Iran National Olympiad Problems 1997 - 2010

iran National Math Olympiad (Second Round) 1997 Day 1 1 Let x, y be positive integers such that 3x2 + x = 4y 2 + y. Pro...

Author: Amir Hossein Parvardi

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iran National Math Olympiad (Second Round) 1997

Day 1 1 Let x, y be positive integers such that 3x2 + x = 4y 2 + y. Prove that x − y is a perfect square. 2 Let segments KN, KL be tangent to circle C at points N, L, respectively. M is a point on the extension of the segment KN and P is the other meet point of the circle C and the circumcircle of ∆KLM . Q is on M L such that N Q is perpendicular to M L. Prove that: ∠M P Q = 2∠KM L. 3 We have a n × n table and weve written numbers 0, +1 or − 1 in each 1 × 1 square such that in every row or column, there is only one +1 and one −1. Prove that by swapping the rows with each other and the columns with each other finitely, we can swap +1s with −1s.

This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

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Day 2 1 Let x1 , x2 , x3 , x4 be positive reals such that x1 x2 x3 x4 = 1. Prove that: 4 X i=1

x3i

≥ max{

4 X i=1

4 X 1 xi , }. xi i=1

2 In triangle ABC, angles B, C are acute. Point D is on the side BC such that AD ⊥ BC. Let the interior bisectors of ∠B, ∠C meet AD at E, F , respectively. If BE = CF , prove that ABC is isosceles. q b 3 Let a, b be positive integers and p = 4 2a−b 2a+b be a prime number. Find the maximum value of p and justify your answer.


Page 2


Day 1 1 If a1 < a2 < · · · < an be real numbers, prove that: a1 a42 + a2 a43 + · · · + an−1 a4n + an a41 ≥ a2 a41 + a3 a42 + · · · + an a4n−1 + a1 a4n . 2 Let ABC be a triangle. I is the incenter of ∆ABC and D is the meet point of AI and the circumcircle of ∆ABC. Let E, F be on BD, CD, respectively such that IE, IF are perpendicular to BD, CD, respectively. If IE + IF = AD 2 , find the value of ∠BAC. P 3 Let n be a positive integer. We call (a1 , a2 , · · · , an ) a good n−tuple if ni=1 ai = 2n and there doesn’t exist a set of ai s such that the sum of them is equal to n. Find all good n−tuple. (For instance, (1, 1, 4) is a good 3−tuple, but (1, 2, 1, 2, 4) is not a good 5−tuple.)


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Day 2 1 Let the positive integer n have at least for positive divisors and 0 < d1 < d2 < d3 < d4 be its least positive divisors. Find all positive integers n such that: n = d21 + d22 + d23 + d24 . 2 Let ABC be a triangle and AB < AC < BC. Let D, E be points on the side BC and the line AB, respectively (A is between B, E) such that BD = BE = AC. The circumcircle of ∆BED meets the side AC at P and BP meets the circumcircle of ∆ABC at Q. Prove that: AQ + CQ = BP. 3 If A = (a1 , · · · , an ) , B = (b1 , · · · , bn ) be 2 n−tuple that ai , bi = 0 or 1 for i = 1, 2, · · · , n, we define f (A, B) the number of 1 ≤ i ≤ n that ai 6= bi . For instance, if A = (0, 1, 1) , B = (1, 1, 0), then f (A, B) = 2. Now, let A = (a1 , · · · , an ) , B = (b1 , · · · , bn ) , C = (c1 , · · · , cn ) be 3 n−tuple, such that for i = 1, 2, · · · , n, ai , bi , ci = 0 or 1 and f (A, B) = f (A, C) = f (B, C) = d. a) Prove that d is even. b) Prove that there exists a n−tuple D = (d1 , · · · , dn ) that di = 0 or 1 for i = 1, 2, · · · , n, such that f (A, D) = f (B, D) = f (C, D) = d2 .


