The Geometry of Numbers

Transcription

1 Involving triangle lengths The Geometry of Numbers 1. A farmer has 50 m of fencing and wishes to build a rectangular yard by adding to an existing 10 m section of fence. What is the maximum area that can be enclosed? 2. A convex quadrilateral is cut by its diagonals into four triangles with integral areas. Prove that the product of the four areas is a perfect square. Also prove that their sum cannot be a prime. 3. Suppose the lengths of the three sides of ΔABC are integers and the inradius of the triangle is 1. Prove that the triangle is a right triangle. (1988 Sichuan Province Mathematics Competition) 4. Prove that the inradius of a right-angled triangle with integer sides is an integer. (Regional Mathematics Olympiad, 1999) 5. Can a scalene triangle have all 3 sides, area, inradius and circumradius all integers? 6. Let be an equilateral triangle and consider a point, X in its interior. Can PQ, QR, PR, PX, PY be all integers? 7. The chord CD is perpendicular to the diameter AB and meets it at H. Let O be the center of the circle. The distances AB and CD are integral. The distance AB has 2 digits and the distance CD is obtained by reversing the digits of AB. The distance OH is a non-zero rational. Find AB. 8. Find the number of all integer-sided isosceles obtuse-angled triangles with perimeter (Regional Mathematics Olympiad, 2008) 9. For any natural number, ( 2), let () denote the number of non-congruent integersided triangles with perimeter. Show that: a) (1999) (1996) b) (2000) (1997). (Indian National Mathematics Olympiad, 2000) 10. How many triangles with ordered integral sides can be formed which have perimeter n (n is an integer)? Here (3,4,5) and (4,3,5) triangles are considered as different 11. What is the maximal number of triangles with perimeter n having integral sides (without ordering)? 12. Let n be a positive integer. Find the maximal number of non-congruent triangles whose side lengths are integers. (Czech proposal for IMO, ILL 1967) 13. Let E be a set of n points in the plane (n 3) whose coordinates are integers such that any three points from E are vertices of a nondegenerate triangle whose centroid does not have both coordinates integers. Determine the maximal n. (ISL 1977) 14. For what n is it possible to construct a closed sequence of segments in the plane with lengths 1,2,, (exactly in this order) if any two neighboring segments are pair-wise perpendicular? 15. In, the bisector of the angle intersects the side BC at D. Suppose that. Find the lengths BC, AC and AB that minimize the perimeter of, given that all the sides of the triangles and have integer lengths. 16. A triple (a, b, c) of positive integers is called quasi-pythagorean if there exists a triangle with lengths of the sides a, b, c and the angle opposite to the side c equal to 120. Prove that if (a, b, c) is a quasi-pythagorean triple then c has a prime divisor bigger than 5. (Baltic Way 1998) 17. Prove that there exists a unique triangle whose side lengths are consecutive natural numbers and one of whose angles is twice the measure of one of the others. (IMO 1968)

2 18. A has integer sides, 2 and 90. What the minimum possible perimeter of? 19. In, 2, is obtuse, and the lengths of the sides are positive integers. Find the minimal value of perimeter of (Morocco National Olympiad 2012) 20. Two non-congruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is 8:7. Find the minimum possible value of their common perimeter.(aime 2010) 21. Does there exist a triangle such that the lengths of all its sides and altitudes are integers and its perimeter is equal to 1995? (Baltic Way 94) 22. Find all triangles with integer sides and having at least one angle equal to If a, b and c are integer sides of a triangle with 60 and. Prove that if a is a prime then the triangle is equilateral. 24. Show that for any integer a5 there exist integers b and c,, such that a, b, c are the lengths of the sides of a right-angled triangle. (Baltic Way 94) 25. Let p be an odd prime. Show that there is at most one non-degenerate integer triangle with perimeter 4p and integer area. Characterize those primes for which such triangle exist. 26. Prove that a right angled triangle with integer sides can't have area as a perfect square. 27. An integer which is the area of a right-angled triangle with integer sides is called Pythagorean. Prove that for every positive integer 12 there exists a Pythagorean number between n and 2n. 28. Show that there are just three right angled triangles with integral side lengths a b c such that ab 4(a + b + c). (Putnam 1965) 29. The lengths of the sides of a quadrilateral are positive integers. The length of each side divides the sum of the other three lengths. Prove that two of the sides have the same length. 30. Let ABCD be a cyclic quadrilateral with integer sides such that is equilateral. Can the lengths BP, PC and PQ be all integers given PQ is prime. 31. Isosceles trapezoid ABCD has side AD parallel to BC. A point E is on AD such that BE and CE divide the trapezoid into three right-angled triangles. These three triangles are similar, but no two are congruent. Given that all the sides of the three triangles are positive integers and the length of 2009 what is the length of BC? 32. Right triangle has integer side lengths and hypotenuse AB. Point D is drawn on AB such that. A perpendicular to AB is drawn from D to a point P such that. If PB is an integer, then prove that + 1 and 1 cannot both be prime. 33. Is there a triangle with angles in ratio of 1:2:4 and the length of its sides are integers; with at least one of them is a prime number? (Indonesia TST 2010) 34. Given a segment AB of the length 1, define the set M of points in the following way: it contains the two points A, B, and also all points obtained from A, B by iterating the following rule: for every pair of points X, Y in M, the set M also contains the point Z of the segment XY for which Y Z 3XZ.Prove that the set M consists of points X from the segment AB for which the distance from the point A is either or where n, k are nonnegative integers.(czech proposal for IMO, ILL 1967) 35. Let a, b, c, d be integers with 0. Suppose that + ( + + )( + + ). Prove that + is not prime. (IMO 2001) 36. Show that there exists no integer k such that ( + + )( + )( + )( + ) where x, y and z are positive integers. 37. Suppose p and q are relatively prime. Then show that: 1) + + ( 1) ( 1)( 1) + + ( 2

