CanadaNational Olympiad

2008

1 ABCD is a convex quadrilateral for which AB is the longest side Points M and N arelocated on sides AB and BC respectively so that each of the segments AN and CM dividesthe quadrilateral into two parts of equal area Prove that the segment MN bisects thediagonal BD

2 Determine all functions f defined on the set of rational numbers that take rational values forwhich

f(2f(x) + f(y)) = 2x + y

for each x and y

3 Let a b c be positive real numbers for which a + b + c = 1 Prove that

aminus bc

a + bc+

bminus ca

b + ca+

cminus ab

c + able 3

2

4 Determine all functions f defined on the natural numbers that take values among the naturalnumbers for which

(f(n))p equiv n mod f(p)

for all n isin N and all prime numbers p

5 A self-avoiding rook walk on a chessboard (a rectangular grid of unit squares) is a path tracedby a sequence of moves parallel to an edge of the board from one unit square to another suchthat each begins where the previous move ended and such that no move ever crosses a squarethat has previously been crossed ie the rookrsquos path is non-self-intersecting

Let R(mn) be the number of self-avoiding rook walks on an m times n (m rows n columns)chessboard which begin at the lower-left corner and end at the upper-left corner For exampleR(m 1) = 1 for all natural numbers m R(2 2) = 2 R(3 2) = 4 R(3 3) = 11 Find a formulafor R(3 n) for each natural number n

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ChinaTeam Selection Test

2008

Day 1

1 Let ABC be a triangle let AB gt AC Its incircle touches side BC at point E Point D isthe second intersection of the incircle with segment AE (different from E) Point F (differentfrom E) is taken on segment AE such that CE = CF The ray CF meets BD at point GShow that CF = FG

2 The sequence xn is defined by x1 = 2 x2 = 12 and xn+2 = 6xn+1 minus xn (n = 1 2 )Let p be an odd prime number let q be a prime divisor of xp Prove that if q 6= 2 3 thenq ge 2pminus 1

3 Suppose that every positve integer has been given one of the colors red bluearbitrarily Provethat there exists an infinite sequence of positive integers a1 lt a2 lt a3 lt middot middot middot lt an lt middot middot middot

such that inifinite sequence of positive integers a1a1 + a2

2 a2

a2 + a3

2 a3

a3 + a4

2 middot middot middot has

the same color

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ChinaTeam Selection Test

2008

Day 2

4 Prove that for arbitary positive integer n ge 4 there exists a permutation of the subsets thatcontain at least two elements of the set Gn = 1 2 3 middot middot middot n P1 P2 middot middot middot P2nminusnminus1 such that|Pi cap Pi+1| = 2 i = 1 2 middot middot middot 2n minus nminus 2

5 For two given positive integers mn gt 1 let aij(i = 1 2 middot middot middot n j = 1 2 middot middot middot m) be nonneg-ative real numbers not all zero find the maximum and the minimum values of f where

f =n

sumni=1(

summj=1 aij)2 + m

summj=1(

sumni=1 aij)2

(sumn

i=1

summj=1 aij)2 + mn

sumni=1

summi=j a2

ij

6 Find the maximal constant M such that for arbitrary integer n ge 3 there exist two sequences

of positive real number a1 a2 middot middot middot an and b1 b2 middot middot middot bn satisfying (1)nsum

k=1

bk = 1 2bk ge bkminus1+

bk+1 k = 2 3 middot middot middot nminus 1 (2)a2k le 1 +

ksumi=1

aibi k = 1 2 3 middot middot middot n an equiv M

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CroatiaTeam Selection Tests

2008

1 Let x y z be positive numbers Find the minimum value of (a)x2 + y2 + z2

xy + yz

(b)x2 + y2 + 2z2

xy + yz

2 For which n isin N do there exist rational numbers a b which are not integers such that botha + b and an + bn are integers

3 Point M is taken on side BC of a triangle ABC such that the centroid Tc of triangle ABMlies on the circumcircle of 4ACM and the centroid Tb of 4ACM lies on the circumcircle of4ABM Prove that the medians of the triangles ABM and ACM from M are of the samelength

4 Let S be the set of all odd positive integers less than 30m which are not multiples of 5 wherem is a given positive integer Find the smallest positive integer k such that each k-elementsubset of S contains two distinct numbers one of which divides the other

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MoldovaNational Olympiad 11-12

2008

Day 1 - 01 March 2008

1 Consider the equation x4 minus 4x3 + 4x2 + ax + b = 0 where a b isin R Determine the largestvalue a + b can take so that the given equation has two distinct positive roots x1 x2 so thatx1 + x2 = 2x1x2

2 Find the exact value of E =int π

2

0cos1003 xdx middot

int π2

0cos1004 xdxmiddot

3 In the usual coordinate system xOy a line d intersect the circles C1 (x+1)2+y2 = 1 and C2

(xminus 2)2 + y2 = 4 in the points AB C and D (in this order) It is known that A

(minus3

2

radic3

2

)and angBOC = 60 All the Oy coordinates of these 4 points are positive Find the slope of d