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Day 1 1 Does there exist a positive integer that is a power of 2 and we get another power of 2 by swapping its digits? Justify your answer. 2 ABC is a triangle with ∠B > 45◦ , ∠C > 45◦ . We draw the isosceles triangles CAM, BAN on the sides AC, AB and outside the triangle, respectively, such that ∠CAM = ∠BAN = 90◦ . And we draw isosceles triangle BP C on the side BC and inside the triangle such that ∠BP C = 90◦ . Prove that ∆M P N is an isosceles triangle, too, and ∠M P N = 90◦ . 3 We have a 100 × 100 garden and weve plant 10000 trees in the 1 × 1 squares (exactly one in each.). Find the maximum number of trees that we can cut such that on the segment between each two cut trees, there exists at least one uncut tree.


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Day 2 1 Find all positive integers m such that there exist positive integers a1 , a2 , . . . , a1378 such that: m=

1378 X k=1

k . ak

2 Let ABC be a triangle and points P, Q, R be on the sides AB, BC, AC, respectively. Now, let A0 , B 0 , C 0 be on the segments P R, QP, RQ in a way that AB||A0 B 0 , BC||B 0 C 0 and AC||A0 C 0 . Prove that: SP QR AB = . 0 0 AB SA0 B 0 C 0 Where SXY Z is the surface of the triangle XY Z. 3 Let A1 , A2 , · · · , An be n distinct points on the plane (n > 1). We consider all the segments Ai Aj where i < j ≤ n and color the midpoints of them. What’s the minimum number of colored points? (In fact, if k colored points coincide, we count them 1.)


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Day 1 1 21 distinct numbers are chosen from the set {1, 2, 3, . . . , 2046}. Prove that we can choose three distinct numbers a, b, c among those 21 numbers such that bc < 2a2 < 4bc 2 The points D, E and F are chosen on the sides BC, AC and AB of triangle ABC, respectively. Prove that triangles ABC and DEF have the same centroid if and only if CE AF BD = = DC EA FB 3 Let M = {1, 2, 3, . . . , 10000}. Prove that there are 16 subsets of M such that for every a ∈ M, there exist 8 of those subsets that intersection of the sets is exactly {a}.


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Day 2 1 Find all positive integers n such that we can divide the set {1, 2, 3, . . . , n} into three sets with the same sum of members. 2 In a tetrahedron we know that sum of angles of all vertices is 180◦ . (e.g. for vertex A, we have ∠BAC + ∠CAD + ∠DAB = 180◦ .) Prove that faces of this tetrahedron are four congruent triangles. 3 Super number is a sequence of numbers 0, 1, 2, . . . , 9 such that it has infinitely many digits at left. For example . . . 3030304 is a super number. Note that all of positive integers are super numbers, which have zeros before they’re original digits (for example we can represent the number 4 as . . . , 00004). Like positive integers, we can add up and multiply super numbers. For example: . . . 3030304 + . . . 4571378

. . . 7601682 And


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. . . 3030304 × . . . 4571378

. . . 4242432 . . . 212128 . . . 90912 . . . 0304 . . . 128 . . . 20 ...6

. . . 5038912 a) Suppose that A is a super number. Prove that there exists a super number B such that ← ← A + B = 0 (Note: 0 means a super number that all of its digits are zero). ←

b) Find all super numbers A for which there exists a super number B such that A × B = 0 1 ← (Note: 0 1 means the super number . . . 00001). ←

←

←

c) Is this true that if A × B = 0 , then A = 0 or B = 0 ? Justify your answer.


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Day 1 1 Let n be a positive integer and p be a prime number such that np + 1 is a perfect square. Prove that n + 1 can be written as the sum of p perfect squares. 2 Let ABC be an acute triangle. We draw 3 triangles B 0 AC, C 0 AB, A0 BC on the sides of ∆ABC at the out sides such that: ∠B 0 AC = ∠C 0 BA = ∠A0 BC = 30◦

,

∠B 0 CA = ∠C 0 AB = ∠A0 CB = 60◦

If M is the midpoint of side BC, prove that B 0 M is perpendicular to A0 C 0 . 3 Find all positive integers n such that we can put n equal squares on the plane that their sides are horizontal and vertical and the shape after putting the squares has at least 3 axises.


