MathDB

2010 CHMMC Winter

Part of CHMMC problems

Subcontests

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Individual
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2010 Winter CHMMC Individual - Caltech Harvey Mudd Mathematics Competition

p1. Compute the degree of the least common multiple of the polynomials x−1x - 1, x2−1x^2 - 1, x3−1x^3 - 1,......, x10−1x^{10} -1.
p2. A line in the xyxy plane is called wholesome if its equation is y=mx+by = mx+b where mm is rational and bb is an integer. Given a point with integer coordinates (x,y)(x,y) on a wholesome line ℓ\ell, let rr be the remainder when xx is divided by 77, and let ss be the remainder when y is divided by 77. The pair (r,s)(r, s) is called an ingredient of the line ℓ\ell. The (unordered) set of all possible ingredients of a wholesome line ℓ\ell is called the recipe of ℓ\ell. Compute the number of possible recipes of wholesome lines.
p3. Let τ(n)\tau (n) be the number of distinct positive divisors of nn. Compute ∑d∣15015τ(d)\sum_{d|15015} \tau (d), that is, the sum of τ(d)\tau (d) for all dd such that dd divides 1501515015.
p4. Suppose 2202010b−22020103=71813265102202010_b - 2202010_3 = 71813265_{10}. Compute bb. (nbn_b denotes the number nn written in base bb.)
p5. Let x=(3−5)/2x = (3 -\sqrt5)/2. Compute the exact value of x8+1/x8x^8 + 1/x^8.
p6. Compute the largest integer that has the same number of digits when written in base 55 and when written in base 77. Express your answer in base 1010.
p7. Three circles with integer radii aa, bb, cc are mutually externally tangent, with a≤b≤ca \le b \le c and a<10a < 10. The centers of the three circles form a right triangle. Compute the number of possible ordered triples (a,b,c)(a, b, c).
p8. Six friends are playing informal games of soccer. For each game, they split themselves up into two teams of three. They want to arrange the teams so that, at the end of the day, each pair of players has played at least one game on the same team. Compute the smallest number of games they need to play in order to achieve this.
p9. Let AA and BB be points in the plane such that AB=30AB = 30. A circle with integer radius passes through AA and BB. A point CC is constructed on the circle such that ACAC is a diameter of the circle. Compute all possible radii of the circle such that BCBC is a positive integer.
p10. Each square of a 3×33\times 3 grid can be colored black or white. Two colorings are the same if you can rotate or reflect one to get the other. Compute the total number of unique colorings.
p11. Compute all positive integers nn such that the sum of all positive integers that are less than nn and relatively prime to nn is equal to 2n2n.
p12. The distance between a point and a line is defined to be the smallest possible distance between the point and any point on the line. Triangle ABCABC has AB=10AB = 10, BC=21BC = 21, and CA=17CA = 17. Let PP be a point inside the triangle. Let xx be the distance between PP and BC↔\overleftrightarrow{BC}, let yy be the distance between PP and CA↔\overleftrightarrow{CA}, and let zz be the distance between PP and AB↔\overleftrightarrow{AB}. Compute the largest possible value of the product xyzxyz.
p13. Alice, Bob, David, and Eve are sitting in a row on a couch and are passing back and forth a bag of chips. Whenever Bob gets the bag of chips, he passes the bag back to the person who gave it to him with probability 13\frac13 , and he passes it on in the same direction with probability 23\frac23 . Whenever David gets the bag of chips, he passes the bag back to the person who gave it to him with probability 14\frac14 , and he passes it on with probability 34\frac34 . Currently, Alice has the bag of chips, and she is about to pass it to Bob when Cathy sits between Bob and David. Whenever Cathy gets the bag of chips, she passes the bag back to the person who gave it to her with probability pp, and passes it on with probability 1−p1-p. Alice realizes that because Cathy joined them on the couch, the probability that Alice gets the bag of chips back before Eve gets it has doubled. Compute pp.
p14. Circle OO is in the plane. Circles AA, BB, and CC are congruent, and are each internally tangent to circle OO and externally tangent to each other. Circle XX is internally tangent to circle OO and externally tangent to circles AA and BB. Circle XX has radius 11. Compute the radius of circle OO. https://cdn.artofproblemsolving.com/attachments/f/d/8ddab540dca0051f660c840c0432f9aa5fe6b0.png
p15. Compute the number of primes pp less than 100 such that pp divides n2+n+1n^2 +n+1 for some integer nn.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
Mixer
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2010 Winter CHMMC Mixer Round - Caltech Harvey Mudd Mathematics Competition