4 Define the sequence (ap)pge0 as follows ap =

(p0

)2 middot 4

minus(p1

)3 middot 5

+

(p2

)4 middot 6

minus +(minus1)p middot(pp

)(p + 2)(p + 4)

Find limnrarrinfin

(a0 + a1 + + an)

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MoldovaNational Olympiad 11-12

2008

Day 2 - 02 March 2008

5 Find the least positive integer n so that the polynomial P (X) =radic

3 middotXn+1 minusXn minus 1 has atleast one root of modulus 1

6 Find limnrarrinfin

an where (an)nge1 is defined by an =1radic

n2 + 8nminus 1+

1radicn2 + 16nminus 1

+1radic

n2 + 24nminus 1+

+1radic

9n2 minus 1

7 Triangle ABC has fixed vertices B and C so that BC = 2 and A is variable Denote by Hand G the orthocenter and the centroid respectively of triangle ABC Let F isin (HG) so

thatHF

FG= 3 Find the locus of the point A so that F isin BC

8 Evaluate I =int π

4

0

(sin6 2x + cos6 2x

)middot ln(1 + tan x)dx

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MoldovaNational Olympiad

2008

1 Let fm R rarr R fm(x) = (m2 + m + 1)x2 minus 2(m2 + 1)x + m2 minusm + 1 where m isin R

1) Find the fixed common point of all this parabolas

2) Find m such that the distance from that fixed point to Oy is minimal

2 Find f(x) (0+infin) rarr R such that

f(x) middot f(y) + f(2008

x) middot f(

2008y

) = 2f(x middot y)

and f(2008) = 1 for forallx isin (0+infin)

3 From the vertex A of the equilateral triangle ABC a line is drown that intercepts the segment[BC] in the point E The point M isin (AE is such that M external to ABC angAMB = 20

and angAMC = 30 What is the measure of the angle angMAB

4 Let n be a positive integer Find all x1 x2 xn that satisfy the relation

radicx1 minus 1 + 2 middot

radicx2 minus 4 + 3 middot

radicx3 minus 9 + middot middot middot+ n middot

radicxn minus n2 =

12(x1 + x2 + x3 + middot middot middot+ xn)

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MoldovaTeam Selection Test

2008

Day 1

1 Let p be a prime number Solve in N0 times N0 the equation x3 + y3 minus 3xy = pminus 1

2 We say the set 1 2 3k has property D if it can be partitioned into disjoint triples sothat in each of them a number equals the sum of the other two

(a) Prove that 1 2 3324 has property D

(b) Prove that 1 2 3309 hasnrsquot property D

3 Let Γ(I r) and Γ(OR) denote the incircle and circumcircle respectively of a triangle ABCConsider all the triangels AiBiCi which are simultaneously inscribed in Γ(OR) and circum-scribed to Γ(I r) Prove that the centroids of these triangles are concyclic

4 A non-zero polynomial S isin R[X Y ] is called homogeneous of degree d if there is a positiveinteger d so that S(λx λy) = λdS(x y) for any λ isin R Let PQ isin R[X Y ] so that Q ishomogeneous and P divides Q (that is P |Q) Prove that P is homogeneous too

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MoldovaTeam Selection Test

2008

Day 2 - 29 March 2008

1 Find all solutions (x y) isin Rtimes R of the following system

x3 + 3xy2 = 49

x2 + 8xy + y2 = 8y + 17x

EDIT Thanks to Silouan for pointing out a mistake in the problem statement

2 Let a1 an be positive reals so that a1 + a2 + + an le n

2 Find the minimal value ofradic

a21 +

1a2

2

+

radica2

2 +1a2

3

+ +

radica2

n +1a2

1

3 Let ω be the circumcircle of ABC and let D be a fixed point on BC D 6= B D 6= C Let Xbe a variable point on (BC) X 6= D Let Y be the second intersection point of AX and ωProve that the circumcircle of XY D passes through a fixed point

4 Find the number of even permutations of 1 2 n with no fixed points

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MoldovaTeam Selection Test

2008

Day 3 - 30 March 2008

1 Determine a subset A sub Nlowast having 5 different elements so that the sum of the squares of itselements equals their product Do not simply post the subset show how you found it

2 Let p be a prime number and k n positive integers so that gcd(p n) = 1 Prove that(

n middot pk

pk

)and p are coprime

3 In triangle ABC the bisector of angACB intersects AB at D Consider an arbitrary circle Opassing through C and D so that it is not tangent to BC or CA Let O cap BC = M andOcapCA = N a) Prove that there is a circle S so that DM and DN are tangent to S in Mand N respectively b) Circle S intersects lines BC and CA in P and Q respectively Provethat the lengths of MP and NQ do not depend on the choice of circle O

4 A non-empty set S of positive integers is said to be good if there is a coloring with 2008 colorsof all positive integers so that no number in S is the sum of two different positive integers(not necessarily in S) of the same color Find the largest value t can take so that the setS = a + 1 a + 2 a + 3 a + t is good for any positive integer a

