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As xx ranges over all real values, what is the minimum value of f(x)=x245+x576+x794 f(x)=|x-245|+|x-576| + |x-794| ?

Details and assumptions

The notation | \cdot | denotes the absolute value. The function is given by x={xx0xx<0 |x | = \begin{cases} x & x \geq 0 \\ -x & x < 0 \\ \end{cases} For example, 3=3,2=2 |3| = 3, |-2| = 2 .

If xx is an integer where 14<x<4314 < x < 43, how many possible values are there for xx?

xx is a real number satisfying x25=20|x^2-5| = 20 . What is the sum of all possible values of xx?

Details and assumptions

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

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 x0x_0, the point of production of the neutrinos at CERN, to x1x_1, the point of measurement at Gran Sasso was roughly 732 km732~\mbox{km}. Now let DD be the shortest distance between x0x_0 and x1x_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 DD than the neutrino path, i.e. what is D732 kmD-732~\mbox{km} in meters?

Details and assumptions

You may assume that x0x_0 and x1x_1 are on the surface of the earth and that the earth is a perfect sphere of radius 6,370 km6,370~\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 ZZ) 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%1\% from the charge of the electron. That is, Q(proton)=Q(electron)(1+1/100).Q(\mbox{proton})=|Q(\mbox{electron})|(1+1/100). Therefore the net electric charge on an atom with atomic number ZZ is QNet=Q(proton)Q(electron)=0.01ZQ(electron)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=9.5×1021N=9.5 \times 10^{21} atoms in one gram of copper and its atomic number is Z=29Z=29. Hint: the answer's big... real big.

Details and assumptions

  • Q(electron)=e=1.6×1019 C|Q(\text{electron})|=e=1.6 \times 10^{-19}~\mbox{C}
  • k=9×109 N m2/C2k=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.

Circle Γ\Gamma with center OO is tangential to PAPA at AA. Line POPO extended intersects Γ\Gamma at BB (i.e. PO<PBPO < PB). CC is a point on ABAB such that PCPC bisects APB \angle APB. What is the measure (in degrees) of PCA \angle PCA?

A cube is inscribed within a sphere such that all the vertices of the cube touch the sphere. If the volume of the cube is equal to 2×9632 \times 96\sqrt{3}, what is the radius of the sphere?

ABCABC is an isosceles triangle with AB=BCAB = BC. The circumcircle of ABCABC has radius 88. Given that ACAC is a diameter of the circumcircle, what is the area of triangle ABCABC?

Details and assumptions

The circumcircle of a triangle ABCABC passes through A,B,CA, B, C.

A 60-Watt light bulb is connected to a 120 V outlet. How long in seconds does it take for 1 billion (10910^{9}) electrons to pass through the light bulb?

Details and assumptions

  • The charge of the electron is e=1.6×1019 Ce=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.

Eight reindeer, each with mass of 400 kg400~\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 kg1200~\mbox{kg}, behind them by their harnesses. On liftoff on Christmas Eve, the reindeer accelerate straight up from the North Pole at 5 m/s25~\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 m/s2-9.8~\mbox{m/s}^2.

How many positive divisors does 2016 have?

Details and assumptions

1 and 2016 are considered divisors of 2016.

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

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

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

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 m/s0.14~\mbox{m/s}. If plaque closes off 80%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 ρ=1050 kg/m3\rho= 1050~\mbox{kg/m}^3.
  • You may neglect the viscosity of blood.

The dipole moment for a system of charges is defined as d=kqkrk. \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 mC\mbox{m} \cdot \mbox{C} for a system of three charges (Q=1 μCQ=1~\mu \mbox{C}, Q=1 μCQ=1~\mu \mbox{C}, and 2Q=2 μC-2Q=-2~\mu \mbox{C}) located on the vertices of an equilateral triangle of side a=1 μm.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).

Consider a 3×33 \times 3 square where each 1×11 \times 1 square is filled with one of the integers 1,2,31, 2, 3 or 44. 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.

Singapore has dollar notes in denominations of 11, 22 and 55. How many ways are there to form exactly $100\$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?

A Boeing 777 is flying due north at v=900km/h v=900km/h (typical cruising speed) and at an altitude where the vertical component of the Earth's magnetic field is B=50μTB=50\mu T. The wingspan of the 777 is about 60m. Determine the induced voltage 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=60mL=60m if you wish.

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.

Given positive real values j,k j, k and ll such that j2+k2+jk=9k2+l2+kl=16l2+j2+lj=25, \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 (j + k + l)^2 has the form a+bc a + b \sqrt{c} , where a,ba, b and cc are integers and c c is not a multiple of any square number. What is a+b+c a + b + c?

How many ordered triples of pairwise distinct, positive integers (a,b,c) ( a, b, c) are there such that abc=106 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} \{1, 2, 2\} is not pairwise distinct, because the last 2 values are the same.

For how many integer values of ii, 1i10001 \leq i \leq 1000 , does there exist an integer jj , 1j1000 1 \leq j \leq 1000 , such that i i is a divisor of 2j1 2^j - 1 ?

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 v0v_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 v1=1m/sv_{1}=1 m/s and follows certain trajectory. What should be the velocity (magnitude) v2v_{2} in m/s of a particle of mass M=10mM=10 m and charge Q=7qQ=7 q at point A in order to follow the same trajectory?

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

s=0tv(t)dts=\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.

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