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Day 2 1 Find all polynomials P with real coefficients such that ∀x ∈ R we have: P (2P (x)) = 2P (P (x)) + 2(P (x))2 . 2 In triangle ABC, AB > AC. The bisectors of ∠B, ∠C intersect the sides AC, AB at P, Q, respectively. Let I be the incenter of ∆ABC. Suppose that IP = IQ. How much isthe value of ∠A? 3 Suppose a table with one row and infinite columns. We call each 1 × 1 square a room. Let the table be finite from left. We number the rooms from left to ∞. We have put in some rooms some coins (A room can have more than one coin.). We can do 2 below operations: a) If in 2 adjacent rooms, there are some coins, we can move one coin from the left room 2 rooms to right and delete one room from the right room. b) If a room whose number is 3 or more has more than 1 coin, we can move one of its coins 1 room to right and move one other coin 2 rooms to left. i) Prove that for any initial configuration of the coins, after a finite number of movements, we cannot do anything more. ii) Suppose that there is exactly one coin in each room from 1 to n. Prove that by doing the allowed operations, we cannot put any coins in the room n + 2 or the righter rooms.


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1 Let n ∈ N and An set of all permutations (a1 , . . . , an ) of the set {1, 2, . . . , n} for which k|2(a1 + · · · + ak ), for all 1 ≤ k ≤ n. Find the number of elements of the set An . 2 A rectangle is partitioned into finitely many small rectangles. We call a point a cross point if it belongs to four different small rectangles. We call a segment on the obtained diagram maximal if there is no other segment containing it. Show that the number of maximal segments plus the number of cross points is 3 more than the number of small rectangles. 3 In a convex quadrilateral ABCD with ∠ABC = ∠ADC = 135◦ , points M and N are taken on the rays AB and AD respectively such that ∠M CD = ∠N CB = 90◦ . The circumcircles of triangles AM N and ABD intersect at A and K. Prove that AK ⊥ KC. 4 Let A and B be two fixed points in the plane. Consider all possible convex quadrilaterals ABCD with AB = BC, AD = DC, and ∠ADC = 90◦ . Prove that there is a fixed point P such that, for every such quadrilateral ABCD on the same side of AB, the line DC passes through P. 5 Let δ be a symbol such that δ 6= 0 and δ 2 = 0. Define R[δ] = {a + bδ|a, b ∈ R}, where a + bδ = c + dδ if and only if a = c and b = d, and define (a + bδ) + (c + dδ) = (a + c) + (b + d)δ, (a + bδ) · (c + dδ) = ac + (ad + bc)δ. Let P (x) be a polynomial with real coefficients. Show that P (x) has a multiple real root if and only if P (x) has a non-real root in R[δ]. 6 Let G be a simple graph with 100 edges on 20 vertices. Suppose that we can choose a pair of disjoint edges in 4050 ways. Prove that G is regular.


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Day 1 1 We call the positive integer n a 3−stratum number if we can divide the set of its positive divisors into 3 subsets such that the sum of each subset is equal to the others. a) Find a 3−stratum number. b) Prove that there are infinitely many 3−stratum numbers. 2 In a village, there are n houses with n > 2 and all of them are not collinear. We want to generate a water resource in the village. For doing this, point A is better than point B if the sum of the distances from point A to the houses is less than the sum of the distances from point B to the houses. We call a point ideal if there doesnt exist any better point than it. Prove that there exist at most 1 ideal point to generate the resource. 3 n volleyball teams have competed to each other (each 2 teams have competed exactly 1 time.). For every 2 distinct teams like A, B, there exist exactly t teams which have lost their match with A, B. Prove that n = 4t + 3. (Notabene that in volleyball, there doesnt exist tie!)


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Day 2 1 Let x, y, z ∈ R and xyz = −1. Prove that: x4 + y 4 + z 4 + 3(x + y + z) ≥

x2 x2 y 2 y 2 z 2 z 2 + + + + + . y z x z x y

2 ∠A is the least angle in ∆ABC. Point D is on the arc BC from the circumcircle of ∆ABC. The perpendicular bisectors of the segments AB, AC intersect the line AD at M, N , respectively. Point T is the meet point of BM, CN . Suppose that R is the radius of the circumcircle of ∆ABC. Prove that: BT + CT ≤ 2R. 3 We have a chessboard and we call a 1 × 1 square a room. A robot is standing on one arbitrary vertex of the rooms. The robot starts to move and in every one movement, he moves one side of a room. This robot has 2 memories A, B. At first, the values of A, B are 0. In each movement, if he goes up, 1 unit is added to A, and if he goes down, 1 unit is waned from A, and if he goes right, the value of A is added to B, and if he goes left, the value of A is waned from B. Suppose that the robot has traversed a traverse (!) which hasnt intersected itself and finally, he has come back to its initial vertex. If v(B) is the value of B in the last of the traverse, prove that in this traverse, the interior surface of the shape that the robot has moved on its circumference is equal to |v(B)|.