PS I have the feeling that Irsquove seen this problem before so if Irsquom right maybe someone canpost some links

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PolandFinals

2008

Day 1

1 In each cell of a matrix ntimesn a number from a set 1 2 n2 is written mdash in the first rownumbers 1 2 n in the second n+1 n+2 2n and so on Exactly n of them have beenchosen no two from the same row or the same column Let us denote by ai a number chosenfrom row number i Show that

12

a1+

22

a2+ +

n2

ange n + 2

2minus 1

n2 + 1

2 A function f R3 rarr R for all reals a b c d e satisfies a condition

f(a b c) + f(b c d) + f(c d e) + f(d e a) + f(e a b) = a + b + c + d + e

Show that for all reals x1 x2 xn (n ge 5) equality holds

f(x1 x2 x3) + f(x2 x3 x4) + + f(xnminus1 xn x1) + f(xn x1 x2) = x1 + x2 + + xn

3 In a convex pentagon ABCDE in which BC = DE following equalities hold

angABE = angCAB = angAED minus 90 angACB = angADE

Show that BCDE is a parallelogram

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PolandFinals

2008

Day 2

4 There is nothing to show The splitting field of a set of polynomials S sube F [X] is simply thefield over F generated by the set of roots of the polynomials in S in some algebraic closureof F And clearly f1 fn and f1 middot middot middot fn have the same sets of roots

5 Let R be a parallelopiped Let us assume that areas of all intersections of R with planescontaining centers of three edges of R pairwisely not parallel and having no common pointsare equal Show that R is a cuboid

6 Let S be a set of all positive integers which can be represented as a2 + 5b2 for some integersa b such that aperpb Let p be a prime number such that p = 4n + 3 for some integer n Showthat if for some positive integer k the number kp is in S then 2p is in S as well

Here the notation aperpb means that the integers a and b are coprime

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SerbiaNational Math Olympiad

2008

Day 1

1 Find all nonegative integers x y z such that 12x + y4 = 2008z

2 Triangle 4ABC is given Points D i E are on line AB such that D minusAminusB minusEAD = ACand BE = BC Bisector of internal angles at A and B intersect BC AC at P and Q andcircumcircle of ABC at M and N Line which connects A with center of circumcircle ofBME and line which connects B and center of circumcircle of AND intersect at X Provethat CX perp PQ

3 Let a b c be positive real numbers such that a + b + c = 1 Prove inequality

1bc + a + 1

a

+1

ac + b + 1b

+1

ab + c + 1c

62731

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SerbiaNational Math Olympiad

2008

Day 2

4 Each point of a plane is painted in one of three colors Show that there exists a trianglesuch that (i) all three vertices of the triangle are of the same color (ii) the radius of thecircumcircle of the triangle is 2008 (iii) one angle of the triangle is either two or three timesgreater than one of the other two angles

5 The sequence (an)nge1 is defined by a1 = 3 a2 = 11 and an = 4anminus1 minus anminus2 for n ge 3 Provethat each term of this sequence is of the form a2 + 2b2 for some natural numbers a and b

6 In a convex pentagon ABCDE let angEAB = angABC = 120 angADB = 30 and angCDE =60 Let AB = 1 Prove that the area of the pentagon is less than

radic3

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TurkeyTeam Selection Tests

2008

Day 1

1 In an ABC triangle such that m(angB) gt m(angC) the internal and external bisectors of verticeA intersects BC respectively at points D and E P is a variable point on EA such that Ais on [EP ] DP intersects AC at M and ME intersects AD at Q Prove that all PQ lineshave a common point as P varies

2 A graph has 30 vertices 105 edges and 4822 unordered edge pairs whose endpoints are disjointFind the maximal possible difference of degrees of two vertices in this graph

3 The equation x3 minus ax2 + bx minus c = 0 has three (not necessarily different) positive real roots

Find the minimal possible value of1 + a + b + c

3 + 2a + bminus c

b

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TurkeyTeam Selection Tests

2008

Day 2

4 The sequence (xn) is defined as x1 = a x2 = b and for all positive integer n xn+2 =2008xn+1 minus xn Prove that there are some positive integers a b such that 1 + 2006xn+1xn isa perfect square for all positive integer n

5 D is a point on the edge BC of triangle ABC such that AD =BD2

AB + AD=

CD2

AC + AD E

is a point such that D is on [AE] and CD =DE2

CD + CE Prove that AE = AB + AC

6 There are n voters and m candidates Every voter makes a certain arrangement list of allcandidates (there is one person in every place 1 2 m) and votes for the first k people inhisher list The candidates with most votes are selected and say them winners A poll profileis all of this n lists If a is a candidate R and Rprime are two poll profiles Rprime is aminus good for Rif and only if for every voter the people which in a worse position than a in R is also in aworse position than a in Rprime We say positive integer k is monotone if and only if for everyR poll profile and every winner a for R poll profile is also a winner for all a minus good Rprime poll

profiles Prove that k is monotone if and only if k gtm(nminus 1)

n

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USAAIME

2008

I

1 Of the students attending a school party 60 of the students are girls and 40 of thestudents like to dance After these students are joined by 20 more boy students all of whomlike to dance the party is now 58 girls How many students now at the party like to dance

