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nước trên thế giới P2 - Cao
Minh Quang

☺ The best problems from around the world Cao Minh Quang

301
13th Mexican 1999

☺ The best problems from around the world Cao Minh Quang

302
14th Mexican 2000

A1. A, B, C, D are circles such that A and B touch externally at P, B and C touch externally

first composite member of the sequence?
B2. Given an n x n board with squares colored alternately black and white like a chessboard.
An allowed move is to take a rectangle of squares (with one side greater than one square, and
both sides odd or both sides even) and change the color of each square in the rectangle. For
which n is it possible to end up with all the squares the same color by a sequence of allowed
moves?
B3. ABC is a triangle with
∠
B > 90
o
. H is a point on the side AC such that AH = BH and
BH is perpendicular to BC. D, E are the midpoints of AB, BC. The line through H parallel to
AB meets DE at F. Show that
∠
BCF =
∠
ACD.
☺ The best problems from around the world Cao Minh Quang

- a
4
+ a
5
- ... + a
2001
. For which n ≥ 5 can we find m such that 2 ≤ m ≤ n/2 and f(m,n)
> 0?
B2. ABC is a triangle with AB < AC and
∠
A = 2
∠
C. D is the point on AC such that CD =
AB. Let L be the line through B parallel to AC. Let L meet the external bisector of
∠
A at M
and the line through C parallel to AB at N. Show that MD = ND.
B3. A collector of rare coins has coins of denominations 1, 2, ... , n (several coins for each
denomination). He wishes to put the coins into 5 boxes so that: (1) in each box there is at
most one coin of each denomination; (2) each box has the same number of coins and the same
denomination total; (3) any two boxes contain all the denominations; (4) no denomination is
in all 5 boxes. For which n is this possible?
many maximum-length chains are there?

B2. A trio is a set of three distinct integers such that two of the numbers are divisors or
multiples of the third. Which trio contained in {1, 2, ... , 2002} has the largest possible sum?
Find all trios with the maximum sum.
B3. ABCD is a quadrilateral with
∠
A =
∠
B = 90
o
. M is the midpoint of AB and
∠
CMD =
90
o
. K is the foot of the perpendicular from M to CD. AK meets BD at P, and BK meets AC
at Q. Show that
∠
AKB = 90
o
and KP/PA + KQ/QB = 1.
☺ The best problems from around the world Cao Minh Quang

305
17th Mexican 2003


n
is the set
of all numbers that can be obtained by a sequence of allowed moves starting with n. For
example, we can form 5 → 11 → 35 so 5, 11 and 35 belong to S
5
. We call m and n
compatible if S
m
∩ S
n
is non-empty. Which members of {1, 2, 3, ... , 2002} are compatible
with 2003?

☺ The best problems from around the world Cao Minh Quang

306
307
34th Polish 1983

A1. The angle bisectors of the angles A, B, C in the triangle ABC meet the circumcircle again
at K, L, M. Show that |AK| + |BL| + |CM| > |AB| + |BC| + |CA|.
A2. For given n, we choose k and m at random subject to 0 ≤ k ≤ m ≤ 2
n
. Let p
n
be the
probability that the binomial coefficient mCk is even. Find lim
n→∞
p
n
.
A3. Q is a point inside the n-gon P
1
P
2
...P
n
which does not lie on any of the diagonals. Show
that if n is even, then Q must lie inside an even number of triangles P
i
P
j
P
k
.
B1. Given a real numbers x


☺ The best problems from around the world Cao Minh Quang

308
35th Polish 1984

A1. X is a set with n > 2 elements. Is there a function f : X → X such that the composition f
n-
1
is constant, but f
n-2
is not constant?
A2. Given n we define a
i,j
as follows. For i, j = 1, 2, ... , n, a
i,j
= 1 for j = i, and 0 for j ≠ i. For
i = 1, 2, ... , n, j = n+1, ... , 2n, a
i,j
= -1/n. Show that for any permutation p of (1, 2, ... , 2n) we
have ∑
i=1
n
|∑
k=1
n
a
i,p(k)
| ≥ n/2.
A3. W is a regular octahedron with center O. P is a plane through the center O. K(O, r

