Download Azijas, Klus¯a oke¯ana olimpi¯ade, Skaitl¸u teorija

Āzijas, Klusā okeāna olimpiāde, Skaitļu teorija
APMO1989.2: Prove that the equation
6(6a2 + 3b2 + c2 ) = 5n2
has no solutions in integers except a = b = c = n = 0.
APMO1991.4: During a break, n children at school sit in a circle around their teacher to
play a game. The teacher walks clockwise close to the children and hands out candies to some
of them according to the following rule: He selects one child and gives him a candy, then he
skips the next child and gives a candy to the next one, then he skips 2 and gives a candy to the
next one, then he skips 3, and so on. Determine the values of n for which eventually, perhaps
after many rounds, all children will have at least one candy each.
APMO1992.3: Let n be an integer such that n > 3. Suppose that we choose three numbers
from the set {1, 2, . . . , n}. Using each of these three numbers only once and using addition,
multiplication, and parenthesis, let us form all possible combinations. (a) Show that if we
choose all three numbers greater than n2 , then the values of these combinations are all distinct.
√
(b) Let p be a prime number such that p ≤ n. Show that the number of ways of choosing three
numbers so that the smallest one is p and the values of the combinations are not all distinct is
precisely the number of positive divisors of p − 1.
APMO1993.2: Find the total number of different integer values the function
5x
f (x) = bxc + b2xc +
+ b3xc + b4xc
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takes for real numbers x with 0 ≤ x ≤ 100.
APMO1993.4: Determine all positive integers n for which the equation
xn + (2 + x)n + (2 − x)n = 0
has an integer as a solution.
APMO1994.3: Let n be an integer of the form a2 + b2 , where a and b are relatively prime
√
integers and such that if p is a prime, p ≤ n, then p divides ab. Determine all such n.
APMO1994.4: Is there an infinite set of points in the plane such that no three points are
collinear, and the distance between any two points is rational?
LV: Vai eksistē bezgalı̄ga plaktnes punktu kopa, tāda ka nekādi trı̄s punkti nav kolineāri, un attālums
starp katriem diviem punktiem ir racionāls skaitlis?
APMO1994.5: You are given three lists A, B, and C. List A contains the numbers of the
form 10k in base 10, with k any integer greater than or equal to 1. Lists B and C contain the
1
same numbers translated into base 2 and 5 respectively:
A
10
100
1000
..
.
B
1010
1100100
1111101000
..
.
C
20
400
13000
..
.
Prove that for every integer n > 1, there is exactly one number in exactly one of the lists B or
C that has exactly n digits.
APMO1995.2: Let a1 , a2 , . . ., an be a sequence of integers with values between 2 and 1995
such that:
(i) Any two of the ai ’s are relatively prime,
(ii) Each ai is either a prime or a product of primes. Determine the smallest possible values of
n to make sure that the sequence will contain a prime number.
APMO1997.2: Find an integer n, where 100 ≤ n ≤ 1997, such that
2n + 2
n
is also an integer.
APMO1998.2: Show that for any positive integers a and b, (36a + b)(a + 36b) cannot be a
power of 2.
APMO1998.5: Find the largest integer n such that n is divisible by all positive integers less
√
than 3 n.
APMO1999.1: Find the smallest positive integer n with the following property: there does
not exist an arithmetic progression of 1999 real numbers containing exactly n integers.
APMO1999.4: Find the smallest positive integer n with the following property: there does
not exist an arithmetic progression of 1999 real numbers containing exactly n integers.
APMO2000.1: Compute the sum:
x3i
i=0 1−3xi +3x2i
P101
for xi =
i
101 .
APMO2001.1: For a positive integer n let S(n) be the sum of digits in the decimal representation of n. Any positive integer obtained by removing several (at least one) digits from the
right-hand end of the decimal representation of n is called a stump of n. Let T (n) be the sum
of all stumps of n. Prove that n = S(n) + 9T (n).
APMO2001.1: Find the largest positive integer N so that the number of integers in the set
{1, 2, . . . , N } which are divisible by 3 is equal to the number of integers which are divisible by
5 or 7 (or both).
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APMO2001.1: A point in the plane with a cartesian coordinate system is called a mixed
point if one of its coordinates is rational and the other one is irrational. Find all polynomials
with real coefficients such that their graphs do not contain any mixed point.
LV: Punktu Dekarta koordinātu sistēmā sauc par jauktu punktu, ja viena no tā koordinātēm ir
racionāla un otra ir iracionāla. Atrast visus polinomus ar reāliem koeficientiem, kuru grafiki nesatur
nevienu jauktu punktu.
APMO2002.2: Find all positive integers a and b such that
a2 + b
b2 − a
and
b2 + a
a2 − b
are both integers.
APMO2003.3: Let k ≥ 14 be an integer, and let pk be the largest prime number which is
strictly less than k. You may assume that pk ≥ 3k/4. Let n be a composite integer. Prove:
(a) if n = 2pk , then n does not divide (n − k)!;
(b) if n > 2pk , then n divides (n − k)!.
APMO2004.1: Determine all finite nonempty sets S of positive integers satisfying
i+j
(i, j)
is an element of S for all i,j in S,
where (i, j) is the greatest common divisor of i and j.