2 Square AIME has sides of length 10 units Isosceles triangle GEM has base EM and thearea common to triangle GEM and square AIME is 80 square units Find the length of thealtitude to EM in 4GEM

3 Ed and Sue bike at equal and constant rates Similarly they jog at equal and constant ratesand they swim at equal and constant rates Ed covers 74 kilometers after biking for 2 hoursjogging for 3 hours and swimming for 4 hours while Sue covers 91 kilometers after jogging for2 hours swimming for 3 hours and biking for 4 hours Their biking jogging and swimmingrates are all whole numbers of kilometers per hour Find the sum of the squares of Edrsquosbiking jogging and swimming rates

4 There exist unique positive integers x and y that satisfy the equation x2 + 84x + 2008 = y2Find x + y

5 A right circular cone has base radius r and height h The cone lies on its side on a flat tableAs the cone rolls on the surface of the table without slipping the point where the conersquos basemeets the table traces a circular arc centered at the point where the vertex touches the tableThe cone first returns to its original position on the table after making 17 complete rotationsThe value of hr can be written in the form m

radicn where m and n are positive integers and

n is not divisible by the square of any prime Find m + n

6 A triangular array of numbers has a first row consisting of the odd integers 1 3 5 99 inincreasing order Each row below the first has one fewer entry than the row above it andthe bottom row has a single entry Each entry in any row after the top row equals the sumof the two entries diagonally above it in the row immediately above it How many entries inthe array are multiples of 67

7 Let Si be the set of all integers n such that 100i le n lt 100(i + 1) For example S4 is theset 400 401 402 499 How many of the sets S0 S1 S2 S999 do not contain a perfectsquare

8 Find the positive integer n such that

arctan13

+ arctan14

+ arctan15

+ arctan1n

=π

4

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USAAIME

2008

9 Ten identical crates each of dimensions 3 ft times 4 ft times 6 ft The first crate is placed flat onthe floor Each of the remaining nine crates is placed in turn flat on top of the previouscrate and the orientation of each crate is chosen at random Let

m

nbe the probability that

the stack of crates is exactly 41 ft tall where m and n are relatively prime positive integersFind m

10 Let ABCD be an isosceles trapezoid with AD||BC whose angle at the longer base AD isπ

3

The diagonals have length 10radic

21 and point E is at distances 10radic

7 and 30radic

7 from verticesA and D respectively Let F be the foot of the altitude from C to AD The distance EFcan be expressed in the form m

radicn where m and n are positive integers and n is not divisible

by the square of any prime Find m + n

11 Consider sequences that consist entirely of Arsquos and Brsquos and that have the property that everyrun of consecutive Arsquos has even length and every run of consecutive Brsquos has odd lengthExamples of such sequences are AA B and AABAA while BBAB is not such a sequenceHow many such sequences have length 14

12 On a long straight stretch of one-way single-lane highway cars all travel at the same speedand all obey the safety rule the distance from the back of the car ahead to the front of the carbehind is exactly one car length for each 15 kilometers per hour of speed or fraction thereof(Thus the front of a car traveling 52 kilometers per hour will be four car lengths behind theback of the car in front of it) A photoelectric eye by the side of the road counts the numberof cars that pass in one hour Assuming that each car is 4 meters long and that the carscan travel at any speed let M be the maximum whole number of cars that can pass thephotoelectric eye in one hour Find the quotient when M is divided by 10

13 Let

p(x y) = a0 + a1x + a2y + a3x2 + a4xy + a5y

2 + a6x3 + a7x

2y + a8xy2 + a9y3

Suppose that

p(0 0) = p(1 0) = p(minus1 0) = p(0 1) = p(0minus1) = p(1 1) = p(1minus1) = p(2 2) = 0

There is a point(

a

cb

c

)for which p

(a

cb

c

)= 0 for all such polynomials where a b and c

are positive integers a and c are relatively prime and c gt 1 Find a + b + c

14 Let AB be a diameter of circle ω Extend AB through A to C Point T lies on ω so that lineCT is tangent to ω Point P is the foot of the perpendicular from A to line CT SupposeAB = 18 and let m denote the maximum possible length of segment BP Find m2

15 A square piece of paper has sides of length 100 From each corner a wedge is cut in thefollowing manner at each corner the two cuts for the wedge each start at distance

radic17 from

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Page 2 httpwwwmathlinksro

USAAIME

2008

the corner and they meet on the diagonal at an angle of 60 (see the figure below) Thepaper is then folded up along the lines joining the vertices of adjacent cuts When the twoedges of a cut meet they are taped together The result is a paper tray whose sides are not atright angles to the base The height of the tray that is the perpendicular distance betweenthe plane of the base and the plane formed by the upper edges can be written in the formnradic

m where m and n are positive integers m lt 1000 and m is not divisible by the nth powerof any prime Find m + n