2
+ ... + α
j
and let p(n) be the probability that the sequence β
1
,
β
2
, ... , β
n
includes the value n. Find p(n) in terms of p(n-1) and p(n-2).
B2. Six disks with diameter 1 are placed so that they cover the edges of a regular hexagon
with side 1. Show that no vertex of the hexagon is covered by two or more disks.
B3. There are 1025 cities, P
1
, ... , P
1025
and ten airlines A
1
, ... , A
10
, which connect some of the
cities. Given any two cities there is at least one airline which has a direct flight between them.
Show that there is an airline which can offer a round trip with an odd number of flights.

i
b
i

≥ 100. Show that the square can be covered with rectangles R
i
with sides length (a
i
, b
i
)
parallel to the square sides.
A3. The function f : R → R satisfies f(3x) = 3f(x) - 4f(x)
3
for all real x and is continuous at x
= 0. Show that |f(x)| ≤ 1 for all x.
B1. P is a point inside the triangle ABC is a triangle. The distance of P from the lines BC,
CA, AB is d
a
, d
b
, d
c
respectively. Show that 2/(1/d
a
+ 1/d
b
+ 1/d
c
) < r < (d


☺ The best problems from around the world Cao Minh Quang

310
37th Polish 1986

A1. A square side 1 is covered with m
2
rectangles. Show that there is a rectangle with
perimeter at least 4/m.
A2. Find the maximum possible volume of a tetrahedron which has three faces with area 1.
A3. p is a prime and m is a non-negative integer < p-1. Show that ∑
j=1
p
j
m
is divisible by p.
B1. Find all n such that there is a real polynomial f(x) of degree n such that f(x) ≥ f '(x) for all
real x.
B2. There is a chess tournament with 2n players (n > 1). There is at most one match between
each pair of players. If it is not possible to find three players who all play each other, show
that there are at most n
2
matches. Conversely, show that if there are at most n

311
38th Polish 1987

A1. There are n ≥ 2 points in a square side 1. Show that one can label the points P
1
, P
2
, ... , P
n

such that ∑
i=1
n
|P
i-1
- P
i
|
2
≤ 4, where we use cyclic subscripts, so that P
0
means P
n
.
A2. A regular n-gon is inscribed in a circle radius 1. Let X be the set of all arcs PQ, where P,
Q are distinct vertices of the n-gon. 5 elements L
1
, L
2
, ... , L

2
-n+11 is the product of four primes (not necessarily
distinct).
B3. A plane is tiled with regular hexagons of side 1. A is a fixed hexagon vertex. Find the
number of paths P such that (1) one endpoint of P is A, (2) the other endpoint of P is a
hexagon vertex, (3) P lies along hexagon edges, (4) P has length 60, and (5) there is no shorter
path along hexagon edges from A to the other endpoint of P. ☺ The best problems from around the world Cao Minh Quang

312

1
, p
2
, ... , p
n
) of (1, 2, ... , n) define X(P) as the number of j such
that p
i
< p
j
for every i < j. What is the expected value of X(P) if each permutation is equally
likely?
A3. W is a polygon. W has a center of symmetry S such that if P belongs to W, then so does
P', where S is the midpoint of PP'. Show that there is a parallelogram V containing W such
that the midpoint of each side of V lies on the border of W.
B1. d is a positive integer and f : [0,d] → R is a continuous function with f(0) = f(d). Show
that there exists x
∈
[0,d-1] such that f(x) = f(x+1).
B2. The sequence a
1
, a
2
, a
3
, ... is defined by a
1
= a
2
= a


☺ The best problems from around the world Cao Minh Quang

313
40th Polish 1989

A1. An even number of politicians are sitting at a round table. After a break, they come back
and sit down again in arbitrary places. Show that there must be two people with the same
number of people sitting between them as before the break.
A2. k
1
, k
2
, k
3
are three circles. k
2
and k
3
touch externally at P, k
3
and k
1
touch externally at Q,

.
B2. Three circles of radius a are drawn on the surface of a sphere of radius r. Each pair of
circles touches externally and the three circles all lie in one hemisphere. Find the radius of a
circle on the surface of the sphere which touches all three circles.
B3. Show that for positive reals a, b, c, d we have ((ab + ac + ad + bc + bd + cd)/6)
1/2
≥ ((abc
+ abd + acd + bcd)/4)
1/3
.

