{N m}^{2}/\mbox{C}^{2} \)
- Since the balls are small and spherical you may treat them as point charges and apply Coulomb's law.

A cube is inscribed within the sphere such that all the vertices of the cube touch the sphere. The volume of the cube is equal to \(2 \times 96\sqrt{3}\), what is the radius of the sphere?

\(ABC\) is an isosceles triangle with \(AB = BC\). The circumcircle of \(ABC\) has radius \(8\). Given that \(AC\) is a diameter of the circumcircle, what is the area of triangle \(ABC\)?

**Details and assumptions**

The **circumcircle** of a triangle \(ABC\) passes through \(A, B, C\).

Circle \(\Gamma\) with center \(O\) is tangential to \(PA\) at \(A\). Line \(PO\) extended intersects \(\Gamma\) at \(B\) (i.e. \(PO < PB\)). \(C\) is a point on \(AB\) such that \(PC\) bisects \( \angle APB\). What is the measure (in degrees) of \( \angle PCA\)?

Eight reindeer, each with mass of \(400~\mbox{kg}\), are attached to Santa's sleigh in four rows of two reindeer each. The last two reindeer, Donner and Blitzen, are attached to the reindeer in front of them and Santa's sleigh, with total mass \(1200~\mbox{kg}\), behind them by their harnesses. On liftoff on Christmas Eve, the reindeer accelerate straight up from the North Pole at \(5~\mbox{m/s}^2\). What is the force **in Newtons** the harness exerts on Santa's sleigh during sleigh liftoff?

**Details and assumptions**

- The acceleration of gravity is \(-9.8~\mbox{m/s}^2\).

A 60-Watt light bulb is connected to a 120 V outlet. How long **in seconds** does it take for 1 billion (\(10^{9}\)) electrons to pass through the light bulb?

**Details and assumptions**

- The charge of the electron is \(e=1.6\times 10^{-19}~\mbox{C}\).
- You may consider the voltage as a constant voltage source. The time you will find is short enough that the fact that wall outlets generally deliver alternating voltages can be ignored.

How many positive divisors does 2016 have?

**Details and assumptions**

1 and 2016 are considered divisors of 2016.

How many 3 digit positive integers \(N\) are there, such that \(N\) is a multiple of both 7 and 13?

What is the smallest positive integer with exactly 12 (positive) divisors?

You may choose to read the summary page on Divisors of an Integer.

Arteriosclerotic plaques forming on the inner walls of arteries can decrease the effective cross-sectional area of an artery. Changes in the effective area of an artery can lead to large changes in the blood pressure in the artery and possibly to the collapse of the blood vessel.

Consider an artery that is horizontal along some stretch. In the healthy portion of the artery the blood flow velocity is \(0.14~\mbox{m/s}\). If plaque closes off \(80\%\) of the area of the artery at one point, what is the magnitude of the pressure drop **in Pascals**?

**Details and assumptions**

- The mass per unit volume of blood is \(\rho= 1050~\mbox{kg/m}^3\).
- You may neglect the viscosity of blood.

The dipole moment for a system of charges is defined as
\[ \vec{d}=\sum_{k} q_{k} \vec{r}_{k}.\] (see here for a more in depth discussion of the dipole moment). Determine the magnitude of the dipole moment **in \(\mbox{m} \cdot \mbox{C} \)** for a system of three charges (\(Q=1~\mu \mbox{C}\), \(Q=1~\mu \mbox{C}\), and \(-2Q=-2~\mu \mbox{C}\)) located on the vertices of an equilateral triangle of side \(a=1~\mu \mbox{m}. \) One can show that for an electrically neutral system, the dipole moment is independent of the origin of coordinates (which makes this problem unambiguous).

Singapore has dollar notes in denominations of \(1\), \(2\) and \(5\). How many ways are there to form exactly \($100\) using just multiples of these notes?

**Details and assumptions**

The question does NOT state that you must use every denomination. Using 20 $5 notes (and 0 $1 and 0 $2 notes) to form exactly $100 is a valid way.

There are 21 people in an orientation group. Each person shakes hands with every other person exactly once. How many handshakes are there in total?

Consider a \(3 \times 3\) square where each \(1 \times 1\) square is filled with one of the integers \(1, 2, 3\) or \(4\). A square is called mystical if each row and column sum up to a multiple of 4. How many mystical squares are there?

**Details and assumptions**

Rotations and reflections of the square are considered distinct solutions.

In Huckleberry Finn, a classic American novel by Mark Twain, much of the story takes place on a raft on the Mississippi river. Huckleberry's raft is a square 5 meters on a side with a vertical height of 0.1 m. If Huckleberry isn't on the raft it floats such that the top of the raft remains 0.07 m above the surface of the water. Huckleberry and his friend Jim, with a combined mass of 130 kg, climb on the raft. How far above the surface of the water **in meters** is the top of the raft after Huckleberry and Jim climb on?

Oh, and the Mississippi river is primarily fresh water...

**Details and assumptions**

- Treat the raft as just a rectangular prism/cuboid. You can ignore for the purposes of the problem that it's made of logs.

**in Volts**across the tips of the wings.

**Details and assumptions**

- Although wings have a complicated shape as in the figure, for the purposes of this problem you may model the wings as a straight conducting rod of length \(L=60m \) if you wish.

For how many integer values of \(i\), \(1 \leq i \leq 1000 \), does there exist an integer \(j \), \( 1 \leq j \leq 1000 \), such that \( i \) is a divisor of \( 2^j - 1 \)?

How many ordered triples of pairwise distinct, positive integers \( ( a, b, c) \) are there such that \( abc = 10^6 \)?

**Details and assumptions**

A set of values is called **pairwise distinct** if no two of them are equal. For example, the set \( \{1, 2, 2\} \) is not pairwise distinct, because the last 2 values are the same.

Given positive real values \( j, k\) and \(l\) such that \[ \begin{array}{l l} j^2 +k^2 + jk & = & 9 \\ k^2 + l^2 + kl & = & 16\\ l^2 + j^2 + lj & = & 25,\\ \end{array} \] \( (j + k + l)^2 \) has the form \( a + b \sqrt{c} \), where \(a, b\) and \(c\) are integers and \( c \) is not a multiple of any square number. What is \( a + b + c\)?

In a region of space are a bunch of stars in circular orbits at different radii and some unknown amount of non-visible matter. You observe the speeds of the stars and notice that the tangential velocity of each star is a constant \(v_0\), independent of radius. If the non-visible matter is distributed spherically symmetrically around the center of the galaxy, what is the ratio of the density of the non-visible matter at some distance 10R to the density at some distance R?

**Details and assumptions**

- You may assume the total mass of the stars is negligible compared to the mass of the non-visible matter.

A charged particle of mass m and charge q moves in an electrostatic field created by an unknown system of static charges. The particle starts from point A with initial velocity is \(v_{1}=1 m/s\) and follows certain trajectory. What should be the velocity (magnitude) \(v_{2}\) **in m/s** of a particle of mass \(M=10 m\) and charge \(Q=7 q\) at point A in order to follow the same trajectory?

Hint: Compute \[\frac{d}{ds} \frac{\vec{v}}{v},\] where s is the arc length along the trajectory:

\(s=\int_0^t v(t') dt'\).

### What are levels?

Levels reflect your problem solving ability in a topic (like "Algebra" or "Number Theory"). The level that you are in determines the difficulty of "popular" problems that appear in your home page from the topics you follow.

### What are ratings?

Ratings are a much more precise measure of your abilities. Only subscribers to
Brilliant^{2}
have access to their personal ratings. Ratings provide data to help you improve and level up.