# Problems of the Week

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# 2018-12-10 Intermediate

Ana tosses a fair coin 50 times. Bastian tosses another fair coin 51 times.

What's the probability that Bastian gets more heads than Ana does? I'm looking for a positive integer $x$ with $n$ positive factors such that the number of positive factors in $x^2$ is $n^2.$

In the system below, the coefficient of friction between all surfaces is the same and equal to 1. When you apply force $F$ on the green body, the three bodies move with different accelerations with respect to the ground.

What is the acceleration of the blue body? $p,q,r,s$ are the 4 roots of the equation

$x^4 - ax^3 +ax^2 + bx +c =0.$

What is the minimum possible value of $p^2+q^2+r^2+s^2?$

Note:

• $p,q,r,s$ are not necessarily real.
• $a,b,c$ are real coefficients.

In the figure below, there are 6 squares: 4 small, 1 medium (in blue), and 1 large. There are 9 circles, which will be filled with all the integers 1 through 9 such that the sum of the vertices of each square is 20.

What is the product of the 4 numbers along the blue square? ×