http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 2 http://www.mathlinks.ro/ vnmath.com... http://www.artofproblems
Trang 11 Let an odd prime p be a given number satisfying 2h 6= 1 (mod p) for all h < p − 1, h ∈ N∗,and an even integer a ∈p
2, p
Let us consider the sequence {an}∞n=0 defined by a0 = a and
an+1= p − bn for n = 0, 1, 2, , where bn is the greatest odd divisor of an Show that {an}
is periodical and find its least positive period
2 Two polynomials f (x) and g(x) with real coefficients are called similar if there exist nonzeroreal number a such that f (x) = q · g(x) for all x ∈ R
I Show that there exists a polynomial P (x) of degree 1999 with real coefficients which satisfiesthe condition: (P (x))2− 4 and (P0(x))2· (x2− 4) are similar
II How many polynomials of degree 1999 are there which have above mentioned property
3 Let a convex polygon H be given Show that for every real number a ∈ (0, 1) there ist 6 distinct points on the sides of H, denoted by A1, A2, , A6 clockwise, satisfying theconditions:
ex-I (A1A2) = (A5A4) = a · (A6A3) II Lines A1A2, A5A4 are equidistant from A6A3
(By (AB) we denote vector AB)
vnmath.com
Trang 21 Let a sequence of positive reals {un}∞n=1 be given For every positive integer n, let kn be theleast positive integer satisfying:
2 Let a triangle ABC inscribed in circle Γ be given Circle Θ lies in angle of triangle andtouches sides AB, AC at M1, N1 and touches internally Γ at P1 The points M2, N2, P2 and
M3, N3, P3 are defined similarly to angles B and C respectively Show that M1N1, M2N2 and
M3N3 intersect each other at their midpoints
3 Let a regular polygon with p vertices be given, where p is an odd prime number At everyvertex there is one monkey An owner of monkeys takes p peanuts, goes along the perimeter
of polygon clockwise and delivers to the monkeys by the following rule: Gives the first peanutfor the leader, skips the two next vertices and gives the second peanut to the monkey at thenext vertex; skip four next vertices gives the second peanut for the monkey at the next vertex after giving the k-th peanut, he skips the 2 · k next vertices and gives k + 1-th for themonkey at the next vertex He does so until all p peanuts are delivered
I How many monkeys are there which does not receive peanuts? II How many edges
of polygon are there which satisfying condition: both two monkey at its vertex receivedpeanut(s)?
http://www.artofproblemsolving.com/
This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page
Page 2 http://www.mathlinks.ro/
vnmath.com
Trang 31 Two circles C1 and C2 intersect at points P and Q Their common tangent, closer to P than
to Q, touches C1 at A and C2 at B The tangents to C1 and C2 at P meet the other circle atpoints E 6= P and F 6= P , respectively Let H and K be the points on the rays AF and BErespectively such that AH = AP and BK = BP Prove that A, H, Q, K, B lie on a circle
2 Let k be a given positive integer Dene x1 = 1 and, for each n > 1, set xn+1to be the smallestpositive integer not belonging to the set {xi, xi+ ik|i = 1, , n} Prove that there is a realnumber a such that xn= [an] for all n ∈ N
3 Two players alternately replace the stars in the expression
∗x2000+ ∗x1999+ + ∗x + 1
by real numbers The player who makes the last move loses if the resulting polynomial has areal root t with |t| < 1, and wins otherwise Give a winning strategy for one of the players
vnmath.com
Trang 41 Let a, b, c be pairwise coprime natural numbers A positive integer n is said to be stubborn
if it cannot be written in the form n = bcx + cay + abz, for some x, y, z ∈ N Determine thenumber of stubborn numbers
2 Let a > 1 and r > 1 be real numbers (a) Prove that if f : R+→ R+ is a function satisfyingthe conditions (i) f (x)2 ≤ axrf (x
a) for all x > 0, (ii) f (x) < 2
Trang 51 Let a sequence of integers {an}, n ∈ N be given, defined by
vnmath.com
Trang 61 Lets consider the real numbers a, b, c satisfying the condition
3 Let a sequence {an}, n ∈ N∗ given, satisfying the condition
0 < an+1− an≤ 2001for all n ∈ N∗
Show that there are infinitely many pairs of positive integers (p, q) such that p < q and ap isdivisor of aq
http://www.artofproblemsolving.com/
This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page
Page 2 http://www.mathlinks.ro/
vnmath.com
Trang 71 Find all triangles ABC for which ∠ACB is acute and the interior angle bisector of BCintersects the trisectors (AX, (AY of the angle ∠BAC in the points N, P respectively, suchthat AB = N P = 2DM , where D is the foot of the altitude from A on BC and M is themidpoint of the side BC.
