Fryer Contest 2017年真题

1. A store sells packages of red pens and packages of blue pens. Red pens are sold only in packages of 6 pens. Blue pens are sold only in packages of 9 pens.

(a) Igor bought 5 packages of red pens and 3 packages of blue pens. How many pens did he buy altogether?

(b) Robin bought 369 pens. She bought 21 packages of red pens. How many packages of blue pens did she buy?

2. By finding a common denominator, we see that is greater than because . Similarly, we see that is less than

because .

(a) Determine the integer n so that is greater than and less than .

(b) Determine all possible integers m so that is greater than

and is less than .

(c) Fiona calculates her win ratio by dividing the number of games that she has won by the total number of games that she has played. At the start of a weekend, Fiona has played 30 games, has w wins, and her win ratio is greater than 0.5.

During the weekend, she plays fifive games and wins three of these games. At the end of the weekend, Fiona’s win ratio is less than 0.7. Determine all possible values of w.

3. When two chords intersect each other inside

a circle, the products of the lengths of their

segments are equal. That is, when chords P Q

and RS intersect at X, (P X)(QX) = (RX)(SX).

(a) In Figure A below, chords DE and F G intersect at X so that EX = 8, F X = 6, and GX = 4. What is the length of DX?

(b) In Figure B, chords JK and LM intersect at X so that JX = 8y, KX = 10, LX = 16, and MX = y + 9. Determine the value of y.

(c) In Figure C, chord ST intersects chords P Q and P R at U and V, respectively, so that P U = m, QU = 5, RV = 8, SU = 3, UV = P V = n, and T V = 6. Determine the values of m and n.

4. Three students sit around a table. Each student has some number of candies. They share their candies using the following procedure:

• Step 1: Each student with an odd number of candies discards one candy. Students with an even number of candies do nothing.

• Step 2: Each student passes half of the candies that they had after Step 1 clockwise to the person beside them.

• Step 1 and Step 2 are repeated until each of the three students has an equal number of candies. The procedure then ends.

On Monday, Dave, Yona and Tam start with 3, 7 and 10 candies, respectively. After Step 1 and Step 2, the number of candies that each student has is given in the following table:

(a) When the procedure in the example above is completed, how many candies does each student have when the procedure ends?

(b) On Tuesday, Dave starts with 16 candies. Each of Yona and Tam starts with zero candies. How many candies does each student have when the procedure ends?

(c) On Wednesday, Dave starts with 2n candies. Each of Yona and Tam starts with 2n+ 3 candies. Determine, with justification, the number of candies in terms of n that each student has when the procedure ends.

(d) On Thursday, Dave starts with 22017 candies. Each of Yona and Tam starts with zero candies. Determine, with justification, the number of candies that each student has when the procedure ends.