Download Selected MOSP Problems 1. (a) Let P(x)

Selected MOSP Problems
1. (a) Let P (x) be a polynomial with rational coefficients such that P (x) = n has a
rational root for each positive integer n. Determine all such polynomials P (x).
(b) Determine if there exists a polynomial Q(x) of degree at least 2 with nonnegative
integer coefficients such that for each prime p, Q(p) is also a prime.
2. (a) For positive integers m and n define
½
φ(n) if n divides m;
φm (n) =
0
otherwise,
where φ(n) counts the number ofPpositive integers between 1 and n which are
relatively prime to n. Show that d|n φm (d) = gcd(m, n).
(b) The function ψ from the set N of positive integers into itself is defined by the
equality
n
X
ψ(n) =
(k, n),
n ∈ N,
k=1
where (k, n) denotes the greatest common divisor of k and n.
(1) Prove that ψ(mn) = ψ(m)ψ(n) for every two relatively prime m, n ∈ N.
(2) Prove that for each a ∈ N the equation ψ(x) = ax has a solution.
(3) Find all a ∈ N such that the equation ψ(x) = ax has a unique solution
(c) Jess has 3 pegs and disks of different sizes. Jess is supposed to transfer the disks
from one peg to another, and the disks have to be sorted so that for any peg the
disk at the bottom is the largest on that peg. (Discs above the bottom one may
be in any order.) There are n disks sorted from largest on bottom to smallest on
top at the start. Determine the minimum number of moves (moving one disk at
a time) needed to move the disks to another peg sorted in the same order.
(d) For each positive integer n, let Dn be the set of all positive divisors of 2n 3n 5n .
Find the maximum size of a subset S of Dn in which no element of S is a proper
divisor of any other.
3. Determine if it is possible to find a cube and a plane such that the distances from the
vertices of the cube to the plane are 0, 1, 2, . . . , 7.
4. Let a1 , a2 , . . . , an be integers such that all the subset sums ai1 + ai2 + · · · + aik , for
1 ≤ i1 < i2 < · · · < ik ≤ n, are nonzero. Prove that it is possible to partition the
set of positive integers into finitely many subsets S1 , S2 , . . . , Sm in a such way that if
Si (1 ≤ i ≤ m) has at least n elements, then a1 x1 + a2 x2 + · · · + an xn 6= 0, where
x1 , x2 , . . . , xn are arbitrary distinct elements in Si .
5. (a) A class of fifteen boys and fifteen girls is seated around a round table. Their
teacher wishes to pair up the students and hand out fifteen tests - one test to
each pair.
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As the teacher is preparing to select the pairs and hand out the tests, he wonders to
himself, “How many seating arrangements would allow me to match up boy/girl
pairs seating next to each other without having to ask any student to change
his or her seat?” Answer the teacher’s question. (Two seating arrangements are
considered the same if one can be obtained from the other by a rotation.)
(b) Let A = (0, 0, 0) be the origin in the three dimensional coordinate space. The
weight of a point is the sum of the absolute values of its coordinates. A point
is a primitive lattice point if all its coordinates are integers with their greatest
common divisor equal to 1. A square ABCD is called a unbalanced primitive
integer square if it has integer side length and the points B and D are primitive
lattice points with different weights.
Show that there are infinitely many unbalanced primitive integer squares ABi Ci Di
such that the plane containing the squares are not parallel to each other.
(c) For a pair of integers a and b, with 0 < a < b < 1000, the set S ⊆ {1, 2, . . . , 2003}
is called a skipping set for (a, b) if for any pair of elements s1 , s2 ∈ S, |s1 − s2 | 6∈
{a, b}. Let f (a, b) be the maximum size of a skipping set for (a, b). Determine
the maximum and minimum values of f .
Let S = {1, 2, 3, . . . , 24, 25}. Compute the number of elements in the largest
subset of S such that no two elements in the subset differ by the square of an
integer.
Let S be a subset of {1, 2, 3, . . . , 1989} such that no two members of S differ by
4 or 7. What is the largest number of elements S can have?
6. (a) Let S be a set with 2002 elements, and let N be an integer with 0 ≤ N ≤ 22002 .
Prove that it is possible to color every subset of S either blue or red so that the
following conditions hold:
(i) the union of any two red subsets is red;
(ii) the union of any two blue subsets is blue;
(iii) there are exactly N red subsets.
(b) In the coordinate plane, color the lattice points which have both coordinates even
black and all other lattice points white. Let P be a polygon with black points
as vertices. Prove that any white point on or inside P lies halfway between two
black points, both of which lie on or inside P .
(c) Let a, b be integers greater than 2. Prove that there exists a positive integer k
and a finite sequence n1 , n2 , . . . , nk of positive integers such that n1 = a, nk = b,
and ni ni+1 is divisible by ni + ni+1 for each i (1 ≤ i < k).
(d) Find all ordered triples of primes (p, q, r) such that
p | q r + 1,
q | rp + 1,
r | pq + 1.
Let n be a positive integer. Consider sequences a0 , a1 , . . . , an such that ai ∈
{1, 2, . . . , n} for each i and an = a0 .
(i) Call such a sequence good if for all i = 1, 2, . . . , n, ai − ai−1 6≡ i
Suppose that n is odd. Find the number of good sequences.
2
(mod n).
(ii) Call such a sequence great if for all i = 1, 2, . . . , n, ai − ai−1 6≡ i, 2i (mod n).
Suppose that n is an odd prime. Find the number of great sequences.
(e) At the vertices of a regular hexagon are written six nonnegative integers whose
sum is 2003. Bert is allowed to make moves of the following form: he may pick a
vertex and replace the number written there by the absolute value of the difference
between the numbers written at the two neighboring vertices. Prove that Bert can
make a sequence of moves, after which the number 0 appears at all six vertices.
(f) A 2004 × 2004 array of points is drawn. Find the largest integer n such that it is
possible to draw a convex n-sided polygon whose vertices lie on the points of the
array.
7. Let n be a positive integer greater than or equal to 4. Let S be a sphere with center
O and radius 1, and let H1 , H2 , . . . Hn be n hemispheres with center O and radius 1.
Sphere S is covered by hemispheres H1 , H2 , . . . , Hn . Prove that one can find positive
integers i1 , i2 , i3 , i4 such that sphere S is covered by Hi1 , Hi2 , Hi3 , Hi4 .
8. Mr. Fat decides to go on a diet. His enemy Mr. Taf attempts to destroy his plan by
walking to him across the plane and feeding him chocolate. Luckily, each of n robotic
bodyguards protects Mr. Fat. If our villain passes through any of the bodyguards’
lines of sight—n fixed rays with the bodyguards at their endpoints—all the candy
he is carrying will melt under the bodyguards’ bright red lasers. (The lasers melt
only chocolate, not bodyguards.) For some value N, whenever Mr. Taf starts at least
distance N from Mr. Fat, it is impossible for him to reach any of the bodyguards or
Mr. Fat with the chocolate. (If a bodyguard or Mr. Fat is on a ray, then Mr. Taf can
still “reach” that person.) What is the smallest value of n for which this is possible?
9. Let P be a 1000-sided regular polygon. Some of its diagonals were drawn to obtain
a triangulation the polygon P . (The region inside P is cut into triangular regions,
and the diagonals drawn only intersect at the vertices of P .) Let n be the number of
different lengths of the drawn diagonals. Determine the minimum value of n.
10. Among a group of 120 people, some pairs are friends. A weak quartet is a set of four
people containing exactly one pair of friends. Determine the maximum possible number
of weak quartets.
11. Let a, b, and c be nonnegative real numbers such that
a2 + b2 + c2 + abc = 4.
Prove that
0 ≤ ab + bc + ca − abc ≤ 2.
12. Each of eight boxes contains six balls. Each ball has been colored with one of n colors,
such that no two balls in the same box are the same color, and no two colors occur
together in more than one box. Determine, with justification, the smallest integer n
for which this is possible.
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13. Let n be a positive integer. A sequence of n positive integers (not necessarily distinct)
is called full if it satisfies the following conditions: for each positive integer k ≥ 2, if
the number k appears in the sequence then so does the number k − 1, and moreover
the first occurrence of k − 1 comes before the last occurrence of k. For each n, how
many full sequences are there?
14. Let n be an even positive integer. Show that there is a permutation (a1 , a2 , . . . , an ) of
(1, 2, . . . , n) such that for every 1 ≤ i ≤ n the number ai+1 is in the set
{2ai , 2ai − 1, 2ai − n, 2ai − n − 1},
where an+1 = a1 .
15. There were n participants in a round-robin tournament. It turned out that for every
non-empty set S of players, there was a player C for whom the number of ties in the
games with the players in S is odd (C may be in S or not). Prove that n is even.
√
√
16. Let k be a positive integer. Prove that k + 1 − k is not the real part of the complex
number z with z n = 1 for some positive integer n.
17. Let p(x) be a polynomial with real coefficients such that p(0) = p(n). Prove that there
are n distinct pairs of real numbers (x, y) such that y − x is a positive integer and
p(x) = p(y).
18. Let n be a positive number, and let x1 , x2 , . . . , xn be positive real numbers such that
x1 + x2 + · · · + xn =
Prove that
1
1
1
+
+ ··· + .
x1 x2
xn
1
1
1
+
+ ··· +
≤ 1.
n − 1 + x1 n − 1 + x2
n − 1 + xn
19. Let n be a positive integer, and let p1 , p2 , . . . , pn be distinct primes greater than 3.
Prove that 2p1 p2 ···pn + 1 has at least 4n divisors.
20. Let p be a prime, and let a1 , a2 , . . . , ap+1 be distinct positive integers. Prove that there
are indices i and j, 1 ≤ i < j ≤ p + 1, such that
max{ai , aj }
≥ p + 1.
gcd(ai , aj )
21. Find all pairs of integers m, n ≥ 3 such that there exist infinitely many positive integers
a for which
am + a − 1
an + a2 − 1
is an integer.
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22. Let n be a positive integer, and let (x1 , . . . , xn ), (y1 , . . . , yn ) be two sequences of positive
real numbers. Suppose (z2 , . . . , z2n ) is a sequence of positive real numbers such that
2
zi+j
≥ xi y j
for all 1 ≤ i, j ≤ n.
Let M = max{z2 , . . . , z2n }. Prove that
µ
¶2 µ
¶µ
¶
M + z2 + · · · + z2n
y1 + · · · + yn
x1 + · · · + xn
≥
.
2n
n
n
23. An integer n is good if |n| is not the square of an integer. Determine all integers m
with the following property:
m can be represented, in infinitely many ways, as a sum of three distinct
good integers whose product is the square of an odd integer.
24. Let p be a prime number. Prove that there exists a prime number q such that for every
integer n, the number np − p is not divisible by q.
25. The sequence a0 , a1 , a2 , . . . is defined as follows:
a0 = 2 and ak+1 = 2a2k − 1 for k ≥ 0.
Prove that if an odd prime p divides an , then 2n+3 divided p2 − 1.
26. Let p be a prime number, and let A be the set of positive integers that satisfies the
following conditions:
(i) the set of prime of the elements in A consists of p − 1 elements;
(ii) for any nonempty subset of A, the product of its elements is not a perfect pth
power of an integer.
What is the largest possible number of elements in A?
27. Find all nondecreasing functions f : R → R such that
(i) f (0) = 0 and f (1) = 1;
(ii) f (a) + f (b) = f (a)f (b) + f (a + b − ab) for all real numbers a and b with a < 1 < b.
28. Find all nondecreasing functions f : R → R such that
(i) f (0) = 0 and f (1) = 1;
(ii) f (a) + f (b) = f (a)f (b) + f (a + b − ab) for all real numbers a and b with a < 1 < b.
29. Consider pairs of sequences of positive real numbers
a1 ≥ a2 ≥ a3 ≥ · · ·
and b1 ≥ b2 ≥ b3 ≥ · · · ,
and the sums
An = a1 + a2 + · · · + an
and Bn = b1 + b2 + · · · + bn ,
for n = 1, 2, . . .. Define sequence {ci }i≥1 with ci = min{ai , bi } and sum Cn = c1 + c2 +
· · · + cn for all positive integers n.
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(1) Does there exists a pair of sequences {a}i≥1 and {b}i≥1 such that the sequences
{A}n≥1 and {B}n≥1 are unbounded while the sequence {C}n≥1 is bounded?
(2) Does the answer to question (1) change by assuming additionally that bi = 1i , for
i = 1, 2, . . .?
30. Find all functions f : R → R satisfying the equation
¡
¢
f x2 + y 2 + 2f (xy = (f (x + y))2 for all x, y ∈ R.
31. Let ABCD be a convex quadrilateral with interior angles 2α, 2β, 2γ, 2δ. Prove that
the area of ABCD is equal to
t2 s2
− ,
q
p
where
p = tan α + tan β + tan γ + tan δ,
q = cot α + cot β + cot γ + cot δ,
AB − BC + CD − DA
s =
,
2
AB + BC + CD + DA
t =
.
2
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MOSP
June 17, 2004
Test 2
r1. [3pts] Let set S = {1, 2, . . . , n} and set T be the set of all subsets of S (including S
and the empty set). One tries to choose three (not necessarily distinct) sets from the
set T such that either two of the chosen sets are subsets of the third set or one of the
chosen set is a subset of both of the other two sets. In how many ways can this be
done?
r2. [5pts] Let ABC be an acute triangle. Let A1 be the foot of the perpendicular from A
to side BC, and let AB and AC be the feet of the perpendiculars from A1 to sides AB
and AC, respectively. Line `A passes through A and is perpendicular to line AB AC .
Lines `B and `C are defined analogously. Prove that lines `A , `B , and `C are concurrent.
r3. [6pts] Circle ω is inscribed in convex quadrilateral ABCD, and it touches sides AB,
BC, CD, and DA at A1 , B1 , C1 , and D1 , respectively. Let E, F, G, and H be the midpoints of A1 B1 , B1 C1 , C1 D1 , and D1 A1 , respectively. Prove that quadrilateral EF GH
is a rectangle if and only if ABCD is cyclic.
r4. [7pts] Let n be a positive integer, and let Sn be the set of all positive integer divisors
of n (including 1 and n). Prove that at most half of the elements of Sn end in the digit
3.
b1. Let ABCD be a cyclic quadrilateral. Diagonals AC and BD meet at P . Points
E, F, G, and H are the feet of perpendiculars from P to sides AB, BC, CD, and DA,
respectively. Prove that lines BD, EH, and F G are either concurrent or parallel to
each other.
b2. Let x1 , x2 , . . . , xn and y1 , y2 , . . . , yn be real numbers. Let M = (aij )1≤i,j≤n be the matrix
with entries
½
1, if xi + yj ≥ 0;
aij =
0, if xi + yj < 0.
Suppose that N is an n × n matrix with entries 0 or 1 in such a way that the sum of
the elements in each row and column of N is equal to the corresponding sum for the
matrix M. Prove that M = N.
b3. Let x, y, and z be real numbers. Prove that
xyz(2x+2y−z)(2y+2z−x)(2z+2x−y)+[x2 +y 2 +z 2 −2(xy+yz+zx)](xy+yz+zx)2 ≥ 0.
k1. Let x1 , x2 , . . . , xn and y1 , y2 , . . . , yn be real numbers. Let M = (aij )1≤i,j≤n be the matrix
with entries
½
1, if xi + yj ≥ 0;
aij =
0, if xi + yj < 0.
Suppose that N is an n × n matrix with entries 0 or 1 in such a way that the sum of
the elements in each row and column of N is equal to the corresponding sum for the
matrix M. Prove that M = N.
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k2. Let ABC be an isosceles triangle with CA = CB, whose incenter is I. Point P ,
in the interior of triangle ABC, lies on the circumcircle of triangle AIB. The lines
passing through P parallel to segments CA and CB meet segment AB at D and E,
respectively. The line passing through P parallel to segment AB meets segments CA
and CB at F and G, respectively. Prove that lines DF and EG meet at a point lying
on the circumcircle of triangle ABC.
k3. Let R+ be the set of all positive real numbers. Find all functions f : R+ → R+ that
satisfy the following conditions:
√
√
√
(i) f (xyz) + f (x) + f (y) + f (z) = f ( xy)f ( yz)f ( zx) for all x, y, and z in R+ ;
(ii) f (x) < f (y) for all 1 ≤ x < y.
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MOSP 2001
Test 9
1. Let ABCD be a convex quadrilateral and let O be the point of intersection of its
diagonals. Prove that
(a) if the perimeters of triangles ABC, BCD, CDA, and DAB are equal then ABCD
is a rectangle; and
(b) if the perimeters of triangles ABO, BCO, CDO, and DAO are equal then ABCD
is a rhombus.
2. Let p be a prime number greater than 2, and let ak be the remainder when k p is divided
by p2 , where k = 1, 2, . . . , p − 1. Prove that
a1 + a2 + · · · + ap−1 =
p3 − p2
.
2
3. A region is the intersection of exactly n circular discs. Prove that its boundary consists
of at most 2n − 2 circular arcs.
4. Let n be a positive integer, and let set
M = {(x, y) | x, y ∈ N, 1 ≤ x, y ≤ n}.
Determine the number of functions f defined on M such that
(i) f (x, y) is a nonnegative integer for any (x, y) ∈ M ;
n
X
f (x, y) = n − 1;
(ii) for 1 ≤ x ≤ n,
y=1
(iii) if f (x1 , y1 )f (x2 , y2 ) > 0, then (x1 − x2 )(y1 − y2 ) ≥ 0.
5. Determine all functions f : Z → Z satisfying
f (x41 + x42 + x43 + x44 ) = (f (x1 ))4 + (f (x2 ))4 + (f (x3 ))4 + (f (x4 ))4
for all integers x1 , x2 , x3 , x4 .
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5th ELMO
Lincoln, Nebraska
June 28, 2003
Experimental Lincoln Math Olympiad
Evil Lagrange Multiplier Olympiad
Ever Lasting Math Olympiad
Emoticon-Loving Math Olympiad
Eerily Looming Math Olympiad
Excitement for Little MOppers
1. Let ABCDEF be a convex equilateral hexagon with sides of length 1. Let R1 be the
area of the region contained within both ACE and BDF , and let R2 be the area of
the region within the hexagon outside both triangles. Prove that:
√
3 3
min {[ACE], [BDF ]} + R2 − R1 ≤
.
4
2. In a set of 30 MOPpers, prove that some two MOPpers have an even number of common
friends.
3. Let k be a positive integer for which the equation
2ab + 2bc + 2ca − a2 − b2 − c2 = k
has some solution in positive integers a, b, c. Prove that the equation has a solution for
which a, b, and c are the sides of a (possibly degenerate) triangle.
4. Let x, y, z ≥ 1 be real numbers such that:
1
1
1
+
+
= 1.
x2 − 1 y 2 − 1 z 2 − 1
Prove that:
1
1
1
+
+
≤ 1.
x+1 y+1 z+1
Even Lincoln has a Math Olympiad
Extremely Ludicrous Math Olympiad
Energetic Lightweight Math Olympiad
ELven Math Olympiad
Earn Lots of MOney
Evil Little Math Omelet
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