# IMO Training 2010 Russian-style Problems Alexander Remorov

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1 Solutions: Combinatorial Geometry 1. No. Call a lattice point even if the sum of its coordinates is even, and call it odd otherwise. Call one of the legs first, and other one second. Then every time a leg is moved from an even point to an even point, or from an odd point to an odd point. If in the beginning one of the legs is at an even point, and the other - at an odd point, the legs cannot switch. If both legs are at even points or both at odd points, consider a new coordinate system on the sheet of paper, with unit length twice as large as unit length in the original coordinate system, and so that the two legs are located at lattice points of the new coordinate system. 2. For each point P on the boundary of A, consider the homothety with centre P and factor 2 3, which carries the set A to a set A P lying inside A. For any three points K, L, M on the boundary of A, the centroid of KLM lies in each of the sets A K, A L, A M so the three sets have nonempty intersection. By Helly s theorem all sets have non-empty intersection; take a point O in this intersection. Consider any two points X, X on the boundary of A such that O lie on XX. By construction O A X, O A X so XX 3 OX, OX 2XX 3 and the result follows. (Iran Math Olympiad 2004) 3. We claim that any such set contains at most 4 points. Assume S has more than 4 points. Take A, B, C such that ABC has the largest possible area over all choices of A, B, C. There is a fourth point D such that ABCD is a parallelogram. It is clear that if a point X is outside ABCD then one of the triangles ABX, BCX, CDX, ADX has area greater than the area of triangle ABC, BCD, CDA, DAB respectively. Each of these triangles has area equal to the area of ABC. Hence there are no points outside of ABCD. Consider a point E inside ABCD. There is a point F such that A, B, E, F are vertices of a parallelogram. It is clear that F cannot be inside ABCD. This gives a contradiction. (USA TST 2005) 4. Let A, B be points in S such that AB is maximal. Consider a point C in S such that the distance from C to AB is maximal. It suffices to show the distance from C to AB is at most 1, since then we can cover S by a strip of length 2 such that line AB passes is the middle line in the strip. We can assume A, B, C are not collinear (otherwise we are done). The triangle ABC can be covered by a strip of length 1, hence one of its altitudes has length at most 1. Since AB is the longest side in the triangle, the altitude to it is the shortest among the three altitudes, so it has length at most 1. Therefore the distance from C to AB is at most 1. (Balkan Math Olympiad 2010) 5. Place the first circle on the second. Then place a painter at a fixed point on the first circle. Rotate the first circle in one direction, and make the painter paint the point on the second circle, under which the painter is located, whenever some marked point lies on a marked arc. It suffices to show that after one full revolution of the first circle, some point on the second arc is not painted. The resulting painting of the second circle will be the same to the one if the first circle is rotated 1000 times, and on the ith rotation the painter paints the point under him whenever the ith marked point lies on a marked arc. During each revolution less than 1 cm is painted on the second circle, hence after 1000 revolutions less than 1000 cm will be painted on the second circle. The result follows. (A Russian Book with Olympiad Problems) 1