3 On lattice points 1. Five lattice points are given in the Euclidean plane. Prove that among all lines passing through these points at least one passes through a lattice point different from the initial five. Generalize! 2. Prove that different lattice points in the space have different distances from2, 3,. Generalize! 3. Prove that no three points with integer coordinates can be the vertices of an equilateral triangle. 4. The vertices of a triangle are lattice points (they have integer coordinates). There are no other lattice points on the boundary of the triangle, but there is exactly one lattice point inside the triangle. Show that it must be the centroid. (Kurschak Competition 1955) 5. A triangle has lattice points as vertices and contains no other lattice points. Prove that its area is ½ 6. An acute-angled triangle has lattice points as vertices. Prove that at least one lattice point is inside it or on one of its sides. 7. A convex pentagon's vertices are all lattice points. Prove that the pentagon contains another lattice point in its interior. 8. The vertices of a convex polyhedron are lattice points. There are no other lattice points on its faces or edges. Prove that it can have at most 8 vertices. 9. We are given a convex pentagon ABCDE in the coordinate plane such that A, B, C, D, E are lattice points. Let Q denote the convex pentagon bounded by the five diagonals of the pentagon ABCDE (so that the vertices of Q are the interior points of intersection of diagonals of the pentagon ABCDE). Prove that there exists a lattice point inside of Q or on the boundary of Q. (Russia 2000) 10. Consider the following one player game. A finite set of selected lattice points and finite set of selected segments is called a position in this game if the following hold: endpoints of each selected segment are lattice points each selected segment is parallel to a coordinate axis, or to the line,or to the line each selected segment contains exactly five lattice points and all of them are selected each of two selected segments have at most one common point. A move in this game consists of selecting a lattice point and a segment such that the new set of selected lattice points and selected segment is a position. Prove or disprove that there exists an initial position such that the game has infinitely many moves. (Austrian Polish Mathematics Competition 1999) 11. Show there do not exist four points in the Euclidean plane such that the pair-wise distances between the points are all odd integers. 12. The vertices of an equilateral closed sequence of segments are lattice points. Prove that it has even number of sides. 13. Consider 37 distinct points in space, all with integer coordinates. Prove that we may find among them three distinct points such that their barycentre has integers coordinates. 14. Prove that if a lattice triangle has no lattice points on its boundary in addition to its vertices, and one point in its interior, then this interior point is its center of gravity. 15. Prove that if all four vertices of a parallelogram are lattice points and there are some other lattice points in or on the parallelogram, then its area exceeds 1.( Eötvös Competition 1941) 16. Find coordinates of a set of eight non-collinear planar points so that each has an integral distance from others.

