MathDB

3

Part of 2001 Tournament Of Towns

Problems(8)

A "Cheesy" Problem

Source: ToT - 2001 Spring Junior O-Level #3

8/17/2011
Twenty kilograms of cheese are on sale in a grocery store. Several customers are lined up to buy this cheese. After a while, having sold the demanded portion of cheese to the next customer, the salesgirl calculates the average weight of the portions of cheese already sold and declares the number of customers for whom there is exactly enough cheese if each customer will buy a portion of cheese of weight exactly equal to the average weight of the previous purchases. Could it happen that the salesgirl can declare, after each of the first 1010 customers has made their purchase, that there just enough cheese for the next 1010 customers? If so, how much cheese will be left in the store after the first 1010 customers have made their purchases? (The average weight of a series of purchases is the total weight of the cheese sold divided by the number of purchases.)
algebra unsolvedalgebra
Geometry Proof

Source: ToT - 2001 Spring Junior A-Level #3

8/17/2011
Point AA lies inside an angle with vertex MM. A ray issuing from point AA is reflected in one side of the angle at point BB, then in the other side at point CC and then returns back to point AA (the ordinary rule of reflection holds). Prove that the center of the circle circumscribed about triangle △BCM\triangle BCM lies on line AMAM.
geometrygeometric transformationreflectionangle bisectorgeometry unsolved
Prove that the Points Lie on a Circle

Source: ToT - 2001 Spring Senior O-Level #3

8/17/2011
Points XX and YY are chosen on the sides ABAB and BCBC of the triangle △ABC\triangle ABC. The segments AYAY and CXCX intersect at the point ZZ. Given that AY=YCAY = YC and AB=ZCAB = ZC, prove that the points BB, XX, ZZ, and YY lie on the same circle.
geometryperpendicular bisectorgeometry unsolved
Congruent Triangles

Source: ToT - 2001 Spring Senior A-Level #3

8/17/2011
Let AHAAH_A, BHBBH_B and CHCCH_C be the altitudes of triangle △ABC\triangle ABC. Prove that the triangle whose vertices are the intersection points of the altitudes of △AHBHC\triangle AH_BH_C, △BHAHC\triangle BH_AH_C and △CHAHB\triangle CH_AH_B is congruent to △HAHBHC\triangle H_AH_BH_C.
geometryparallelogramgeometry unsolved
Fake Coins

Source: ToT - 2001 Fall Junior O-Level #3

8/17/2011
Kolya is told that two of his four coins are fake. He knows that all real coins have the same weight, all fake coins have the same weight, and the weight of a real coin is greater than that of a fake coin. Can Kolya decide whether he indeed has exactly two fake coins by using a balance twice?
combinatorics unsolvedcombinatorics
Rows in an Array

Source: ToT - 2001 Fall Junior A-Level #3

8/17/2011
Let n≥3n\ge3 be an integer. Each row in an (n−2)×n(n-2)\times n array consists of the numbers 1,2,...,nn in some order, and the numbers in each column are all different. Prove that this array can be expanded into an n×nn\times n array such that each row and each column consists of the numbers 1,2,...,nn.
graph theorycombinatorics unsolvedcombinatorics
Sailing Ships

Source: ToT - 2001 Fall Senior O-Level #3

8/17/2011
On an east-west shipping lane are ten ships sailing individually. The first five from the west are sailing eastwards while the other five ships are sailing westwards. They sail at the same constant speed at all times. Whenever two ships meet, each turns around and sails in the opposite direction. When all ships have returned to port, how many meetings of two ships have taken place?
combinatorics unsolvedcombinatorics
Diagonal of an Array

Source: ToT - 2001 Fall Senior A-Level #3

8/17/2011
An 8×88\times8 array consists of the numbers 1,2,...,641,2,...,64. Consecutive numbers are adjacent along a row or a column. What is the minimum value of the sum of the numbers along the diagonal?
algebra unsolvedalgebra