2 On a blackboard a positive integer n0 is written Two players, A and B are playing a game,which respects the following rules:
− acting alternatively per turn, each player deletes the number written on the blackboard nkand writes instead one number denoted with nk+1 from the set nnk− 1,jnk
2002+ 1
2 .
3 Let m be a given positive integer which has a prime divisor greater than √2m + 1 Find theminimal positive integer n such that there exists a finite set S of distinct positive integerssatisfying the following two conditions:
I m ≤ x ≤ n for all x ∈ S;
II the product of all elements in S is the square of an integer
vnmath.com
Trang 81 Let n ≥ 2 be an integer and consider an array composed of n rows and 2n columns Half ofthe elements in the array are colored in red Prove that for each integer k, 1 < k ≤jn
2
k+ 1,there exist k rows such that the array of size k × 2n formed with these k rows has at least
k!(n − 2k + 2)(n − k + 1)(n − k + 2) · · · (n − 1)columns which contain only red cells
2 Find all polynomials P (x) with integer coefficients such that the polynomial
Q(x) = (x2+ 6x + 10) · P2(x) − 1
is the square of a polynomial with integer coefficients
3 Prove that there exists an integer n, n ≥ 2002, and n distinct positive integers a1, a2, , an
such that the number N = a21a22· · · a2n− 4(a21+ a22+ · · · + a2n) is a perfect square
http://www.artofproblemsolving.com/
This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page
Page 2 http://www.mathlinks.ro/
vnmath.com
Trang 91 Let be four positive integers m, n, p, q, with p < m given and q < n Take four pointsA(0; 0), B(p; 0), C(m; q) and D(m; n) in the coordinate plane Consider the paths f from A
to D and the paths g from B to C such that when going along f or g, one goes only inthe positive directions of coordinates and one can only change directions (from the positivedirection of one axe coordinate into the the positive direction of the other axe coordinate) atthe points with integral coordinates Let S be the number of couples (f, g) such that f and
g have no common points Prove that
S =
n
m + n
·
q
m + q − p
−
q
m + q
·
n
m + n − p
2 Given a triangle ABC Let O be the circumcenter of this triangle ABC Let H, K, L be thefeet of the altitudes of triangle ABC from the vertices A, B, C, respectively Denote by A0,
B0, C0 the midpoints of these altitudes AH, BK, CL, respectively The incircle of triangleABC has center I and touches the sides BC, CA, AB at the points D, E, F , respectively.Prove that the four lines A0D, B0E, C0F and OI are concurrent (When the point O concideswith I, we consider the line OI as an arbitrary line passing through O.)
3 Let f (0, 0) = 52003, f (0, n) = 0 for every integer n 6= 0 and f (m, n) = f (m − 1, n) − 2 ·
f (m − 1, n)
2
+ f (m − 1, n − 1)
2
+ f (m − 1, n + 1)
2
for every natural number m > 0 andfor every integer n
Prove that there exists natural number M such that f (M, n) = 1 for all integers n such that
Trang 101 On the sides of triangle ABC take the points M1, N1, P1 such that each line M M1, N N1, P P1divides the perimeter of ABC in two equal parts (M, N, P are respectively the midpoints ofthe sides BC, CA, AB).