2 6. If (x i, y i ) and (x i+1, y i+1 ) are the coordinates of two consecutive vertices in the polygon, then x i + y i x i+1 + y i+1 mod 2, since the length of the segment joining the two vertices is an odd integer. Therefore n cannot be odd (otherwise we will have x 1 + y 1 x n + y n mod 2 which is impossible. We will show any even n 4 works. For every n we will construct a polygon with n vertices X 1,..., X n satisfying the conditions of the problem, and furthermore the side X i X n+1 i is parallel to the x axis. We proceed by induction. For n = 4 take the vertices of the polygon to have coordinates (0, 0), (69, 12), (100, 12), (109, 0). Assume we have the polygon with n vertices, we need to add two new vertices A, B. Notice that we can shift all vertices X n 2 +1, X n 2 +2,..., X n in the positive x direction by an arbitrary number of units m. The resulting polygon with the new vertices X 1, X 2,..., X n still satisfies the conditions of the problem. Hence it suffices to prove that there exists a trapezoid ABCD with points A, B on the x axis and CD AB, so that all its sides are pairwise distinct odd integers, which are arbitrarily large, and the angles CAB, DBA are arbitrarily small; since then we can shift vertices X n 2 +1, X n 2 +2,..., X n, and then let X n 2 X n 2 +1 be side AB and CD be above AB so that C, D are the two new vertices. Construction of ABCD is not hard. Let x be the height of the trapezoid. We want to find integers y, z and s, t such that x 2 = z 2 y 2, x 2 = s 2 t 2 so that AC, BD have lengths y, s; and tan( CAB), tan( DBA) are x y, x t ; we want these to be arbitrarily small. Take x = 6k + 3, y = 18k k + 4, z = 18k k + 5, s = 6k 2 + 6k, t = 6k 2 + 6k + 3. Then since we can take k arbitrarily large, the integers x, y, z, s, t satisfy the conditions we want. (Chinese TST Quiz 6, 2008; most of the solution is from Adrei Frimu s post on AOPS) 7. Assume the contrary. Notice that for any point of intersection of two lines, there is a red line passing through it. Consider a line l and points A and B on l, which are points of intersections of l with two red lines m and n respectively. Choose A and B such that AB is maximal. Let C be the point of intersection of m and n. There is a blue line passing through C and intersecting l at a point D; D must be a point on line segment AB (since AB is maximal). Look at all configurations of 4 lines l, m, n, p that look like the configuration of l, m, n, p; with lines l, p being of one color, m, n of the other color; m, l, n concurrent and the point of intersection of p with l is between the points of intersection of m, n with l. Find the configuration for which the area of the triangle formed by lines l, m, n is minimal. Let C, A, B be the points of intersection of m, n ; m, l ; l, n respectively. Through the point of intersection of p, l there is another line q of the same color as m and n. It will intersect the line segment C A or C B ; wolog it is line segment C A. Then lines m, p, l, q form a configuration with a triangle of smaller area. This is a contradiction. (Russian Math Olympiad 2002) 8. Call such drawing of diagonals a triangulation, and call a triangle special if two of its sides coincide with two of the sides of the polygon. Let the number of sides of the polygon be n. The sum of the angles in the polygon is (n 2)π. The sum of the angles in a triangle is π so there are (n 2) triangles in the triangulation. Hence there are at least 2 distinct special triangles. Furthermore, there are at most 3 acute angles in any convex polygon, since otherwise the sum of the angles in the polygon is less than (n 2)π. Consider two distinct triangulations T 1, T 2. There are two triangles t 1, t 2 that are special in triangulation T 1. Each of them has an angle that coincides with an angle of the polygon; let the two 2

3 angles be α 1, α 2. Similarly define two special triangles s 1, s 2 in triangulation T 2, and the two angles β 1, β 2. Since angles α i, β i are all acute, it follows that two of them are the same; wolog α 1, β 1 are the same. Then triangles t 1, s 1 are the same. Cut them off and consider the remaining convex polygon with n 1 sides. Finish the problem with induction. (Russian Math Olympiad 2003) 9. Let the sides of the squares be equal to a i for i = 1, 2,..., N(N is the number of squares). If some a k > 1 then kth square will cover the unit square. Assume a i < 1 for all i. Each a i must belong to some interval [2 k i, 2 ki+1 ) where k i are positive integers for all i. Let us decrease every ith square to the square with side length b i = 1. Its area will decrease by at most 4 times, because 2 k i 1 a i b i < 2. Therefore the area of the new squares will be greater than 1. Let us prove we can tile the unit square with the new squares. Divide the unit square into 4 squares with side length 1 2. First place the squares with side 1 2 (if they exist). On the non-tiled squares with side 1 2 (if they exist) place the squares with side 1 4 (if they exist), by dividing each non-tiled square with side 1 2 into 4 equal squares. We will continue this procedure for k = 3, 4,... by placing squares with side length 1 1 on the on-tiled squares with side length, each time dividing them into 4 2 k 2 k 1 equal squares. Because the sum of areas of squares is greater than 1, then eventually we will cover the unit square. Increasing the ith square with side b i to the square with side a i, we will get the tiling of the unit square with the given squares. (Russian Math Olympiad 1979) 10. Call the 2n angles into which the lines divide the angle around O - basic, and two lines that form a basic angle - adjacent. Since no line passes inside a basic angle, then for any two adjacent lines, there is a third line bisecting the other pair of vertical angles (complementary with the basic angles). If this line is rotated by 90 around O, it will bisect the simple angles. There are as many bisecting lines, as all the lines, hence if the whole figure is rotated by 90 around O, each of the rotated n lines will be bisect two vertical simple angles. Let α be the largest vertical simple angle, and β, θ be two adjacent simple angles, the angle bisectors of which become after the 90 degree rotation the sides of α. Then α = β+θ 2 hence α = β = θ. Rotating by 90 in opposite direction the angles β, θ, we see that the two angles adjacent to α are equal to α. Hence all simple angles are equal. (Russian Math Olympiad 2003) 11. We use induction on the number of colors, n. Base case will be done for n = 2. Let S be the left-most square. If it is of color 1, all squares of color 2 have a point in common with it, hence each of the squares of color 2 contains one of the right vertices of S; hence all squares of color 2 can be pinned using 2 pins at those two vertices. Assume the result holds for n colors; we need to prove it for n + 1 colors. Consider all squares and find the left-most square S; wolog it has color n + 1. All squares intersecting S contain one of its right vertices, hence they can all be pinned using 2 pins. Remove from the table all squares of color n + 1 and all squares intersecting S. Among the remaining squares, each square has one of n colors. Among any n of these squares of pairwise distinct colors, there are two that intersect; or else add in square S and we get n + 1 squares of pairwise distinct colors, no two of which intersect. Hence there is a color i, such that all the remaining squares of color i can be pinned using 2n 2 pins. The squares of color i that are not pinned all intersect square S and can be pinned using 2 pins at the two right vertices of S. (Russian Math Olympiad 2000) 3

