MathDB

4

Part of 2003 Tournament Of Towns

Problems(8)

Maximal number of successive odd terms in such a sequence

Source: Tournament of Towns Spring 2003 - Junior O-Level - Problem 4

6/14/2011
Each term of a sequence of positive integers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence?
number theory proposednumber theory
Chocolate bar in the shape of an equilateral triangle

Source: Tournament of Towns Spring 2003 - Junior A-Level - Problem 4

6/14/2011
A chocolate bar in the shape of an equilateral triangle with side of the length nn, consists of triangular chips with sides of the length 11, parallel to sides of the bar. Two players take turns eating up the chocolate. Each player breaks off a triangular piece (along one of the lines), eats it up and passes leftovers to the other player (as long as bar contains more than one chip, the player is not allowed to eat it completely).
A player who has no move or leaves exactly one chip to the opponent, loses. For each nn, find who has a winning strategy.
combinatorics unsolvedcombinatorics
The maximal number of terms that could remain on their place

Source: Tournament of Towns Spring 2003 - Senior O-Level - Problem 4

6/14/2011
In the sequence 00,01,02,03,…,9900, 01, 02, 03,\ldots , 99 the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by 11 (for example, 2929 can be followed by 19,3919, 39, or 2828, but not by 3030 or 2020). What is the maximal number of terms that could remain on their places?
number theory proposednumber theory
Prove that < EMK = 90

Source: Tournament of Towns Spring 2003 - Senior A-Level - Problem 4

6/15/2011
A right triangle ABCABC with hypotenuse ABAB is inscribed in a circle. Let KK be the midpoint of the arc BCBC not containing A,NA, N the midpoint of side ACAC, and MM a point of intersection of ray KNKN with the circle. Let EE be a point of intersection of tangents to the circle at points AA and CC. Prove that ∠EMK=90∘\angle EMK = 90^\circ.
geometrygeometry proposed
Colouring line segments using two colours

Source: ToT 2003-JO-4

6/18/2011
There are NN points on the plane; no three of them belong to the same straight line. Every pair of points is connected by a segment. Some of these segments are colored in red and the rest of them in blue. The red segments form a closed broken line without self-intersections(each red segment having only common endpoints with its two neighbors and no other common points with the other segments), and so do the blue segments. Find all possible values of NN for which such a disposition of NN points and such a choice of red and blue segments are possible.
graph theorycombinatorics unsolvedcombinatorics
Squares on chessboard so that bishop attacks 2 squares.

Source: ToT 2003-JA-4

6/19/2011
Several squares on a 15×1515 \times 15 chessboard are marked so that a bishop placed on any square of the board attacks at least two of marked squares. Find the minimal number of marked squares.
combinatorics unsolvedcombinatorics
Prove that AO is parallel HK if IO is parallel to BC.

Source: ToT 2003 SA-4

7/3/2011
In a triangle ABCABC, let HH be the point of intersection of altitudes, II the center of incircle, OO the center of circumcircle, KK the point where the incircle touches BCBC. Given that IOIO is parallel to BCBC, prove that AOAO is parallel to HKHK.
geometrycircumcirclegeometric transformationparallelogramrectanglegeometry proposed
Inequality on area of quadrilaterals

Source: ToT 2003 SO-4

6/26/2011
Each side of 1×11 \times 1 square is a hypothenuse of an exterior right triangle. Let A,B,C,DA, B, C, D be the vertices of the right angles and O1,O2,O3,O4O_1, O_2, O_3, O_4 be the centers of the incircles of these triangles. Prove that a)a) The area of quadrilateral ABCDABCD does not exceed 22; b)b) The area of quadrilateral O1O2O3O4O_1O_2O_3O_4 does not exceed 11.
inequalitiesgeometrygeometry unsolved