[asy]import math unitsize(5mm) defaultpen(fontsize(9pt)+Helvetica()+linewidth(07))

pair O=(00) pair A=(0sqrt(17)) pair B=(sqrt(17)0) pair C=shift(sqrt(17)0)(sqrt(34)dir(75))pair D=(xpart(C)8) pair E=(8ypart(C))

draw(Ondash(08)) draw(Ondash(80)) draw(OndashC) draw(AndashCndashB) draw(DndashCndashE)

label(quot36radic

1736 quot (0 2)W ) label(quot 36radic

1736 quot (2 0) S) label(quot cutquot midpoint(AminusminusC) NNW ) label(quot cutquot midpoint(B minus minusC) ESE) label(quot foldquot midpoint(C minusminusD)W ) label(quot foldquot midpoint(CminusminusE) S) label(quot 36 3036 quot shift(minus06minus06)lowastCWSW ) label(quot 36 3036 quot shift(minus12minus12) lowast CSSE) [asy]

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USAAIME

2008

II

1 Let N = 1002 + 992 minus 982 minus 972 + 962 + middot middot middot + 42 + 32 minus 22 minus 12 where the additions andsubtractions alternate in pairs Find the remainder when N is divided by 1000

2 Rudolph bikes at a constant rate and stops for a five-minute break at the end of every mileJennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes butJennifer takes a five-minute break at the end of every two miles Jennifer and Rudolph beginbiking at the same time and arrive at the 50-mile mark at exactly the same time How manyminutes has it taken them

3 A block of cheese in the shape of a rectangular solid measures 10 cm by 13 cm by 14 cm Tenslices are cut from the cheese Each slice has a width of 1 cm and is cut parallel to one faceof the cheese The individual slices are not necessarily parallel to each other What is themaximum possible volume in cubic cm of the remaining block of cheese after ten slices havebeen cut off

4 There exist r unique nonnegative integers n1 gt n2 gt middot middot middot gt nr and r unique integers ak

(1 le k le r) with each ak either 1 or minus1 such that

a13n1 + a23n2 + middot middot middot+ ar3nr = 2008

Find n1 + n2 + middot middot middot+ nr

5 In trapezoid ABCD with BC AD let BC = 1000 and AD = 2008 Let angA = 37angD = 53 and m and n be the midpoints of BC and AD respectively Find the length MN

6 The sequence an is defined by

a0 = 1 a1 = 1 and an = anminus1 +a2

nminus1

anminus2for n ge 2

The sequence bn is defined by

b0 = 1 b1 = 3 and bn = bnminus1 +b2nminus1

bnminus2for n ge 2

Findb32

a32

7 Let r s and t be the three roots of the equation

8x3 + 1001x + 2008 = 0

Find (r + s)3 + (s + t)3 + (t + r)3

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USAAIME

2008

8 Let a = π2008 Find the smallest positive integer n such that

2[cos(a) sin(a) + cos(4a) sin(2a) + cos(9a) sin(3a) + middot middot middot+ cos(n2a) sin(na)]

is an integer

9 A particle is located on the coordinate plane at (5 0) Define a move for the particle as acounterclockwise rotation of π4 radians about the origin followed by a translation of 10 unitsin the positive x-direction Given that the particlersquos position after 150 moves is (p q) findthe greatest integer less than or equal to |p|+ |q|

10 The diagram below shows a 4 times 4 rectangular array of points each of which is 1 unit awayfrom its nearest neighbors [asy]unitsize(025inch) defaultpen(linewidth(07))

int i j for(i = 0 i iexcl 4 ++i) for(j = 0 j iexcl 4 ++j) dot(((real)i (real)j))[asy]Define a growingpath to be a sequence of distinct points of the array with the property that the distancebetween consecutive points of the sequence is strictly increasing Let m be the maximumpossible number of points in a growing path and let r be the number of growing pathsconsisting of exactly m points Find mr

11 In triangle ABC AB = AC = 100 and BC = 56 Circle P has radius 16 and is tangent toAC and BC Circle Q is externally tangent to P and is tangent to AB and BC No point ofcircle Q lies outside of 4ABC The radius of circle Q can be expressed in the form mminusn

radick

where m n and k are positive integers and k is the product of distinct primes Find m+nk

12 There are two distinguishable flagpoles and there are 19 flags of which 10 are identical blueflags and 9 are identical green flags Let N be the number of distinguishable arrangementsusing all of the flags in which each flagpole has at least one flag and no two green flags oneither pole are adjacent Find the remainder when N is divided by 1000