Page 2


Day 1 1 ABC is a triangle and ∠A = 90◦ . Let D be the meet point of the interior bisector of ∠A and BC. And let Ia be the A−excenter of ∆ABC. Prove that: √ AD ≤ 2 − 1. DIa 2 Let f : R≥0 → R be a function such that f (x) − 3x and f (x) − x3 are ascendant functions. Prove that f (x) − x2 − x is an ascendant function, too. (We call the function g(x) ascendant, when for every x ≤ y we have g(x) ≤ g(y).) 3 The road ministry has assigned 80 informal companies to repair 2400 roads. These roads connect 100 cities to each other. Each road is between 2 cities and there is at most 1 road between every 2 cities. We know that each company repairs 30 roads that it has agencies in each 2 ends of them. Prove that there exists a city in which 8 companies have agencies.


Page 1


Day 2 1 N is the set of positive integers. Determine all functions f : N → N such that for every pair (m, n) ∈ N2 we have that: f (m) + f (n) | m + n. 2 The interior bisector of ∠A from ∆ABC intersects the side BC and the circumcircle of ∆ABC at D, M , respectively. Let ω be a circle with center M and radius M B. A line passing through D, intersects ω at X, Y . Prove that AD bisects ∠XAY . 3 We have a m × n table and m ≥ 4 and we call a 1 × 1 square a room. When we put an alligator coin in a room, it menaces all the rooms in his column and his adjacent rooms in his row. What’s the minimum number of alligator coins required, such that each room is menaced at least by one alligator coin? (Notice that all alligator coins are vertical.)


Page 2


Day 1 1 Let n, p > 1 be positive integers and p be prime. We know that n|p − 1 and p|n3 − 1. Prove that 4p − 3 is a perfect square. 2 In triangle ABC, ∠A = 60◦ . The point D changes on the segment BC. Let O1 , O2 be the circumcenters of the triangles ∆ABD, ∆ACD, respectively. Let M be the meet point of BO1 , CO2 and let N be the circumcenter of ∆DO1 O2 . Prove that, by changing D on BC, the line M N passes through a constant point. 3 In one galaxy, there exist more than one million stars. Let M be the set of the distances between any 2 of them. Prove that, in every moment, M has at least 79 members. (Suppose each star as a point.)


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Day 2 1 We have a 2 × n rectangle. We call each 1 × 1 square a room and we show the room in the ith row and j th column as (i, j). There are some coins in some rooms of the rectangle. If there exist more than 1 coin in each room, we can delete 2 coins from it and add 1 coin to its right adjacent room OR we can delete 2 coins from it and add 1 coin to its up adjacent room. Prove that there exists a finite configuration of allowable operations such that we can put a coin in the room (1, n). 2 BC is a diameter of a circle and the points X, Y are on the circle such that XY ⊥ BC. The points P, M are on XY, CY (or their stretches), respectively, such that CY ||P B and CX||P M . Let K be the meet point of the lines XC, BP . Prove that P B ⊥ M K. 3 Let R+ be the set of positive real numbers. Find all functions f : R+ → R+ such that for all positive real numbers x, y the equation holds: (x + y)f (f (x)y) = x2 f (f (x) + f (y))


Page 2


Day 1 1 Let C1 , C2 be two circles such that the center of C1 is on the circumference of C2 . Let C1 , C2 intersect each other at points M, N . Let A, B be two points on the circumference of C1 such that AB is the diameter of it. Let lines AM, BN meet C2 for the second time at A0 , B 0 , respectively. Prove that A0 B 0 = r1 where r1 is the radius of C1 . 2 Determine all polynomials P (x, y) with real coefficients such that P (x + y, x − y) = 2P (x, y)

∀x, y ∈ R.

3 In the night, stars in the sky are seen in different time intervals. Suppose for every k stars (k > 1), at least 2 of them can be seen in one moment. Prove that we can photograph k − 1 pictures from the sky such that each of the mentioned stars is seen in at least one of the pictures. (The number of stars is finite. Define the moments that the nth star is seen as [an , bn ] that an < bn .)


