MathDB

Round 1
p1. A party is attended by ten people (men and women). Among them is Pat, who always lies to people of the opposite gender and tells the truth to people of the same gender. Pat tells five of the guests: “There are more men than women at the party.” Pat tells four of the guests: “There are more women than men at the party.” Is Pat a man or a woman?
p2. Once upon a time in a land far, far away there lived

100

knights,

99

princesses, and

101

dragons. Over time, knights beheaded dragons, dragons ate princesses, and princesses poisoned knights. But they always obeyed an ancient law that prohibits killing any creature who has killed an odd number of others. Eventually only one creature remained alive. Could it have been a knight? A dragon? A princess?
p3. The numbers

1 \circ 2 \circ 3 \circ 4 \circ 5 \circ 6 \circ 7 \circ 8 \circ 9 \circ 10

are written in a line. Alex and Vicky play a game, taking turns inserting either an addition or a multiplication symbol between adjacent numbers. The last player to place a symbol wins if the resulting expression is odd and loses if it is even. Alex moves first. Who wins? (Remember that multiplication is performed before addition.)
p4. A chess tournament had

8

participants. Each participant played each other participant once. The winner of a game got

1

point, the loser

0

points, and in the case of a draw each got

1/2

a point. Each participant scored a different number of points, and the person who got

2

nd prize scored the same number of points as the

5

th,

6

th,

7

th and

8

th place participants combined. Can you determine the result of the game between the

3

rd place player and the

5

th place player?
p5. One hundred gnomes sit in a circle. Each gnome gets a card with a number written on one side and a different number written on the other side. Prove that it is possible for all the gnomes to lay down their cards so that no two neighbors have the same numbers facing up.
Round 2
p6. A casino machine accepts tokens of

32

different colors, one at a time. For each color, the player can choose between two fixed rewards. Each reward is up to

\$10

cash, plus maybe another token. For example, a blue token always gives the player a choice of getting either

\$5

plus a red token or

\$3

plus a yellow token; a black token can always be exchanged either for

\$10

(but no token) or for a brown token (but no cash). A player may keep playing as long as he has a token. Rob and Bob each have one white token. Rob watches Bob play and win

\$500

. Prove that Rob can win at least

\$1000

. https://cdn.artofproblemsolving.com/attachments/6/6/e55614bae92233c9b2e7d66f5f425a18e6475a.png
p7. Each of the

100

residents of Pleasantville has at least

30

friends in town. A resident votes in the mayoral election only if one of her friends is a candidate. Prove that it is possible to nominate two candidates for mayor so that at least half of the residents will vote.

PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement