A. Number Theory B. Calculus and Analysis

TIM OLIMPIADE MATEMATIKA tk. MAHASISWA
PEMBINAAN PESERTA IMC 2015
Sesi Mandiri
24-25 Juli 2015
A.
Number Theory
1. (i) Show that n + 1 divides 2n
.
n
(ii) Show that for any prime p and integer k such that 1 ≤ k ≤ p − 1, p divides kp .
(iii) Show that for all primes p and integers x, y, we have (x+y)p ≡ xp +y p (mod p).
2. Let f be a polynomial with positive integer coefficients. Prove that if n is a positive
integer, then f (n) divides f (f (n) + 1) if and only if n = 1.
3. Misalkan f adalah polinomial berkoefisien real sehingga f (x) rasional untuk setiap
x rasional. Haruskah f polinomial berkoefisien rasional?
4. Misalkan f polinomial berkoefisien bilangan bulat sehingga terdapat a, b ∈ Z dengan
31 habis membagi f (a) dan 65 habis membagi f (b). Tunjukkan bahwa terdapat
c ∈ Z sehingga 2015 habis membagi f (c).
5. Misalkan Sn menyatakan jumlah n bilangan prima pertama. Buktikan bahwa untuk
setiap bilangan asli n terdapat bilangan kuadrat sempurna antara Sn dan Sn+1 .
6. Determine the set of all pairs (a, b) of positive integers for which the set of positive
integers can be decomposed into 2 sets A and B so that a · A = b · B.
7. (Putnam 1998) Misalkan A1 = 0, A1 = 1. Untuk n ≥ 2, An didefinisikan sebagai
bilangan yang didapatkan dengan menuliskan secara berurutan An−1 dengan An−2 .
Jadi, A3 = 10, A4 = 101, A5 = 10110, dan seterusnya. Tentukan semua bilangan
asli n sehingga 11 | An .
8. Is the set of positive integers n such that n! + 1 divides (2015n)! finite or infinite?
B.
Calculus and Analysis
9. Let 0 < a < b. Prove that
Rb
a
2
2
2
(x2 + 1)e−x dx ≥ e−a − e−b .
10. Let p(x) be a polynomial that is nonnegative for all real x. Prove that for some k,
there are polynomials f1 (x), f2 (x), . . . , fk (x) such that
p(x) =
k
X
j=1
1
(fj (x))2 .
11. Barisan (xn ) didefinisikan dengan x0 = 0, x1 = 1 dan
1
1
xn + 1 −
xn−1
xn+1 =
n+1
n+1
untuk setiap n ≥ 1. Buktikan bahwa xn konvergen dan tentukan limitnya.
12. Misalkan a bilangan real pada interval terbuka (0, 1), n bilangan bulat positif lebih
2
dari 1 dan fn : R → R dengan fn (x) = x + xn . Buktikan bahwa
an + a2
(fn ◦ fn ◦ · · · ◦ fn )(a) <
.
|
{z
}
(1 − a)n + a
n
13. Define the sequence a0 , a1 , . . . inductively by a0 = 1, a1 = 12 , and
an+1
Show that the series
∞
X
ak+1
k=0
ak
na2n
,
=
1 + (n + 1)an
∀n ≥ 1.
converges and determine its value.
14. Let (an ) ⊂ ( 21 , 1). Define the sequence
x0 = 0,
xn+1 =
an+1 + xn
,
1 + an+1 xn
for n = 0, 1, 2, . . . .
Is this sequence convergent? If yes find the limit.
15. Find all functions f : R → R such that for any a < b, f ([a, b]) is an interval of
length b − a.
16. Let f : R → R be a continuous function. A point x is called a shadow point if there
exists a point y ∈ R with y > x such that f (y) > f (x). Let a < b be real numbers
and suppose that:
(i) all the points of the open interval I = (a, b) are shadow points, and
(ii) a and b are not shadow points.
Prove that:
a) f (x) ≤ f (b) for all a < x < b, and
b) f (a) = f (b).
17. Let all roots of n degree polynomial P (z) with complex coefficients lie on the unit
circle. Show that the roots of 2zP 0 (z) − P (z) lie on the same circle.