☺ The best problems from around the world Cao Minh Quang

2
2
+x
3
x
4
) + ... +
x
n
2
/(x
n
2
+x
1
x
2
) ≤ n-1.
A3. In a tournament there are n players. Each pair of players play each other just once. There
are no draws. Show that either (1) one can divide the players into two groups A and B, such
that every player in A beat every player in B, or (2) we can label the players P
1
, P
2
, ... , P
n
such
that P
i
beat P

k+1
.
B3. Show that ∑
k=0
[n/3]
(-1)
k
nC3k is a multiple of 3 for n > 2. (nCm is the binomial
coefficient).

n
on the line y = 0, such that each P
i
is a lattice point and for each i < n, P
i
and
P
i+1
are adjacent lattice points a distance 1 apart. Show that F(n) = (2n)Cn.
A3. N is a number of the form ∑
k=1
60
a
k
k
kk
, where each a
k
= 1 or -1. Show that N cannot be a
5th power.
B1. Let V be the set of all vectors (x,y) with integral coordinates. Find all real-valued
functions f on V such that (a) f(v
) = 1 for all v of length 1; (b) f(v + w) = f(v) + f(w) for all
perpendicular v
, w
∈
V. (The vector (0,0) is considered to be perpendicular to any vector.)
B2. k
1
, k


☺ The best problems from around the world Cao Minh Quang

316
43rd Polish 1992

A1. Segments AC and BD meet at P, and |PA| = |PD|, |PB| = |PC|. O is the circumcenter of the
triangle PAB. Show that OP and CD are perpendicular.
A2. Find all functions f : Q
+
→ Q
+

, f
2
, ... are defined on the reals by f
0
(x) = 8 for all x, f
n+1
(x) = √(x
2
+
6f
n
(x)). For all n solve the equation f
n
(x) = 2x.
B2. The base of a regular pyramid is a regular 2n-gon A
1
A
2
...A
2n
. A sphere passes through the
apex S of the pyramid and cuts the edge SA
i
at B
i
(for i = 1, 2, ... , 2n). Show that ∑ SB
2i-1
= ∑
SB
2i


317
44th Polish 1993 A1. Find all rational solutions to:
t
2
- w
2
+ z
2
= 2xy
t
2
- y
2
+ w
2
= 2xz
t
2
- w
2
+ x
2
= 2yz.
A2. A circle center O is inscribed in the quadrilateral ABCD. AB is parallel to and longer
than CD and has midpoint M. The line OM meets CD at F. CD touches the circle at E. Show
that DE = CF iff AB = 2CD.

☺ The best problems from around the world Cao Minh Quang

318
45th Polish 1994

A1. Find all triples (x,y,z) of positive rationals such that x + y + z, 1/x + 1/y + 1/z and xyz are
all integers.
A2. L, L' are parallel lines. C is a circle that does not intersect L. A is a variable point on L.
The two tangents to C from A meet L' in two points with midpoint M. Show that the line AM
passes through a fixed point (as A varies).
A3. k is a fixed positive integer. Let a
n
be the number of maps f from the subsets of {1, 2, ... ,
n} to {1, 2, ... , k} such that for all subsets A, B of {1, 2, ... , n} we have f(A ∩ B) = min(f(A),
f(B)). Find lim

(n > 3) satisfy ∑ x
i
= 0, &sum x
i
2
= 1. Show that four of
the numbers a, b, c, d must satisfy a + b + c + nabc ≤ ∑ x
i
3
≤ a + b + d + nabd.
3
mod p.
B1. The positive reals x
1
, x
2
, ... , x
n
have harmonic mean 1. Find the smallest possible value
of x
1
+ x
2
2
/2 + x
3
3
/3 + ... + x
n
n
/n.
B2. An urn contains n balls labeled 1, 2, ... , n. We draw the balls out one by one (without
replacing them) until we obtain a ball whose number is divisible by k. Find all k such that the
expected number of balls removed is k.
B3. PA, PB, PC are three rays in space. Show that there is just one pair of points B', C' with
B' on the ray PB and C' on the ray PC such that PC' + B'C' = PA + AB' and PB' + B'C' = PA +
AC'.

