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2015 Math Hour Olympiad

Part of Math Hour Olympiad

Subcontests

(2)
8-10
1

2015 Math Hour Olympiad - University of Washington - Grades 8-10

Round 1
p1. Six pirates – Captain Jack and his five crewmen – sit in a circle to split a treasure of 9999 gold coins. Jack must decide how many coins to take for himself and how many to give each crewman (not necessarily the same number to each). The five crewmen will then vote on Jack's decision. Each is greedy and will vote “aye” only if he gets more coins than each of his two neighbors. If a majority vote “aye”, Jack's decision is accepted. Otherwise Jack is thrown overboard and gets nothing. What is the most coins Captain Jack can take for himself and survive?
p2. Rose and Bella take turns painting cells red and blue on an infinite piece of graph paper. On Rose's turn, she picks any blank cell and paints it red. Bella, on her turn, picks any blank cell and paints it blue. Bella wins if the paper has four blue cells arranged as corners of a square of any size with sides parallel to the grid lines. Rose goes first. Show that she cannot prevent Bella from winning. https://cdn.artofproblemsolving.com/attachments/d/6/722eaebed21a01fe43bdd0dedd56ab3faef1b5.png
p3. A 25×2525\times 25 checkerboard is cut along the gridlines into some number of smaller square boards. Show that the total length of the cuts is divisible by 44. For example, the cuts shown on the picture have total length 1616, which is divisible by 44. https://cdn.artofproblemsolving.com/attachments/c/1/e152130e48b804fe9db807ef4f5cd2cbad4947.png
p4. Each robot in the Martian Army is equipped with a battery that lasts some number of hours. For any two robots, one's battery lasts at least three times as long as the other's. A robot works until its battery is depleted, then recharges its battery until it is full, then goes back to work, and so on. A battery that lasts NN hours takes exactly NN hours to recharge. Prove that there will be a moment in time when all the robots are recharging (so you can invade the planet).
p5. A casino machine accepts tokens of 3232 different colors, one at a time. For each color, the player can choose between two fixed rewards. Each reward is up to $10\$10 cash, plus maybe another token. For example, a blue token always gives the player a choice of getting either $5\$5 plus a red token or $3\$3 plus a yellow token; a black token can always be exchanged either for $10\$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\$500. Prove that Rob can win at least $1000\$1000. https://cdn.artofproblemsolving.com/attachments/6/6/e55614bae92233c9b2e7d66f5f425a18e6475a.png

Round 2
p6. The sum of 20152015 rational numbers is an integer. The product of every pair of them is also an integer. Prove that they are all integers. (A rational number is one that can be written as m/nm/n, where mm and nn are integers and n≠0n\ne 0.)
p7. An N×NN \times N table is filled with integers such that numbers in cells that share a side differ by at most 11. Prove that there is some number that appears in the table at least NN times. For example, in the 5×55 \times 5 table below the numbers 11 and 22 appear at least 55 times. https://cdn.artofproblemsolving.com/attachments/3/8/fda513bcfbe6834d88fb8ca0bfcdb504d8b859.png
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
5-7
1

2015 Math Hour Olympiad - University of Washington - Grades 5-7

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 100100 knights, 9999 princesses, and 101101 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∘2∘3∘4∘5∘6∘7∘8∘9∘101 \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 88 participants. Each participant played each other participant once. The winner of a game got 11 point, the loser 00 points, and in the case of a draw each got 1/21/2 a point. Each participant scored a different number of points, and the person who got 22nd prize scored the same number of points as the 55th, 66th, 77th and 88th place participants combined. Can you determine the result of the game between the 33rd place player and the 55th 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 3232 different colors, one at a time. For each color, the player can choose between two fixed rewards. Each reward is up to $10\$10 cash, plus maybe another token. For example, a blue token always gives the player a choice of getting either $5\$5 plus a red token or $3\$3 plus a yellow token; a black token can always be exchanged either for $10\$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\$500. Prove that Rob can win at least $1000\$1000. https://cdn.artofproblemsolving.com/attachments/6/6/e55614bae92233c9b2e7d66f5f425a18e6475a.png
p7. Each of the 100100 residents of Pleasantville has at least 3030 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.