13 A regular hexagon with center at the origin in the complex plane has opposite pairs of sidesone unit apart One pair of sides is parallel to the imaginary axis Let R be the region outside

the hexagon and let S = 1z|z isin R Then the area of S has the form aπ +

radicb where a and

b are positive integers Find a + b

14 Let a and b be positive real numbers with a ge b Let ρ be the maximum possible value ofa

bfor which the system of equations

a2 + y2 = b2 + x2 = (aminus x)2 + (bminus y)2

has a solution in (x y) satisfying 0 le x lt a and 0 le y lt b Then ρ2 can be expressed as afraction

m

n where m and n are relatively prime positive integers Find m + n

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USAAIME

2008

15 Find the largest integer n satisfying the following conditions (i) n2 can be expressed as thedifference of two consecutive cubes (ii) 2n + 79 is a perfect square

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USAHarvard-MIT Mathematics Tournament

Cambridge MA 2008

Algebra

1 Positive real numbers x y satisfy the equations x2 + y2 = 1 and x4 + y4 =1718

Find xy

2 Let f(n) be the number of times you have to hit the radic key on a calculator to get a numberless than 2 starting from n For instance f(2) = 1 f(5) = 2 For how many 1 lt m lt 2008is f(m) odd

3 Determine all real numbers a such that the inequality |x2 + 2ax + 3a| le 2 has exactly onesolution in x

4 The function f satisfies

f(x) + f(2x + y) + 5xy = f(3xminus y) + 2x2 + 1

for all real numbers x y Determine the value of f(10)

5 Let f(x) = x3 + x + 1 Suppose g is a cubic polynomial such that g(0) = minus1 and the rootsof g are the squares of the roots of f Find g(9)

6 A root of unity is a complex number that is a solution to zn = 1 for some positive integern Determine the number of roots of unity that are also roots of z2 + az + b = 0 for someintegers a and b

7 Computeinfinsum

n=1

nminus1sumk=1

k

2n+k

8 Compute arctan (tan 65 minus 2 tan 40) (Express your answer in degrees)

9 Let S be the set of points (a b) with 0 le a b le 1 such that the equation

x4 + ax3 minus bx2 + ax + 1 = 0

has at least one real root Determine the area of the graph of S

10 Evaluate the infinite suminfinsum

n=0

(2n

n

)15n

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USAHarvard-MIT Mathematics Tournament

Cambridge MA 2008

Calculus

1 Let f(x) = 1 + x + x2 + middot middot middot+ x100 Find f prime(1)

2 (3) Let ` be the line through (0 0) and tangent to the curve y = x3 + x + 16 Find the slopeof `

3 (4) Find all y gt 1 satisfyingint y

1x lnx dx =

14

4 (4) Let a b be constants such that limxrarr1

(ln(2minus x))2

x2 + ax + b= 1 Determine the pair (a b)

5 (4) Let f(x) = sin6(x

4

)+ cos6

(x

4

)for all real numbers x Determine f (2008)(0) (ie f

differentiated 2008 times and then evaluated at x = 0)

6 (5) Determine the value of limnrarrinfin

nsumk=0

(n

k

)minus1

7 (5) Find p so that limxrarrinfin

xp(

3radic

x + 1 + 3radic

xminus 1minus 2 3radic

x)

is some non-zero real number

8 (7) Let T =int ln 2

0

2e3x + e2x minus 1e3x + e2x minus ex + 1

dx Evaluate eT

9 (7) Evaluate the limit limnrarrinfin

nminus12(1+ 1

n) (11 middot 22 middot middot middot middot middot nn

) 1n2

10 (8) Evaluate the integralint 1

0lnx ln(1minus x) dx

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USAHarvard-MIT Mathematics Tournament

Cambridge MA 2008

Combinatorics

1 A 3times3times3 cube composed of 27 unit cubes rests on a horizontal plane Determine the numberof ways of selecting two distinct unit cubes from a 3 times 3 times 1 block (the order is irrelevant)with the property that the line joining the centers of the two cubes makes a 45 angle withthe horizontal plane

2 Let S = 1 2 2008 For any nonempty subset A isin S define m(A) to be the median ofA (when A has an even number of elements m(A) is the average of the middle two elements)Determine the average of m(A) when A is taken over all nonempty subsets of S

3 Farmer John has 5 cows 4 pigs and 7 horses How many ways can he pair up the animalsso that every pair consists of animals of different species (Assume that all animals aredistinguishable from each other)

4 Kermit the frog enjoys hopping around the innite square grid in his backyard It takes him1 Joule of energy to hop one step north or one step south and 1 Joule of energy to hop onestep east or one step west He wakes up one morning on the grid with 100 Joules of energyand hops till he falls asleep with 0 energy How many different places could he have gone tosleep

5 Let S be the smallest subset of the integers with the property that 0 isin S and for any x isin Swe have 3x isin S and 3x + 1 isin S Determine the number of non-negative integers in S lessthan 2008

6 A Sudoku matrix is dened as a 9 times 9 array with entries from 1 2 9 and with theconstraint that each row each column and each of the nine 3 times 3 boxes that tile the arraycontains each digit from 1 to 9 exactly once A Sudoku matrix is chosen at random (so thatevery Sudoku matrix has equal probability of being chosen) We know two of the squares inthis matrix as shown What is the probability that the square marked by contains thedigit 3

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USAHarvard-MIT Mathematics Tournament