18. Misalkan polinomial P (x) memiliki koefisien real dengan akar-akar kompleks yang
berbeda-beda z1 , z2 , . . . , zn . Tunjukkan bahwa untuk setiap akar kompleks z dari
polinomial P 0 (x), terdapat bilangan real a1 , a2 , . . . , an dengan 0 ≤ ai ≤ 1 untuk
setiap i = 1, 2, . . . , n sehingga
z = a1 z1 + a2 z2 + · · · + an zn .
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C.
Combinatorics
19. For every positive integer n, let p(n) denote the number of ways to express n as a
sum of positive integers. For instance, p(4) = 5 because
4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1.
Also define p(0) = 1.
Prove that p(n) − p(n − 1) is the number of ways to express n as a sum of integers
each of which is strictly greater than 1.
20. 200 students participated in a math contest. They had 6 problems to solve. Each
problem was correctly solved by at least 120 participants. Prove that there must
be 2 participants such that every problem was solved by at least one of these two
students.
21. For a partition π of {1, 2, 3, 4, 5, 6, 7, 8, 9}, let π(x) be the number of elements in the
part containing x. Prove that for any two partitions π and π 0 , there are two distinct
numbers x and y in {1, 2, 3, 4, 5, 6, 7, 8, 9} such that π(x) = π(y) and π 0 (x) = π 0 (y).
22. Given a positive integer n, what is the largest k such that the numbers 1, 2, . . . , n
can be put into k boxes so that the sum of the numbers in each box is the same?
(When n = 8, the example {1, 2, 3, 6}, {4, 8}, {5, 7} shows that the largest k is at
least 3.)
23. In a round robin tournament, a cyclic 3−sets occurs occasionally, that is a set
{a, b, c} of three teams where a beats b, b beats c, and c beats a. If 23 teams play
in a round robin tournament (without ties), what is the largest number of cyclic
3−sets that occur?
24. Call a subset S of {1, 2, . . . , n} mediocre if it has the following property: Whenever a
and b are elements of S whose average is an integer, that average is also an element
of S. Let A(n) be the number of mediocre subsets of {1, 2, . . . , n}. [For instance,
every subset of {1, 2, 3} except {1, 3} is mediocre, so A(3) = 7.] Find all positive
integers n such that A(n + 2) − 2A(n + 1) + A(n) = 1.
25. Misalkan n, m adalah bilangan bulat positif dengan n ≥ 2m dan f (n, m) menyatakan
banyaknya string biner dengan panjang n yang memuat blok 01 sebanyak m buah.
Sebagai contoh, barisan 100 01 01 11 01 1 memuat 3 buah blok 01. Buktikan bahwa
n+1
f (n, m) =
.
2m + 1
26. An alien race has three genders: male, female and emale. A married triple consists
of three persons, one from each gender who all like each other. Any person is allowed
to belong to at most one married triple. The feelings are always mutual ( if x likes
3
y then y likes x). The race wants to colonize a planet and sends n males, n females
and n emales. Every expedition member likes at least k persons of each of the two
other genders. The problem is to create as many married triples so that the colony
could grow.
a) Prove that if n is even and k ≥ 1/2 then there might be no married triple.
b) Prove that if k ≥ 3n/4 then there can be formed n married triple ( i.e. everybody
is in a triple).
D.
Matrix
27. a) Show that ∀n ∈ N0 , ∃A ∈ Rn×n : A3 = A + I.
b) Show that det(A) > 0, ∀A fulfilling the above condition.
28. Let n ≥ 2 be an integer. What is the minimal and maximal possible rank of an
n × n matrix whose n2 entries are precisely the numbers 1, 2, . . . , n2 ?
29. Let A, B ∈ Cn×n with ρ(AB − BA) = 1. Show that (AB − BA)2 = 0.
30. Let A, B ∈ Rn×n satisfying A2 + B 2 = AB. Show that if AB − BA is invertible,
then n is divisible by 3.
31. Let n be a fixed positive integer. Determine the smallest possible rank of an n × n
matrix that has zeros along the main diagonal and strictly positive real numbers off
the main diagonal.
32. Does there exist a real 3 × 3 matrix A such that tr(A) = 0 and A2 + At = I? (tr(A)
denotes the trace of A, At the transpose of A, and I is the identity matrix.)
33. Let A, B ∈ Mn (C) be two n × n matrices such that
A2 B + BA2 = 2ABA
Prove there exists k ∈ N such that
(AB − BA)k = 0n
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