I Prove that the lines M M1, N N1, P P1 are concurrent at a point K II Prove that amongthe ratios KA
For each a = (a1, a2, , a2003) ∈ A, let d(a) =
3 Let n be a positive integer Prove that the number 2n+ 1 has no prime divisor of the form
8 · k − 1, where k is a positive integer
http://www.artofproblemsolving.com/
This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page
Page 2 http://www.mathlinks.ro/
vnmath.com
Trang 111 Let us consider a set S = {a1 < a2 < < a2004}, satisfying the following properties:
f (ai) < 2003 and f (ai) = f (aj) ∀i, j from {1, 2, , 2004}, where f (ai) denotes number of
elements which are relatively prime with ai Find the least positive integer k for which in
every k-subset of S, having the above mentioned properties there are two distinct elements
with greatest common divisor greater than 1
2 Find all real values of α, for which there exists one and only one function f : R 7→ R and
satisfying the equation
f (x2+ y + f (y)) = (f (x))2+ α · yfor all x, y ∈ R
3 In the plane, there are two circles Γ1, Γ2 intersecting each other at two points A and B
Tangents of Γ1 at A and B meet each other at K Let us consider an arbitrary point M
(which is different of A and B) on Γ1 The line M A meets Γ2 again at P The line M K
meets Γ1 again at C The line CA meets Γ2 again at Q Show that the midpoint of P Q lies
on the line M C and the line P Q passes through a fixed point when M moves on Γ1
{[Moderator edit: This problem was also discussed on http://www.mathlinks.ro/Forum/viewtopic.php?t=21414.]
vnmath.com
Trang 121 Let {xn}, with n = 1, 2, 3, , be a sequence defined by x1 = 603, x2 = 102 and xn+2 =
xn+1+ xn+ 2pxn+1· xn− 2 ∀n ≥ 1 Show that:
(1) The number xn is a positive integer for every n ≥ 1
(2) There are infinitely many positive integers n for which the decimal representation of xnends with 2003
(3) There exists no positive integer n for which the decimal representation of xn ends with2004
2 Let us consider a convex hexagon ABCDEF Let A1, B1, C1, D1, E1, F1 be midpoints of thesides AB, BC, CD, DE, EF, F A respectively Denote by p and p1, respectively, the perimeter
of the hexagon ABCDEF and hexagon A1B1C1D1E1F1 Suppose that all inner angles ofhexagon A1B1C1D1E1F1 are equal Prove that
p ≥ 2 ·
√3
3 · p1.When does equality hold ?
3 Let S be the set of positive integers in which the greatest and smallest elements are relativelyprime For natural n, let Sn denote the set of natural numbers which can be represented assum of at most n elements (not necessarily different) from S Let a be greatest element from
S Prove that there are positive integer k and integers b such that |Sn| = a · n + b for all
Trang 131 Let (I), (O) be the incircle, and, respectiely, circumcircle of ABC (I) touches BC, CA, AB
in D, E, F respectively We are also given three circles ωa, ωb, ωc, tangent to (I), (O) in D, K(for ωa), E, M (for ωb), and F, N (for ωc)
a) Show that DK, EM, F N are concurrent in a point P ;
b) Show that the orthocenter of DEF lies on OP
2 Given n chairs around a circle which are marked with numbers from 1 to n There are k,
k ≤ 4 · n students sitting on those chairs Two students are called neighbours if there is nostudent sitting between them Between two neighbours students ,there are at less 3 chairs.Find the number of choices of k chairs so that k students can sit on those and the condition
is satisfied
3 Find all functions f : Z 7→ Z satisfying the condition: f (x3+ y3+ z3) = f (x)3+ f (y)3+ f (z)3
vnmath.com
Trang 141 Let be given positive reals a, b, c Prove that: a
3 n is called diamond 2005 if n = ab999 99999cd , e.g 2005 × 9 Let {an} : an< C · n, {an}
is increasing Prove that {an} contain infinite diamond 2005
Compare with [url=http://www.mathlinks.ro/Forum/topic-15091.html]this problem.[/url]
http://www.artofproblemsolving.com/
This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page
Page 2 http://www.mathlinks.ro/
vnmath.com
Trang 151 Given an acute angles triangle ABC, and H is its orthocentre The external bisector of theangle ∠BHC meets the sides AB and AC at the points D and E respectively The internalbisector of the angle ∠BAC meets the circumcircle of the triangle ADE again at the point
K Prove that HK is through the midpoint of the side BC
2 Find all pair of integer numbers (n, k) such that n is not negative and k is greater than 1,and satisfying that the number:
A = 172006n+ 4.172n+ 7.195ncan be represented as the product of k consecutive positive integers
3 In the space, given 2006 distinct points such that no 4 of them are coplanar One draws eachpair of points by a segment A natural number m is called ”good” if one can put on each
of these segments a number not greater than m sothat every triangle whose three points are
in the 2006 points given has the following property: Two of this triangle’s sides are put twoequal numbers, and the other a greater number Find the minimum value of the ”good”number m
vnmath.com
Trang 161 Prove that for all real numbers x, y, z ∈ [1, 2] the following inequality always holds:
When does the equality occur?