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Day 2 1 a.) Let m > 1 be a positive integer. Prove there exist finite number of positive integers n such that m + n|mn + 1. b.) For positive integers m, n > 2, prove that there exists a sequence a0 , a1 , · · · , ak from positive integers greater than 2 that a0 = m, ak = n and ai + ai+1 |ai ai+1 + 1 for i = 0, 1, · · · , k − 1. 2 Let ABCD be a convex cyclic quadrilateral. Prove that: a) the number of points on the MA MD circumcircle of ABCD, like M , such that M B = M C is 4. b) The diagonals of the quadrilateral which is made with these points are perpendicular to each other. 3 Some books are placed on each other. Someone first, reverses the upper book. Then he reverses the 2 upper books. Then he reverses the 3 upper books and continues like this. After he reversed all the books, he starts this operation from the first. Prove that after finite number of movements, the books become exactly like their initial configuration.


Page 2


Day 1 1 In triangle ABC, ∠A = 90◦ and M is the midpoint of BC. Point D is chosen on segment AC such that AM = AD and P is the second meet point of the circumcircles of triangles ∆AM C, ∆BDC. Prove that the line CP bisects ∠ACB. 2 Two vertices of a cube are A, O such that AO is the diagonal of one its faces. A n−run is a sequence of n + 1 vertices of the cube such that each 2 consecutive vertices in the sequence are 2 ends of one side of the cube. Is the 1386−runs from O to itself less than 1386−runs from O to A or more than it? 3 In a city, there are some buildings. We say the building A is dominant to the building B if the line that connects upside of A to upside of B makes an angle more than 45◦ with earth. We want to make a building in a given location. Suppose none of the buildings are dominant to each other. Prove that we can make the building with a height such that again, none of the buildings are dominant to each other. (Suppose the city as a horizontal plain and each building as a perpendicular line to the plain.)


Page 1


Day 2 1 Prove that for every positive integer n, there exist n positive integers such that the sum of them is a perfect square and the product of them is a perfect cube. 2 Tow circles C, D are exterior tangent to each other at point P . Point A is in the circle C. We draw 2 tangents AM, AN from A to the circle D (M, N are the tangency points.). The ME second meet points of AM, AN with C are E, F , respectively. Prove that PP E F = NF . 3 Farhad has made a machine. When the machine starts, it prints some special numbers. The property of this machine is that for every positive integer n, it prints one of the numbers n, 2n, 3n. We know that the machine prints 2. Prove that it doesn’t print 13824.


Page 2


Day 1 1 In how many ways, can we draw n − 3 diagonals of a n-gon with equal sides and equal angles such that: i) none of them intersect each other in the polygonal. ii) each of the produced triangles has at least one common side with the polygonal. 2 Let Ia be the A-excenter of ∆ABC and the A-excircle of ∆ABC be tangent to the lines AB, AC at B 0 , C 0 , respectively. Ia B, Ia C meet B 0 C 0 at P, Q, respectively. M is the meet point of BQ, CP . Prove that the length of the perpendicular from M to BC is equal to r where r is the radius of incircle of ∆ABC. 3 a, b, c, d ∈ R and at least one of c, d is non-zero. Let f : R → R be a function and f (x) = ax+b cx+d . 1387 We know for every x ∈ R, f (x) isn’t equal to x. Prove that if ∃a ∈ R ; f (a) = a, then ∀x ∈ Df 1387 ; f 1387 (x) = x. (f 1387 (x) = f (f (· · · (f (x))) · · · )) | {z } 1387 times

Guidance. Prove that for every function g(x) = 2 roots, then ∀x ∈ R − { −v u } ; g(x) = x.

sx+t ux+v ,

if the equation g(x) = x has more than


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Day 2 1 N is the set of positive integers and a ∈ N. We know that for every n ∈ N, 4(an + 1) is a perfect cube. Prove that a = 1. 2 We want to choose telephone numbers for a city. The numbers have 10 digits and 0 isnt used in the numbers. Our aim is: We dont choose some numbers such that every 2 telephone numbers are different in more than one digit OR every 2 telephone numbers are different in a digit which is more than 1. What is the maximum number of telephone numbers which can be chosen? In how many ways, can we choose the numbers in this maximum situation? 3 In triangle ABC, H is the foot of perpendicular from A to BC. O is the circumcenter of ∆ABC. T, T 0 are the feet of perpendiculars from H to AB, AC, respectively. We know that AC = 2OT . Prove that AB = 2OT 0 .