Cambridge MA 2008

1

2

7 Let P1 P2 P8 be 8 distinct points on a circle Determine the number of possible configu-rations made by drawing a set of line segments connecting pairs of these 8 points such that(1) each Pi is the endpoint of at most one segment and (2) no two segments intersect (Theconfiguration with no edges drawn is allowed An example of a valid configuration is shownbelow)

[asy]unitsize(1cm) pair[] P = new pair[8] align[] A = E NE N NW W SW S SE for (int i= 0 i iexcl 8 ++i) P[i] = dir(45i) dot(P[i]) label(quot36Pquot+((string)i)+quot36quotP [i]A[i]fontsize(8pt))draw(unitcircle) draw(P [0]minusminusP [1]) draw(P [2]minusminusP [4]) draw(P [5]minusminusP [6]) [asy]Determinethenumberofwaystoselectasequenceof8sets A1 A2 A8 such that each is a subset (possibly empty) of 1 2 and Am contains An

if m divides n

89 On an innite chessboard (whose squares are labeled by (x y) where x and y range over allintegers) a king is placed at (0 0) On each turn it has probability of 01 of moving to eachof the four edge-neighboring squares and a probability of 005 of moving to each of the fourdiagonally-neighboring squares and a probability of 04 of not moving After 2008 turnsdetermine the probability that the king is on a square with both coordinates even An exactanswer is required

10 Determine the number of 8-tuples of nonnegative integers (a1 a2 a3 a4 b1 b2 b3 b4) satisfying0 le ak le k for each k = 1 2 3 4 and a1 + a2 + a3 + a4 + 2b1 + 3b2 + 4b3 + 5b4 = 19

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USAHarvard-MIT Mathematics Tournament

Cambridge MA 2008

General 1

1 Let ABCD be a unit square (that is the labels AB C D appear in that order around thesquare) Let X be a point outside of the square such that the distance from X to AC is equal

to the distance from X to BD and also that AX =radic

22

Determine the value of CX2

2 Find the smallest positive integer n such that 107n has the same last two digits as n

3 There are 5 dogs 4 cats and 7 bowls of milk at an animal gathering Dogs and cats aredistinguishable but all bowls of milk are the same In how many ways can every dog and catbe paired with either a member of the other species or a bowl of milk such that all the bowlsof milk are taken

4 Positive real numbers x y satisfy the equations x2 + y2 = 1 and x4 + y4 =1718

Find xy

5 The function f satisfies

f(x) + f(2x + y) + 5xy = f(3xminus y) + 2x2 + 1

for all real numbers x y Determine the value of f(10)

6 In a triangle ABC take point D on BC such that DB = 14 DA = 13 DC = 4 and thecircumcircle of ADB is congruent to the circumcircle of ADC What is the area of triangleABC

7 The equation x3 minus 9x2 + 8x + 2 = 0 has three real roots p q r Find1p2

+1q2

+1r2

8 Let S be the smallest subset of the integers with the property that 0 isin S and for any x isin Swe have 3x isin S and 3x + 1 isin S Determine the number of non-negative integers in S lessthan 2008

9 A Sudoku matrix is dened as a 9 times 9 array with entries from 1 2 9 and with theconstraint that each row each column and each of the nine 3 times 3 boxes that tile the arraycontains each digit from 1 to 9 exactly once A Sudoku matrix is chosen at random (so thatevery Sudoku matrix has equal probability of being chosen) We know two of the squares inthis matrix as shown What is the probability that the square marked by contains thedigit 3

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USAHarvard-MIT Mathematics Tournament

Cambridge MA 2008

1

2

10 Let ABC be an equilateral triangle with side length 2 and let Γ be a circle with radius12

centered at the center of the equilateral triangle Determine the length of the shortest paththat starts somewhere on Γ visits all three sides of ABC and ends somewhere on Γ (notnecessarily at the starting point) Express your answer in the form of

radicpminus q where p and q

are rational numbers written as reduced fractions

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USAHarvard-MIT Mathematics Tournament

Cambridge MA 2008

General 2

1 Four students from Harvard one of them named Jack and five students from MIT one ofthem named Jill are going to see a Boston Celtics game However they found out that only5 tickets remain so 4 of them must go back Suppose that at least one student from eachschool must go see the game and at least one of Jack and Jill must go see the game howmany ways are there of choosing which 5 people can see the game

2 Let ABC be an equilateral triangle Let Ω be its incircle (circle inscribed in the triangle) andlet ω be a circle tangent externally to Ω as well as to sides AB and AC Determine the ratioof the radius of Ω to the radius of ω

3 A 3times3times3 cube composed of 27 unit cubes rests on a horizontal plane Determine the numberof ways of selecting two distinct unit cubes from a 3 times 3 times 1 block (the order is irrelevant)with the property that the line joining the centers of the two cubes makes a 45 angle withthe horizontal plane

4 Suppose that a b c d are real numbers satisfying a ge b ge c ge d ge 0 a2 + d2 = 1 b2 + c2 = 1and ac + bd = 13 Find the value of abminus cd