2 Given a non-isoceles triangle ABC inscribes a circle (O, R) (center O, radius R) Consider
a varying line l such that l ⊥ OA and l always intersects the rays AB, AC and these sectional points are called M, N Suppose that the lines BN and CM intersect, and if theintersectional point is called K then the lines AK and BC intersect 1, Assume that P isthe intersectional point of AK and BC Show that the circumcircle of the triangle M N P isalways through a fixed point 2, Assume that H is the orthocentre of the triangle AM N De-note BC = a, and d is the distance between A and the line HK Prove that d ≤p4R2− a2
inter-and the equality occurs iff the line l is through the intersectional point of two lines AO inter-andBC
3 The real sequence {an|n = 0, 1, 2, 3, } defined a0= 1 and
Denote
An= 3
3 · a2
n− 1.Prove that An is a perfect square and it has at least n distinct prime divisors
http://www.artofproblemsolving.com/
This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page
Page 2 http://www.mathlinks.ro/
vnmath.com
Trang 171 Given two sets A, B of positive real numbers such that: |A| = |B| = n; A 6= B and S(A) =S(B), where |X| is the number of elements and S(X) is the sum of all elements in set X.Prove that we can fill in each unit square of a n × n square with positive numbers and somezeros such that:
a) the set of the sum of all numbers in each row equals A;
b) the set of the sum of all numbers in each column equals A
c) there are at least (n − 1)2+ k zero numbers in the n × n array with k = |A ∩ B|
2 Let ABC be an acute triangle with incricle (I) (KA) is the cricle such that A ∈ (KA)and AKA ⊥ BC and it in-tangent for (I) at A1, similary we have B1, C1 a) Prove that
AA1, BB1, CC1are concurrent, called point-concurrent is P b) Assume circles (JA), (JB), (JC)are symmetry for excircles (IA), (IB), (IC) across midpoints of BC, CA, AB ,resp Prove that
PP/(JA)= PP/(JB)= PP/(JC)
Note If (O; R) is a circle and M is a point then PM/(O)= OM2− R2
3 Given a triangle ABC Find the minimum of
cos2 A
2 cos2 B 2
cos2 C 2
+ cos
2 B
2 cos2 C 2
cos2 A 2
+ cos
2 C
2 cos2 A 2
cos2 B 2
vnmath.com
Trang 184 Find all continuous functions f : R → R such that for all real x we have
5 Let A ⊂ {1, 2, , 4014}, |A| = 2007, such that a does not divide b for all distinct elements
a, b ∈ A For a set X as above let us denote with mX the smallest element in X Find min mA(for all A with the above properties)
6 Let A1A2 A9 be a regular 9−gon Let {A1, A2, , A9} = S1∪ S2 ∪ S3 such that |S1| =
|S2| = |S3| = 3 Prove that there exists A, B ∈ S1, C, D ∈ S2, E, F ∈ S3 such that