Problems Submitted to the IMO Problem Committee
39th International Mathematical Olympiad, Taiwan 1998
1. A convex quadrilateral ABCD has perpendicular diagonals. The perpendicular
bisectors of AB and CD meet at a unique point P inside ABCD. Prove that ABCD
is cyclic if and only if the triangles ABP and CDP have equal areas.
2. Let ABCD be a circumscribed quadrilateral. Given that AE/EB = CF/FD and
PE/PF = AB/CD prove that area(APD)/area(BPC) is constant with respect to the
choice of E and F.
3. Let I be the incenter of a triangle ABC. Let K, L and M be the points of tangency
of the incircle of ABC with AB, BC and CA. The line t passes through B and is
parallel to KL. The lines MK and ML intersect t at the points R and S. Prove that
angle RIS is acute.
4. Let ABC be a triangle and M and N two internal isogonic points. Prove that:
1=
⋅
⋅
�
sym ACAB
ANAM
5. Let ABC be a triangle, H its orthocenter, O its circumcenter and R its
circumradius. Let D be the reflection of A across BC, E be that of B across CA
and F that of C across AB. Prove that D, E and F are collinear if and only if OH =
2R.
6. Let ABCDEF be a convex hexagon. If B + D + F = 360 and 1=
FA
EF
DE
CD
BC
AB
then
1=
BD
FD
EF
AE
CA
BC
.
7. Let ABC be a triangle such that angle ACB is twice angle ABC. Let D be the
point on the side BC such that CD = 2BD. The segment AD is extended to E so
that AD = DE. Prove that ECB + 180 = 2EBC.
8. Let L be a circle and ABC an inscribed triangle so that BC is a diameter. BC
intersects the tangent through A to L in D. E is the reflection of A across BC, X
the foot of the perpendicular from A to BE, and Y the midpoint of AX. Let the
line BY meet L again at Z. Prove that BD is tangent to the circumcircle of ADZ.
9. Prove that if a1,
, an are positive real numbers with sum less than 1 then the
inequality holds:
1
2121
2121 1
)1)...(1)(1)(...(
)...1(...
+
≤
−−−+++
−−−−
n
nn
nn
naaaaaa
aaaaaa
.
10. Prove that if r1, r2,
, rn ≥ 1 then:
n
n
n
i i rrr
n
r ...11
1
211 +
≥
+
�
=
.
11. Prove that if xyz = 1 then:
4
3
)1)(1(
3
≥
++
�
zy
x
.
12. Define the numbers c(n, k), k ≤ n this way :
c(n, 0) = c(n, n) = 1, n nonnegative integer.
c(n+1, k) = 2kc(n, k) + c(n, k-1), k ≤ n.
Prove that c(n, k) = c(n-k, k).
13. Find the least possible value of f(1998) where f is a function defined from the set
of nonnegative integers to the same set so that for all nonnegative integers m, n we
have f(n2f(m)) = m[f(n)]2.
14. Find all pairs of whole numbers (x, y) so that x2y + x + y divides xy2 + y + 7.
15. Find all pairs of real numbers (a, b) so that a[bn] = b[an] for all whole positive n.
16. Find the smallest n ≥ 4 so that for any n integer numbers one can find among these
four: a, b, c and d so that a + b = c + d (mod 20).
17. The sequence ai is defined this way: an+1 is the smallest whole number greater
than an so that ai + aj is different from 3ak (i, j, k not necessarily distinct). Find
a1998.
18. Determine all positive integers n for which there exists an integer m so that 2n - 1
divides m2 + 9.
19. For any positive integer n let d(n) denote the number of n's positive divisors
(including 1 and itself). Determine all positive integers m for which there exists a
positive integer n so that d(n2)/d(n) = m.
20. Prove that for any n positive integer there is an n digit number, with none of its
digits 0, so that it is divisible by the sum of its digits.
21. The numbers ai are some positive integer numbers so that any positive integer n
can be uniquely written as n = ai+2aj+4ak (i, j, k not necessarily distinct). Find
a1998.
22. A matrix with real entries has the property that the sum of the elements on each
row and column is integer. Prove that all the elements can be changed into either
their lower or upper integer part so that the sum of elements on each row or
column remains unchanged.
23. Let n be an integer greater than 1. A number is called attainable if is equal to 1 or
can be obtained from 1 using a finite sequence of the following operations:
the operation is either + or *.
+ and * alternate.
+ is done either with 2 or with n.
is done either with 2 or with n.
Prove that if n ≥ 9 then there is an infinite number of numbers that are not
attainable. If n = 3 then all positive integers except 7 are attainable.
24. The numbers from 1 to 9 are written randomly in a row. In a move one may
choose any block of consecutive numbers that are either in an increasing or a
decreasing order and reverse the order of the numbers in it. For example
916532748 may be changed into 913562748 (block 653). Prove that in at most 12
moves one can arrange the numbers increasingly.
25. U is the set of all whole positive numbers less than or equal to n. S, a subset of U,
is called split by a set of elements if a number not in S appears in the set between
two elements of S. (e.g. 1 3 5 4 2 splits 1,2,3 but does not split 3,4,5). Prove that
for any n - 2 subsets of U with at least 2 and at most n-1 elements there exists a set
from U that splits all of them.
26. In a contest there are m candidates and n judges, where n ≥ 3 is odd. Each
candidate is evaluated by each judge with either pass or fail. Suppose that each
pair of judges agrees on at most k candidates. Prove that
n
n
m
k
2
1−≥ .
27. Let K be a 10 complete graph. Each side is colored in one of k colors so that for
any k of the 10 vertices there are k sides with ends in the k vertices that are colored
distinctly. Find k for which this is possible. (Prove that the least k for which it is
possible is 5).
28. On an m x n board there are mn cards with one side white and the other black. At
first all cards have the white side upwards except for one card in the upper left
corner. In a move one can take out a card with black side up and turn all adjacent
cards upside-down. Find all m and n for which all cards may be removed from the
board.
本文档为【IMO预选题1998】,请使用软件OFFICE或WPS软件打开。作品中的文字与图均可以修改和编辑,
图片更改请在作品中右键图片并更换,文字修改请直接点击文字进行修改,也可以新增和删除文档中的内容。
该文档来自用户分享,如有侵权行为请发邮件ishare@vip.sina.com联系网站客服,我们会及时删除。
[版权声明] 本站所有资料为用户分享产生,若发现您的权利被侵害,请联系客服邮件isharekefu@iask.cn,我们尽快处理。
本作品所展示的图片、画像、字体、音乐的版权可能需版权方额外授权,请谨慎使用。
网站提供的党政主题相关内容(国旗、国徽、党徽..)目的在于配合国家政策宣传,仅限个人学习分享使用,禁止用于任何广告和商用目的。