A2. P is a point inside the triangle ABC such that
∠
PBC =
∠
PCA <
∠
PAB. The line PB
meets the circumcircle of ABC again at E. The line CE meets the circumcircle of APE again
at F. Show that area APEF/area ABP does not depend on P.
A3. a
i
, x
i
are positive reals such that a
1
+ a
2
+ ... + a
n
= x
1
+ x
2
+ ... + x
n
= 1. Show that 2 ∑
i<j

x
i

n+1
= xnp(x
n
)/q(x
n
).
Find all n such that x
n
= 111111.
B3. Let S be the set of permutations a
1
a
2
...a
n
of 123...n such that a
i
≥ i. An element of S is
chosen at random. Find all n such that the probability that the chosen permutation satisfies a
i

≤ i+1 exceeds 1/3.
(x
n+1
+x
n
) for n = 1, 2, 3, 4.
Find x
7
.
A2. Find all real solutions to 3(x
2
+ y
2
+ z
2
) = 1, x
2
y
2
+ y
2
z
2
+ z
2
x
2
= xyz(x + y + z)
3
.
A3. ABCD is a tetrahedron. DE, DF, DG are medians of triangles DBC, DCA, DAB. The

. F is a point on
the side AB such that AF/BF = AE/BC. Show that
∠
FCE =
∠
FDE and
∠
FEC =
∠
BDC.
B3. Given any n points on a unit circle show that at most n
2
/3 of the segments joining two
points have length > √2.


, ... defined by x
0
= F
k
/F
m
and x
n+1
= (2x
n
- 1)/(1 - x
n
) for x
n
≠ 1, or 1
if x
n
= 1, contains the number 1.
A3. PABCDE is a pyramid with ABCDE a convex pentagon. A plane meets the edges PA,
PB, PC, PD, PE in points A', B', C', D', E' distinct from A, B, C, D, E and P. For each of the
quadrilaterals ABB'A', BCC'B, CDD'C', DEE'D', EAA'E' take the intersection of the
diagonals. Show that the five intersections are coplanar.
B1. Define the sequence a
1
, a
2
, a
3
, ... by a
1


☺ The best problems from around the world Cao Minh Quang

323
50th Polish 1999

A1. D is a point on the side BC of the triangle ABC such that AD > BC. E is a point on the
side AC such that AE/EC = BD/(AD-BC). Show that AD > BE.
A2. Given 101 distinct non-negative integers less than 5050 show that one choose four a, b,
c, d such that a + b - c - d is a multiple of 5050.
A3. Show that one can find 50 distinct positive integers such that the sum of each number
and its digits is the same.
B1. For which n do the equations have a solution in integers:
x
1

n

x
n
2
+ x
1
2
+ 50 = 16x
n
+ 12x
1

B2. Show that ∑
1≤i<j≤n
(|a
i
-a
j
| + |b
i
-b
j
|) ≤ ∑
1≤i<j≤n
|a
i
-b
j
| for all integers a☺ The best problems from around the world Cao Minh Quang

324
51st Polish 2000

A1. How many solutions in non-negative reals are there to the equations:
x
1
+ x
n
2
= 4x
n

x
2
+ x
1
2
= 4x
1

...
x
n
+ x

n-1
+ a
n-2
+ 2000. Show that the sequence is bounded.
B1. PA
1
A
2
...A
n
is a pyramid. The base A
1
A
2
...A
n
is a regular n-gon. The apex P is placed so
that the lines PA
i
all make an angle 60
o
with the plane of the base. For which n is it possible to
find B
i
on PA
i
for i = 2, 3, ... , n such that A
1
B
2