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2017-05-01 Intermediate

         

\[ \begin{array} { l l l } & A & B \\ & C & D \\ + & E & F \\ \hline & G & H \\ \end{array} \]

In the above cryptogram, all the letters represent distinct digits from 1 to 9.

What is the minimum possible value of \( \overline{GH} \)?


Inspiration.

For a greater challenge, try Cryptotastic #2.

\[ \begin{array} { l l l l l l l l } & A & B & C & D & E & 9 \\ \times & & & & & & 4 \\ \hline & 9 & A & B & C & D & E \\ \end{array}\]

I have a 6-digit positive integer with a last digit 9. If I move its last digit directly to the front without shifting the other digits, then this new 6-digit number is 4 times the original one.

What is the original 6-digit number?

Three regular heptagons with different side lengths form a right triangle, using one side of each. The smallest heptagon has area \(2017 \text{ cm}^2,\) while the largest one has area \(7102 \text{ cm}^2.\)

Find the area of the remaining heptagon in \(\text{cm}^2.\)

4 objects \(A, B, C, D\) are made up of the same material. They are released, simultaneously from rest, from the starting line on the top of a rough inclined plane, so that they roll without slipping. \(A\) and \(B\) are cylinders, and \(C\) and \(D\) are spheres.

What is the order relation of the times taken by the objects to cross the finish line?
(A, B, C, D in the answer options denote the times taken by corresponding objects.)

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Details and Assumptions:

  • The objects are released with all their centers above the same starting line, which is the top edge of the ramp. They are considered to have crossed the finish line--the red line near the bottom edge--only if their centers have crossed the line when viewed from the side.

  • Moment of inertia of a cylinder is \(\frac{1}{2} MR^2\) and that of a sphere is \(\frac{2}{5} MR^2\), where \(M\) is the mass and \(R\) is the radius of the respective objects.

  • \(B\) has twice the radius of \(A,\) and \(D\) has twice the radius of \(C\).

Consider concentric circles with respective radii \(\pi\) and \(2\pi\). Choose two points on the larger circle independently and uniformly at random, and then join them to form a chord. If \(P\) is the probability that this chord intersects the smaller circle at at least one point, then find \(\lfloor 1000P \rfloor\).

\(\)
Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.

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