4 12. Let A 1, A 2,..., A N be the points, and each of the distances A i A j is equal to one of the numbers r 1, r 2,..., r n. Then for each i, all of the points except A i lie on one of the circles ω(a i, r 1 ), ω(a i, r 2 ),..., ω(a i, r n ) where ω(o, r) denotes the circle with centre O and radius r. Introduce a system of coordinates so that the coordinates axes are not parallel to any of the lines A i A j. Wolog A 1 has the smallest x coordinate among the points. Among the lines A 1 A i find the one with the largest absolute value of the slope; wolog it is A 1 A 2. Then all points A 3, A 4,..., A N lie in the same half-plane π with respect to line A 1 A 2. Each of the points A 3, A 4,..., A N is the intersection of circles ω(a 1, r k ), ω(a 2, r l ) for some k, l {1, 2,..., n}. Each of the n 2 pairs of these circles has at most one point of intersection in π. Hence among N 2 points A 3, A 4,..., A N at most n 2 are distinct. Hence N 2 n 2 so N n = (n + 1) 2. (Russian Math Olympiad 2004) Processes 1. Among all grids we can get (there are finitely many of them) take the one with the largest possible sum of all numbers. Assume the sum of the numbers in one of the rows in this grid is negative. If we switch the sign of the numbers in the grid, we get a grid with a larger sum of numbers. The result follows. (Russian Math Olympiad 1961) 2. Call a characteristic of a deck the number of cards of the most frequently occurring suit. Every time the characteristic stays the same or decreases by 1. If it decreases by 1, Igor guessed the right suit. In the beginning the characteristic is 13, in the end it is 0. Hence it decreased by 1 exactly 13 times. 3. Let us write numbers on a second board. Every time numbers x, y appear on the first board, we will write down numbers ab x, ab y on the second board. Then when numbers x, y are on the first board, x < y, and the operation is performed, then on the other board the pair ( ab x, ab y ) will be replaced by ( ab y, ab x ab y ) (verify this). This is just Euclid s algorithm, so eventually both numbers on the second board will be equal to gcd(a, b). At that point both numbers on the first board will be equal. 4. Call a move internal if the checker jumps into the n n square S, and external if the checker jumps outside of S. Assume we got to a position from which it is impossible to make any moves, and k internal and l external moves have been made. There are at least n2 2 empty squares in S (or else two checkers are in adjacent squares); an internal move increases the number of empty squares in S by at most 1; an external move increases this number by at most 2. Hence 2l + k n2 2 (1) Assume n is even. Divide S into n squares. In every such square there were at least 2 moves which involved the checkers from this square (either the checker was used to jump in a move, or it was jumped over). In every internal move, the checkers used were from at most 2 squares; in every external moves, the checkers used were from at most 1 square; hence 2k + l 2 n2 4 (2) 4