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Day 1 1 Let p(x) be a quadratic polynomial for which : |p(x)| ≤ 1 Prove that: |p(x)| ≤

∀x ∈ {−1, 0, 1} 5 4

∀x ∈ [−1, 1]

2 In some of the 1 × 1 squares of a square garden 50 × 50 we’ve grown apple, pomegranate and peach trees (At most one tree in each square). We call a 1 × 1 square a room and call two rooms neighbor if they have one common side. We know that a pomegranate tree has at least one apple neighbor room and a peach tree has at least one apple neighbor room and one pomegranate neighbor room. We also know that an empty room (a room in which theres no trees) has at least one apple neighbor room and one pomegranate neighbor room and one peach neighbor room. Prove that the number of empty rooms is not greater than 1000. 3 Let ABC be a triangle and the point D is on the segment BC such that AD is the interior bisector of ∠A. We stretch AD such that it meets the circumcircle of ∆ABC at M . We draw a line from D such that it meets the lines M B, M C at P, Q, respectively (M is not between B, P and also is not between C, Q). Prove that ∠P AQ ≥ ∠BAC.


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Day 2 1 We have a (n + 2) × n rectangle and weve divided it into n(n + 2) 1 × 1 squares. n(n + 2) soldiers are standing on the intersection points (n + 2 rows and n columns). The commander shouts and each soldier stands on its own location or gaits one step to north, west, east or south so that he stands on an adjacent intersection point. After the shout, we see that the soldiers are standing on the intersection points of a n × (n + 2) rectangle (n rows and n + 2 columns) such that the first and last row are deleted and 2 columns are added to the right and left (To the left 1 and 1 to the right). Prove that n is even. 2 Let a1 < a2 < · · · < an be positive integers such that for every distinct 1 ≤ i, j ≤ n we have aj − ai divides ai . Prove that iaj ≤ jai

for 1 ≤ i < j ≤ n

3 11 people are sitting around a circle table, orderly (means that the distance between two adjacent persons is equal to others) and 11 cards with numbers 1 to 11 are given to them. Some may have no card and some may have more than 1 card. In each round, one [and only one] can give one of his cards with number i to his adjacent person if after and before the round, the locations of the cards with numbers i − 1, i, i + 1 dont make an acute-angled triangle. (Card with number 0 means the card with number 11 and card with number 12 means the card with number 1!) Suppose that the cards are given to the persons regularly clockwise. (Mean that the number of the cards in the clockwise direction is increasing.) Prove that the cards cant be gathered at one person.


Page 2


1 Let a, b be two positive integers and a > b.We know that gcd(a − b, ab + 1) = 1 and gcd(a + b, ab − 1) = 1. Prove that (a − b)2 + (ab + 1)2 is not a perfect square. 2 There are n points in the page such that no three of them are collinear.Prove that number of triangles that vertices of them are chosen from these n points and area of them is 1,is not greater than 23 (n2 − n). 3 Circles W1 , W2 meet at Dand P .A and B are on W1 , W2 respectively,such that AB is tangent to W1 and W2 .Suppose D is closer than P to the line AB. AD meet circle W2 for second time at C.If M be the midpoint of BC,prove that ˆ DPˆM = BDC 4 Let P (x) = ax3 + bx2 + cx + d be a polynomial with real coefficients such that min{d, b + d} > max{|c|, |a + c|} Prove that P (x) do not have a real root in [−1, 1]. 5 In triangle ABC,Aˆ = π3 .Construct E and F on continue of AB and AC respectively such that BE = CF = BC.EF meet circumcircle of 4ACE in K.(K 6≡ E).Prove that K is on ˆ the bisector of A. 6 A school has n students and some super classes are provided for them. Each student can participate in any number of classes that he/she wants. Every class has at least two students participating in it. We know that if two different classes have at least two common students, then the number of the students in the first of these two classes is different from the number of the students in the second one. Prove that the number of classes is not greater that (n − 1)2 .


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