4 17. A cube of integral dimensions is given in space so that all four vertices of one of the faces are lattice points. Prove that the other four vertices are also lattice points. (Eötvös-Kürschák Competition 1982) 18. Is it possible to choose a set of 100 (or 200) points on the boundary of a cube such that this set is fixed under each isometry of the cube into itself? Justify your answer. (USSR proposal for IMO, ILL 1966) 19. Does there exist an infinite set of lattice points such that: i) no three are collinear ii) no four are concyclic iii) the distance between any two is an integer 20. Let P be a convex polygon with integer sides and odd perimeter. Prove that the area of 21. The side lengths of a polygon with 1994 sides are + 4( 1,2,3,, 1994). Prove that its vertices are not all on lattice points. 22. Prove that no n points with rational coordinates can be chosen in the Euclidean plane to form the vertices of a regular polygon with n sides except the case of Let B be a set of more than distinct points with coordinates of the form (±1,±1,...,±1) in n-dimensional space with n 3. Show that there are three distinct points in B which are the vertices of an equilateral triangle. (Putnam 2000) 24. Let ABCDE be a convex pentagon that satisfies all of the following: There is a circle ξ tangent to each of the sides The lengths of the sides are all positive integers At least one of the sides of the pentagon has length 1 The side AB has length 2 Let P be the point of tangency of ξ with AB. Determine the lengths of the segments AP and BP 25. Prove that on a coordinate plane it is impossible to draw a closed broken line such that (i) coordinates of each vertex are rational, (ii) the length of its every edge is equal to 1 (iii) the line has an odd number of vertices. 26. A point in the plane with a Cartesian coordinate system is called a mixed point if one of its coordinates is rational and the other one is irrational. Find all polynomials with real coefficients such that their graphs do not contain any mixed point. (APMO 2001) 27. For any positive real r, let d(r) be the distance of the nearest lattice point from the circle center the origin and radius r. Show that () 0. ( Eötvös-Kürschák Competition 1973) Rectangle tiling 1. A 66 square is dissected into 9 rectangles by lines parallel to its sides such that all these rectangles have only integer sides. Prove that there are always two congruent triangles. (Regional Mathematics Olympiad 2006) 2. A rectangle can be divided into n equal squares. The same rectangle can also be divided into n + 76 equal squares. Find n. (Baltic Way 1997) 3. A rectangular piece of paper measures 4 units by 5 units. Several lines are drawn parallel to the edges of the paper. A rectangle determined by the intersections of some of these lines is called basic if (i) all four sides of the rectangle are segments of drawn line segments, and (ii) no segments of drawn lines lie inside the rectangle. Given that the total length of all lines

5 drawn is exactly 2007 units, let N be the maximum possible number of basic rectangles determined. Find the remainder when N is divided by (AIME 2007) 4. A long thin strip of paper is 1024 units in length, 1 unit in width, and is divided into 1024 unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is a 512 by 1 strip of double thickness. Next, the right end of this strip is folded over to coincide with and lie on top of the left end, resulting in a 256 by 1 strip of quadruple thickness. This process is repeated 8 more times. After the last fold, the strip has become a stack of 1024 unit squares. How many of these squares lie below the square that was originally the 942nd square counting from the left?(aime 2004) 5. In a village 1998 persons volunteered to clean up, for a fair, a rectangular field with integer sides and perimeter equal to 3996 feet. For this purpose, the field was divided into 1998 equal parts. If each part had an integer area (measured in sq. ft.), find the length of the field. (Indian National Mathematics Olympiad, 1999) 6. A small square is constructed inside a square of area 1 by dividing each side of the unit square into n equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of n if the area of the small square is exactly (AIME, 1985) 7. Is it possible to cut a rectangular-shaped cake into three parts of equal area having a long straight knife? (O. Tolesnikov) 8. Some barrels of radius 1 are stored inside a square of side 100, each barrel standing on its circular bottom. The barrels are placed such that any line segment of length 10 inside the yard hits at least one barrel. Prove that there are at least 400 barrels inside the yard. 9. In a square lattice, consider any triangle of minimum area that is similar to a given triangle. Prove that the centre of its circumscribed circle is not a lattice point. ( Eötvös-Kürschák Competition 2001) 10. Show that there exists a convex hexagon in the such that: a) all its interior angles are equal. b) all its sides are 1,2,3,4,5 and 6 in some order. (Indian National Mathematics Olympiad, 1993) 11. Is it possible to draw a hexagon with vertices in the knots of an integer lattice so that the squares of the lengths of the sides are six consecutive positive integers? (Baltic Way, 1992) 12. Prove that there exists a convex 1990-gon with the following two properties: a) all angles are equal b) the lengths of the 1990 sides are the numbers 1, 2, 3,, 1990 in some order (IMO 1990) 13. Let 5 be an integer. Find the maximal integer k such that there exists a polygon with n vertices (convex or not, but not self-intersecting) having k internal 90 angles. (Germany TST and ISL 2003!) 14. Find the number of unit squares through which the diagonal of rectangle with integer sides passes. 15. Show that we cannot place 10 unit squares in the plane so that no two have an interior point in common and one has a point in common with each of the others. (Putnam 1958) 16. Prove that in each parallelogram there is exactly one point, which connected with the corner points, divides parallelogram into four equal-area triangles. Determine all quadrangles, convex and not-convex, with this property. (ANMO 1996) 17. A regular 2004-gon is divided into a) 2002; b) 2003 triangles. Find the maximal number of obtuse triangles in this dissection. (B. Rublyov, Kyiv)