5 Kermit the frog enjoys hopping around the innite square grid in his backyard It takes him1 Joule of energy to hop one step north or one step south and 1 Joule of energy to hop onestep east or one step west He wakes up one morning on the grid with 100 Joules of energyand hops till he falls asleep with 0 energy How many different places could he have gone tosleep

6 Determine all real numbers a such that the inequality |x2 + 2ax + 3a| le 2 has exactly onesolution in x

7 A root of unity is a complex number that is a solution to zn = 1 for some positive integern Determine the number of roots of unity that are also roots of z2 + az + b = 0 for someintegers a and b

8 A piece of paper is folded in half A second fold is made at an angle φ (0 lt φ lt 90) tothe first and a cut is made as shown below [img]12881[img] When the piece of paper isunfolded the resulting hole is a polygon Let O be one of its vertices Suppose that all theother vertices of the hole lie on a circle centered at O and also that angXOY = 144 whereX and Y are the the vertices of the hole adjacent to O Find the value(s) of φ (in degrees)

9 Let ABC be a triangle and I its incenter Let the incircle of ABC touch side BC at D andlet lines BI and CI meet the circle with diameter AI at points P and Q respectively GivenBI = 6 CI = 5 DI = 3 determine the value of (DPDQ)2

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USAHarvard-MIT Mathematics Tournament

Cambridge MA 2008

10 Determine the number of 8-tuples of nonnegative integers (a1 a2 a3 a4 b1 b2 b3 b4) satisfying0 le ak le k for each k = 1 2 3 4 and a1 + a2 + a3 + a4 + 2b1 + 3b2 + 4b3 + 5b4 = 19

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USAHarvard-MIT Mathematics Tournament

Cambridge MA 2008

Geometry

1 How many different values can angABC take where AB C are distinct vertices of a cube

2 Let ABC be an equilateral triangle Let Ω be its incircle (circle inscribed in the triangle) andlet ω be a circle tangent externally to Ω as well as to sides AB and AC Determine the ratioof the radius of Ω to the radius of ω

3 Let ABC be a triangle with angBAC = 90 A circle is tangent to the sides AB and AC at Xand Y respectively such that the points on the circle diametrically opposite X and Y bothlie on the side BC Given that AB = 6 find the area of the portion of the circle that liesoutside the triangle

[asy]import olympiad import math import graph

unitsize(20mm) defaultpen(fontsize(8pt))

pair A = (00) pair B = A + right pair C = A + up

pair O = (13 13)

pair Xprime = (1323) pair Yprime = (2313)

fill(Arc(O13090)ndashXprimendashYprimendashcycle07white)

draw(AndashBndashCndashcycle) draw(Circle(O 13)) draw((013)ndash(2313)) draw((130)ndash(1323))

label(quot36A36quotA SW) label(quot36B36quotB down) label(quot36C36quotCleft) label(quot36X36quot(130) down) label(quot36Y36quot(013) left)[asy]

4 In a triangle ABC take point D on BC such that DB = 14 DA = 13 DC = 4 and thecircumcircle of ADB is congruent to the circumcircle of ADC What is the area of triangleABC

5 A piece of paper is folded in half A second fold is made at an angle φ (0 lt φ lt 90) tothe first and a cut is made as shown below [img]12881[img] When the piece of paper isunfolded the resulting hole is a polygon Let O be one of its vertices Suppose that all theother vertices of the hole lie on a circle centered at O and also that angXOY = 144 whereX and Y are the the vertices of the hole adjacent to O Find the value(s) of φ (in degrees)

6 Let ABC be a triangle with angA = 45 Let P be a point on side BC with PB = 3 andPC = 5 Let O be the circumcenter of ABC Determine the length OP

7 Let C1 and C2 be externally tangent circles with radius 2 and 3 respectively Let C3 be acircle internally tangent to both C1 and C2 at points A and B respectively The tangents toC3 at A and B meet at T and TA = 4 Determine the radius of C3

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USAHarvard-MIT Mathematics Tournament

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8 Let ABC be an equilateral triangle with side length 2 and let Γ be a circle with radius12

centered at the center of the equilateral triangle Determine the length of the shortest paththat starts somewhere on Γ visits all three sides of ABC and ends somewhere on Γ (notnecessarily at the starting point) Express your answer in the form of

radicpminus q where p and q

are rational numbers written as reduced fractions

9 Let ABC be a triangle and I its incenter Let the incircle of ABC touch side BC at D andlet lines BI and CI meet the circle with diameter AI at points P and Q respectively GivenBI = 6 CI = 5 DI = 3 determine the value of (DPDQ)2

10 Let ABC be a triangle with BC = 2007 CA = 2008 AB = 2009 Let ω be an excircle ofABC that touches the line segment BC at D and touches extensions of lines AC and AB atE and F respectively (so that C lies on segment AE and B lies on segment AF ) Let O bethe center of ω Let ` be the line through O perpendicular to AD Let ` meet line EF at GCompute the length DG

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