MathDB

15

Part of 2000 AIME Problems

Problems(2)

Stack of 2000 Cards

Source:

1/5/2007
A stack of 20002000 cards is labelled with the integers from 11 to 2000,2000, with different integers on different cards. The cards in the stack are not in numerical order. The top card is removed from the stack and placed on the table, and the next card is moved to the bottom of the stack. The new top card is removed from the stack and placed on the table, to the right of the card already there, and the next card in the stack is moved to the bottom of the stack. The process - placing the top card to the right of the cards already on the table and moving the next card in the stack to the bottom of the stack - is repeated until all cards are on the table. It is found that, reading from left to right, the labels on the cards are now in ascending order: 1,2,3,…,1999,2000.1,2,3,\ldots,1999,2000. In the original stack of cards, how many cards were above the card labelled 1999?
Trig Summation

Source: AIME 2 2000 #15

12/9/2005
Find the least positive integer nn such that 1sin⁡45∘sin⁡46∘+1sin⁡47∘sin⁡48∘+⋯+1sin⁡133∘sin⁡134∘=1sin⁡n∘. \frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.
trigonometryinductionAMCAIMEMATHCOUNTSfunction