6 18. A polygon (not necessarily convex) with vertices in the lattice points of a rectangular grid is given. The area of the polygon is S. If I is the number of lattice points that are strictly in the interior of the polygon and B the number of lattice points on the border of the polygon, find the number T 2S B 2I We are given an infinite set of rectangles in the plane, each with vertices of the form (0, 0), (0, m), (n, 0) and (n, m) where m and n are positive integers. Prove that there exist two rectangles in the set such that one contains the other.(eötvös Competition 1934) 20. Let n be an even positive integer. We consider rectangles with integers side lengths k and k + 1, where k is greater than and at most equal to n. Show that for all even positive integers n the sum of the areas of these rectangles equals ()() (Austrian Mathematical Olympiad 2006) 21. Show that if a rectangle is tiled with smaller rectangles that have the property that at least one side is of integer length then show that the big rectangle has at least one side of integer length.(rectangle tiling theorem) 22. Does there exist a real number L such that, if m and n are integers greater than L, then an mn rectangle may be expressed as a union of 46 and 57 rectangles, any two of which intersect at most along their boundaries?(putnam 1991) 23. A lattice rectangle with sides parallel to the coordinate axes is divided into lattice triangles, each of area ½. Prove that the number of right triangles among them is at least twice the length of the shorter side of rectangle.( Eötvös-Kürschák Competition 1995) Pick s Theorem and Minkowski s Theorem 1. Given three numbers,, such that a and b are co-prime. Let x be a integer such that ( ). Let y be an integer such that ( ). Then, the line through the points (a, x) and (b, y) contains the point (ab, z) for some intger z which satisfies ( ). 2. A lattice line is a line that passes through at least two lattice points. A lattice line segment is a line segment with endpoints at lattice points. Let l be a lattice line through the origin. Then the visible points on l are the two non-zero lattice points on l that have minimum positive distance to the origin. Show that a lattice point p (m, n) is visible if and only if m and n are relatively prime. 3. Given any set of n 3 points in the plane, not all on one line, there is always a line that contains exactly two of the points. (Sylvester) 4. S is an infinite set of points in the plane. The distance between any two points of S is integral. Prove that S is a subset of a straight line. (Putnam 1958) 5. S is a finite set of collinear points. Let k be the maximum distance between any two points of S. Given a pair of points of S a distance dk apart, we can find another pair of points of S also a distance d apart. Prove that if two pairs of points of S are distances a and b apart, then is rational. (Putnam 1964) 6. A primitive lattice polygon is a lattice polygon with no lattice points in its interior and with no lattice points on its sides other than its vertices. (i) Show that every lattice polygon can be dissected into primitive lattice triangles. (ii) A primitive parallelogram has area If p (m, n) is a visible point on the lattice line l, then the lattice points on l are each of the form tp for some integer t. Let m and n be nonnegative integers. There are exactly gcd(m, n)1 lattice points on the lattice line segment between the origin and the point (m, n) not including the endpoints.