p1. Compute xx such that 20092010≡x2009^{2010} \equiv x (mod 20112011) and 0≤x<20110 \le x < 2011.
p2. Compute the number of "words" that can be formed by rearranging the letters of the word "syzygy" so that the y's are evenly spaced. (The yy's are evenly spaced if the number of letters (possibly zero) between the first yy and the second yy is the same as the number of letters between the second yy and the third yy.)
p3. Let AA and BB be subsets of the integers, and let A+BA + B be the set containing all sums of the form a+ba + b, where aa is an element of AA, and bb is an element of BB. For example, if A={0,4,5}A = \{0, 4, 5\} and B={−3,−1,2,6}B =\{-3,-1, 2, 6\}, then A+B={−3,−1,1,2,3,4,6,7,10,11}A + B = \{-3,-1, 1, 2, 3, 4, 6, 7, 10, 11\}. If AA has 19551955 elements and BB has 18911891 elements, compute the smallest possible number of elements in A+BA + B.
p4. Compute the sum of all integers of the form pnp^n where pp is a prime, n≥3n \ge 3, and pn≤1000p^n \le 1000.
p5. In a season of interhouse athletics at Caltech, each of the eight houses plays each other house in a particular sport. Suppose one of the houses has a 1/31/3 chance of beating each other house. If the results of the games are independent, compute the probability that they win at least three games in a row.
p6. A positive integer nn is special if there are exactly 20102010 positive integers smaller than nn and relatively prime to nn. Compute the sum of all special numbers.
p7. Eight friends are playing informal games of ultimate frisbee. For each game, they split themselves up into two teams of four. They want to arrange the teams so that, at the end of the day, each pair of players has played at least one game on the same team. Determine the smallest number of games they need to play in order to achieve this.
p8. Compute the number of ways to choose five nonnegative integers a,b,c,da, b, c, d, and ee, such that a+b+c+d+e=20a + b + c + d + e = 20.
p9. Is 2323 a square mod 4141? Is 1515 a square mod 4141?
p10. Let ϕ(n)\phi (n) be the number of positive integers less than or equal to nn that are relatively prime to nn. Compute ∑d∣15015ϕ(d) \sum_{d|15015} \phi (d).
p11. Compute the largest possible volume of an regular tetrahedron contained in a cube with volume 11.
p12. Compute the number of ways to cover a 4×44 \times 4 grid with dominoes.
p13. A collection of points is called mutually equidistant if the distance between any two of them is the same. For example, three mutually equidistant points form an equilateral triangle in the plane, and four mutually equidistant points form a regular tetrahedron in three-dimensional space. Let AA, BB, CC, DD, and EE be five mutually equidistant points in four-dimensional space. Let PP be a point such that AP=BP=CP=DP=EP=1AP = BP = CP = DP = EP = 1. Compute the side length ABAB.
p14. Ten turtles live in a pond shaped like a 1010-gon. Because it's a sunny day, all the turtles are sitting in the sun, one at each vertex of the pond. David decides he wants to scare all the turtles back into the pond. When he startles a turtle, it dives into the pond. Moreover, any turtles on the two neighbouring vertices also dive into the pond. However, if the vertex opposite the startled turtle is empty, then a turtle crawls out of the pond and sits at that vertex. Compute the minimum number of times David needs to startle a turtle so that, by the end, all but one of the turtles are in the pond.
p15. The game hexapawn is played on a 3×33 \times 3 chessboard. Each player starts with three pawns on the row nearest him or her. The players take turns moving their pawns. Like in chess, on a player's turn he or she can either ∙\bullet move a pawn forward one space if that square is empty, or ∙\bullet capture an opponent's pawn by moving his or her own pawn diagonally forward one space into the opponent's pawn's square. A player wins when either ∙\bullet he or she moves a pawn into the last row, or ∙\bullet his or her opponent has no legal moves. Eve and Fred are going to play hexapawn. However, they're not very good at it. Each turn, they will pick a legal move at random with equal probability, with one exception: If some move will immediately win the game (by either of the two winning conditions), then he or she will make that move, even if other moves are available. If Eve moves first, compute the probability that she will win.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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2010 Winter Team #8

Alice and Bob are going to play a game called extra tricky double rock paper scissors (ETDRPS). In ETDRPS, each player simultaneously selects two moves, one for his or her right hand, and one for his or her left hand. Whereas Alice can play rock, paper, or scissors, Bob is only allowed to play rock or scissors. After revealing their moves, the players compare right hands and left hands separately. Alice wins if she wins strictly more hands than Bob. Otherwise, Bob wins. For example, if Alice and Bob were to both play rock with their right hands and scissors with their left hands, then both hands would be tied, so Bob would win the game. However, if Alice were to instead play rock with both hands, then Alice would win the left hand. The right hand would still be tied, so Alice would win the game. Assuming both players play optimally, compute the probability that Alice will win the game.
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2010 Winter Team #1

A matrix MM is called idempotent if M2=MM^2 = M. Find an idempotent 2×22 \times 2 matrix with distinct rational entries or write “none” if none exist.

2010 CHMMC Winter Tiebreaker 1 - f(f(\sqrt5 -\sqrt2)) if f(\sqrt5 +\sqrt2)= 0

The monic polynomial ff has rational coefficients and is irreducible over the rational numbers. If f(5+2)=0f(\sqrt5 +\sqrt2)= 0, compute f(f(5−2))f(f(\sqrt5 -\sqrt2)). (A polynomial is monic if its leading coeffi cient is 11. A polynomial is irreducible over the rational numbers if it cannot be expressed as a product of two polynomials with rational coefficients of positive degree. For example, x2−2x^2 - 2 is irreducible, but x2−1=(x+1)(x−1)x^2 - 1 = (x + 1)(x - 1) is not.)