Fryer Contest 2014年真题

1.

(a) The positive integers from 1 to 99 are written in order next to each other to form the integer 123456789101112 . . . 9899. How many digits does this integer have?

(b) The positive integers from 1 to 199 are written in order next to each other to form the integer 123456789101112 . . . 198199. How many digits does this integer have?

(c) The positive integers from 1 to n are written in order next to each other. If the resulting integer has 1155 digits, determine n.

(d) The positive integers from 1 to 1000 are written in order next to each other. Determine the 1358th digit of the resulting integer.

2.

(a) In ∆ABC, ∠ABC = 60° and ∠ACB = 50°.

What is the measure of ∠BAC?

(b) An angle bisector is a line segment that

divides an angle into two equal angles.

In ∆ABC, ∠ABC = 60° and ∠ACB = 50°.

If BD and CD are angle bisectors of ∠ABC

and ∠ACB, respectively, what is the measure

of ∠BDC?

(c) Point S is inside ∆PQR so that QS and RS are angle bisectors of ∠P QR and ∠P RQ,

respectively, with QS = RS. If

∠QSR = 140° , determine with

justification, the measure of ∠QPR.

(d) In ∆PQR, QS and RS are angle bisectors of ∠PQR and ∠PRQ, respectively, with QS = RS (as in part (c)). Explain why it is not possible that ∠QSR = 80°. 

3.       Triangle ABC begins with vertices A(6, 9),

B(0, 0), C(10, 0), as shown. Two players

play a game using ∆ABC. On each turn a

player can move vertex A one unit, either

to the left or down. The x- and y-coordinates of A cannot be made negative. The person who makes the area of ∆ABC equal to 25 wins the game.

(a) What is the area of 4ABC before the first move in the game is made?

(b) Dexter and Ella play the game. After several moves have been made, vertex A is at (2, 7). It is now Dexter’s turn to move. Explain how Ella can always win the game from this point.

(c) Faisal and Geoff play the game, with Faisal always going first. There is a winning strategy for one of these players; that is, by following the rules in a certain way, he can win the game every time no matter how the other player plays.

(i) Which one of the two players has a winning strategy?

(ii) Describe a winning strategy for this player.

(iii) Justify why this winning strategy described in (ii) always results in a win.

4. The set A = {1, 2} has exactly four subsets: {}, {1}, {2}, and {1, 2}. The four subset sums of A are 0, 1, 2 and 3 respectively. The sum of the subset sums of A is 0 + 1 + 2 + 3 = 6. Note that {} is the empty set and {1, 2} is the same as {2, 1}.

(a) The set {1, 2, 3} has exactly eight subsets and therefore it has eight subset sums. List all eight subset sums of {1, 2, 3}.

(b) Determine, with justification, the sum of all of the subset sums of {1, 2, 3, 4, 5}.

(c) Determine, with justification, the sum of all of the subset sums of {1, 3, 4, 5, 7, 8, 12, 16} that are divisible by 4.