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2017-03-06 Intermediate


Is 12345554321 {\color{#E81990}1}{\color{grey}{2}}{\color{#20A900}{3}}{\color{#3D99F6}{4}}{\color{#D61F06}555}{\color{#3D99F6}{4}}{\color{#20A900}{3}}{\color{grey}{2}}{\color{#E81990}1} prime?

Hint: You may use the fact that 111111×111111=12345654321 111111 \times 111111 = {\color{#E81990}1}{\color{grey}{2}}{\color{#20A900}{3}}{\color{#3D99F6}{4}}{\color{#D61F06}5}6{\color{#D61F06}5}{\color{#3D99F6}{4}}{\color{#20A900}{3}}{\color{grey}{2}}{\color{#E81990}1} . This is similar to, but not identical to the above number, due to the middle 6 6 .

A vending machine is designed to return change in the following way:

Keep returning the coin with the largest denomination smaller than or equal to the remaining amount until all of the change has been returned.

However, this approach doesn't always yield the least number of coins. With the denominations 10, 40, and 50 cents, which of the following changes (in cents) could be returned using fewer coins than what the algorithm requires?

As an explicit example of the algorithm, when required to produce a 110 cents change, the vending machine would return the coins 50, 50, 10 cents, in that order.

Drake is driving a bike on a straight road to his college at a speed of vv. He realizes that he might be late for his exam.

To rush, he accelerates his bike at a constant acceleration of aa for a distance of L1L_1. Then, for the remaining distance L2L_2, he slows down at a constant retardation of aa to just reach his college.

Given that L2=kL1,L_2 = k L_1, what is the maximum velocity attained during his travel?

In the diagram, AC=1AC=1, where CC is the midpoint of side ABAB of triangle ABJABJ. Square BCDEBCDE is drawn, where D D lies on side AJAJ and EE is in the interior of the triangle. Square FGIHFGIH is drawn, where G G lies on side BJ BJ, HH lies on side AJAJ, II is in the interior of the square, and E E is the midpoint of FGFG.

Find the area of triangle ABJABJ .

A bowl contains NN strands of noodles. You reach into the bowl and grab two free ends at random and attach them. You do this NN times until there are no free ends left.

What is the minimum NN such that the expected number of loops generated by the NN steps described above exceeds 3?


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