APMO2004.4: For a real number x, let bxc stand for the largest integer that is less than or
equal to x. Prove that
(n − 1)!
n(n + 1)
is even for every positive integer n.
APMO2005.1: Prove that for every irrational real number a, there are irrational real numbers
b and b0 so that a + b and ab0 are both rational while ab and a + b0 are both irrational.
LV: Pierādı̄t, ka ikvienam iracionālam skaitlim a, eksistē iracionāli reāli skaitļi b un b0 , ka a + b un
ab0 abi ir racionāli, bet ab un a + b0 abi ir iracionāli.
APMO2006.2: Prove that every positive integer can be written as a finite sum of distinct
integral powers of the golden ratio.
APMO2006.3: Let p ≥ 5 be a prime and let r be the number of ways of placing p checkers
on a p × p checkerboard so that not all checkers are in the same row (but they may all be in
3
the same column). Show that r is divisible by p5 . Here, we assume that all the checkers are
identical.
APMO2007.1: Let S be a set of 9 distinct integers all of whose prime factors are at most 3.
Prove that S contains 3 distinct integers such that their product is a perfect cube.
APMO2008.5: Let a, b, c be integers satisfying 0 < a < c − 1 and 1 < b < c. For each k,
0 ≤ k ≤ a, Let rk , 0 ≤ rk < c be the remainder of kb when divided by c. Prove that the two
sets {r0 , r1 , r2 , · · · , ra } and {0, 1, 2, · · · , a} are different.
APMO2009.4: Prove that for any positive integer k, there exists an arithmetic sequence
ak
a1 a2 a3
b1 , b2 , b3 , ..., bk of rational numbers, where ai , bi are relatively prime positive integers for each
i = 1, 2, ..., k such that the positive integers a1 , b1 , a2 , b2 , ..., ak , bk are all distinct.
APMO2009.5: Larry and Rob are two robots travelling in one car from Argovia to Zillis.
Both robots have control over the steering and steer according to the following algorithm: Larry
makes a 90 degrees left turn after every ` kilometer driving from start, Rob makes a 90 degrees
right turn after every r kilometer driving from start, where ` and r are relatively prime positive
integers. In the event of both turns occurring simultaneously, the car will keep going without
changing direction. Assume that the ground is flat and the car can move in any direction. Let
the car start from Argovia facing towards Zillis. For which choices of the pair (`, r) is the car
guaranteed to reach Zillis, regardless of how far it is from Argovia?
APMO2010.2: For a positive integer k, call an integer a pure k-th power if it can be represented as mk for some integer m. Show that for every positive integer n, there exists n distinct
positive integers such that their sum is a pure 2009-th power and their product is a pure 2010-th
power.
APMO2011.1: Let a, b, c be positive integers. Prove that it is impossible to have all of the
three numbers a2 + b + c, b2 + c + a, c2 + a + b to be perfect squares.
APMO2012.3: Determine all the pairs (p, n) of a prime number p and a positive integer n
p +1
for which npn +1
is an integer.
n2 + 1
APMO2013.2: Determine all positive integers n for which √ 2
is an integer. Here brc
b nc + 2
denotes the greatest integer less than or equal to r.
APMO2013.3:
Xn by
For 2k real numbers a1 , a2 , ..., ak , b1 , b2 , ..., bk define a sequence of numbers
Xn =
k
X
bai n + bi c
(n = 1, 2, ...).
i=1
If the sequence XN forms an arithmetic progression, show that
Here brc denotes the greatest integer less than or equal to r.
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Pk
i=1 ai
must be an integer.
APMO2014.3: Find all positive integers n such that for any integer k there exists an integer
a for which a3 + a − k is divisible by n.
APMO2014.4: Let n and b be positive integers. We say n is b-discerning if there exists a set
consisting of n different positive integers less than b that has no two different subsets U and V
such that the sum of all elements in U equals the sum of all elements in V .
(a) Prove that 8 is 100-discerning.
(b) Prove that 9 is not 100-discerning.
APMO2015.3: A sequence of real numbers a0 , a1 , ... is said to be good if the following three
conditions hold.
(i) The value of a0 is a positive integer.
(ii) For each non-negative integer i we have ai+1 = 2ai + 1 or ai+1 =
ai
ai +2
(iii) There exists a positive integer k such that ak = 2014.
Find the smallest positive integer n such that there exists a good sequence a0 , a1 , ... of real
numbers with the property that an = 2014.
APMO2015.5: Determine all sequences a0 , a1 , a2 , . . . of positive integers with a0 ≥ 2015 such
that for all integers n ≥ 1:
(i) an+2 is divisible by an ;
(ii) |sn+1 − (n + 1)an | = 1, where sn+1 = an+1 − an + an−1 − · · · + (−1)n+1 a0 .
APMO2016.2: A positive integer is called fancy if it can be expressed in the form
2a1 + 2a2 + · · · + 2a100 ,
where a1 , a2 , · · · , a100 are non-negative integers that are not necessarily distinct. Find the
smallest positive integer n such that no multiple of n is a fancy number.
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