7 8. Let (, ) 1,2, ; 1,2,3. A rook tour of S is a polygonal path made up of line segments connecting points,,, in the sequence such that: (i) (ii) and are a unit distance apart (iii) for each : How many rook tours are there that begin from (1,1) and end at (n,1)? (Putnam) 9. Let P be a set of n 3 points in the plane, not all on a line. Then the set C of lines passing through at least two points contains at least n lines. 10. Pick proved a simple formula for the area of the () of any lattice polygon P. Prove it for simple polygons in the following way: Prove the formula for any lattice rectangle with sides p and q. Prove the formula for a right triangle with one horizontal and one vertical leg. Prove that the function () is additive. Show that () gives the correct formula for any lattice triangle P Finally, show that () is the area of any simple lattice polygon P 11. Prove that Pick s Theorem holds for primitive parallelograms. 12. A convex planar polygon M with perimeter l and area S is given. Let M(R) be the set of all points in space that lie a distance at most R from a point of M. Show that the volume V(R) of this set is () (USSR proposal for IMO, ILL ) 13. Let P be a parallelogram whose vertices have integer coordinates. Prove that the area of P is an integer. 14. Let Ɽ be a convex region symmetrical about the origin with area greater than 4. Then Ɽ must contain a lattice point different form the origin. (Minkowski s Theorem) 15. Show using Minkowski's Theorem that: a) 1( 4) + b) 1 3( 8) + 2 (Two-squares theorem) c) Similarly prove Lagrange s Four Squares Theorem 16. Consider positive integers such that Then the equation (2 + 1) + 1 has integer solutions. (Austrian Polish Mathematics Olympiad) 17. Suppose that a, b are rational numbers such that the equation + 1 has at least one rational solution. Then it has infinitely many rational solutions. (Kurschak Competition) 18. Suppose that n is a natural number for which the equation + + has rational solutions. Then show that this equation has integer solutions as well. (Komal) 19. Suppose that a natural number is the sum of three squares of rational numbers. Then prove that it is also a sum of squares of three natural numbers (the use of three squares theorem is forbidden!). Davenport-Cassels lemma 20. Say that an integer n has the property P k if n can expressed as a sum of k positive squares. For any given m, prove that there exist infinitely many integers having all the properties P 1, P 2, P 3 P m 21. Show that the number r(n) of representations of n as a sum of two squares has an average value, that is () 22. Consider a circular orchard with radius fifty feet that has its center at the origin. A tree is planted at each lattice point that lies in the orchard. Assuming all trees have the same radius, we must show a person standing at the origin cannot see out of the orchard no matter which direction he or she looks when the trees have radius greater than units.(the orchard problem)

8 23. The distances between any two trees in a forest is less than the difference between their altitudes. Any tree has altitude100m. Prove that the forest can surrounded a fence length 200m. 24. A mathematician got lost in the woods. He knows its area S. He knows nothing about its shape. Show that he can walk out of the woods walking not more than 2 miles. He knows that it is convex. Show that he can walk out of the woods walking not more than 2 miles. He knows the way. Show that he can walk out of the woods walking at most miles. 25. A point x of a set S in the plane is called an interior point of S if there is an 0 such that S contains all the points within a distance r of x. Show that if S is convex and contains no interior points then S is a subset of a line. 26. Prove that a finite point set cannot have more than one center of symmetry. ( Eötvös Competition 1935) 27. Suppose that a, b, c are positive integers such that + 1. Then there exist x, y, z, t integers such that +, + and +. (ISL) 28. Let a, b, c be real numbers with 0. Put 4 and suppose that 0. Show that there exist integers x, y not both 0, such that Show that any lattice Ɽ in the plane contains a non-zero point (x, y) such that: a) + ( ) b) ( ) 30. Given,, define the Minkowski sum + +,. Prove: i. If A and B are convex then A+B is convex ii. iii. iv. If A and B are two bounded if and only A+B is bounded If A and B are compact then A+B compact If A and B are 2 connected subsets then A+B is connected in v. If A and B are path-connected then A+B is connected vi. vii. viii. If A and B have center of symmetry then A+B has center of symmetry If A and B have axis of symmetry such that these two axis are parallel then A+B has axis of symmetry If A and B are star-shaped then A+B is star-shaped 31. Given,, define the Minkowski sum + +,. Suppose that A is compact and B is a closed ball of radius r. Prove that if A is convex, then the volume of A+B is a polynomial in r. Is the converse true? 32. Let P be a lattice n-gon, with vertices (, ), (, ) (, ). Let gcd(, ). Show that the number of lattice points on the boundary. of P is () 33. Let be a triangle in xy-plane. Then prove that for any 0, there are three points,, belonging to - neighborhood of A, B, C respectively, such that: i) P, Q, R are rational points. ii) the lengths of PQ, PR, QR are all rational. iii) the area of is rational.

9 34. Let n be a positive integer such that n pq, and A 1A 2A 3A n be a regular polygon inscribed in a unit circle. Prove that there exists a permutation,,, of 1,2,, such that for any point M in the plane, we always have () 35. Label n equally spaced points on a circle as A 0, A 1,, A n-1 with 2. If A ia j by definition when ( ) and line segments A 0A m, A ma 2m, are constructed indefinitely for some positive integer m, then under what condition(s) will each point A be reached by the line segments? 36. Suppose, is a matrix with m, n real numbers and 1 is a real number. Then one can find x 1, x 2, x 3,, x n between and, not all zero, such that Show that if 0( ), prime, and if n is any positive integer, then there exists integers x and y such that ( ), 0,. 38. A polygon of area greater than n is given in a plane. Prove that it contains + 1 points (, ) such that, 1, + 1 (China TST 1988)