5 Adding (1) and (2) the result follows for all even n. For odd n = 2m + 1 consider the cross formed by taking the third column from the left and third row from the top. Divide the cross into one unit square and 2m dominoes made up of two squares. Divide the remaining part into m squares. Every internal move uses at checkers from at most 2 figures among the figures we selected; and every external move uses at most one such figure. Hence 2k + l 2m 2 + 2m (3) Then add (1) and (2) and get the result for all odd n (you might need to consider some cases of n mod 3). (Russian Math Olympiad 1999) 5. Cynthia wins. Let us divide the points into 4 equal groups A, B, C, D and index the points from 1 to 500 in each group. We will prove Cynthia can always make a move so that after her move the number of line segments exiting from the points in each group is the same. If Danny erases a line segment connecting two points from the same group, i.e. A i A j, Cynthia will erase B i B j, C i C j, D i D j. If Danny erases a line segment connecting two points from different groups and having different indices, i.e. A i B j Cynthia will erase A j B i, C i D j, D i C j. If Danny erases a line segment connecting two points from different groups but with different indices, i.e. A k B k Cynthia does the following. There are 6 line segments connecting pairs of points from A k, B k, C k, D k. Cynthia can always erase two other line segments among these 6 so that the remaining three have an endpoint in common. For example she can erse A k C k, B k C k so the remaining three segments are A k D k, B k D k, C k D k. If Danny ever erases one of these line segments, there must exist l k such that A l D k, B l D k or C l D k is not yet erased (otherwise no line segments are exiting from D k ). Then the same thing will hold for B k, A k, C k so Cynthia can erase the other two segments from A k D k, B k D k, C k D k and not lose. (Russian Math Olympiad 2000) 6. No. We first prove there is a 2 2 square in the original grid with three zeroes and one 1. This follows from the fact that there is a row containing only zeroes, hence there is a row containing only zeroes adjacent to a row containing a 1. Consider any 2 2 square K in the grid; let a, b, c, d be the numbers in top left, top right, bottom left, bottom right corners of this square, respectively. After the operation is performed let these numbers be a, b, c, d respectively. Let S = (a + d) (b + c); S = (a + d ) (b + c ). We will show D D mod 3. Call the square on which the operation is performed cool. If the cool square is outside K, it is easy to verify D = D. If the cool square is inside K, wolog it is in the top left corner of K; then after operation is performed a = a 1, b = b + 1, c = c + 1, d = d hence D = D 3. Hence D D mod 3. The result follows. 7. Let r be the positive root of the equation x 2 x 1 = 0. For any configuration A of the stones, let a i be the number of stones in square i and let w(a) = a i r i. Then w(a) stays constant (verify this yourself). We now show the process cannot repeat forever by induction on n, the total number of stones. For sufficiently large M, r M > w(a) since r > 1. Hence no stones can ever be in a square with index greater than M. Hence eventually a stone will appear in a square and will not be moved from there 5

6 anymore. Throw out this stone when this happens. Now use the induction hypothesis. Assume it is possible to get two distinct final configurations A, B with a i, b i respectively being the number of stones in square i; so that for some j, a j b j. Assume w(a) = w(b). Choose the largest k for which a k b k.; wolog a k = 0, b k = 1. Throw out stones on squares k + 1, k + 2,... in both configurations A, B (they are identical in A and B). For the remaining configurations A, B we have w(b ) r k = rk 1 1 r 2 = rk 1 + r k > w(a ) since in configuration A it is impossible to make any more moves, hence no two stones in A are in one square, and no two are in adjacent squares. Hence w(a) w(b) and the result follows. (Russian Math Olympiad 1997) 8. Label the two rooms X and Y. Let 2n be the size of the largest clique. Place all students in one such clique C in X and everyone else in Y. Let s(x), s(y ) be the sizes of the largest cliques in rooms X and Y respectively. Move students from X to Y one by one. Every time a student is moved, d = s(x) s(y ) decreases by 1 or 2. Hence at one point it will become 0 or 1. It suffices to consider the case d = 1. Then s(x) = k, s(y ) = k + 1 for some positive integer k. If some student from C is in Y and is not in one of the cliques in Y of size k + 1, move this student to X so that d(x) = d(y ) = k + 1. Hence we can assume each clique in Y of size k + 1 contains 2n k students from C. Then in each such clique there are 2(k n) students no in C. Move these students one by one to X. We will show every time the size of the largest clique in X is k. Assume not, then the last student that has been moved is friends with all students in C, which contradicts the fact that 2n is the size of the largest clique. Hence at some point d(x) = d(y ) = k and we are done. (IMO 2007/3; proposed by Russia. Surprising, eh?) Graphs 1. Assume there is a graph G in which the length of every cycle is divisible by 3. Take such graph with the smallest possible number of vertices, and look at any cycle C in this graph containing the cities A 1, A 2,..., A 3k. Assume there is a path connecting cities A m, A n which does not use any of the edges in C. The union of this path with C gives two cycles C 1, C 2 each of which has length (i.e. number of edges in it) divisible by 3. Hence the length of the path is also divisible by 3. Then for any vertex B not in C, it cannot be connected by an edge with two cities from C. Replace the cycle C by a vertex A. (!) Connect it by an edge with those vertices that were previously connected by an edge with some vertex in C. It is clear the new graph G has less vertices, there are at least 3 edges coming out of every vertex, and every cycle in the new graph has length which is divisible by 3. This contradicts the choice of G as the smallest possible graph. 2. (a) The pandemic will last forever if for example on the first day one person is sick, one is healthy, and one is immune, and every two people are friends. (b) Let us prove nobody can get sick twice. Divide the people into groups G 1, G 2,... so that G 1 is the group of all people who got sick on day 1; G 2 contains all people who are not in G 1 and are friends with somebody from G 1, G 3 contains all people who are not in G 1 or G 2 and are friends with somebody from G 2, etc. If two people are friends, the indices of the groups they belong to differ by 1. By induction it is easy to prove that on day i only people from G i are sick, and only people from G i 1 are immune to the flu. 6

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