MathDB

2

Part of 2006 Tournament of Towns

Problems(8)

n x n Table With Numbers

Source: Spring 2006 Tournament of Towns Junior O-Level #2

4/15/2015
A n×nn \times n table is filled with the numbers as follows: the first column is filled with 11’s, the second column with 22’s, and so on. Then, the numbers on the main diagonal (from top-left to bottom-right) are erased. Prove that the total sums of the numbers on both sides of the main diagonal differ in exactly two times.
(3 points)
100 Distinct Pairs

Source: Spring 2006 Tournament of Towns Junior A-Level #2

4/15/2015
Prove that one can find 100 distinct pairs of integers such that every digit of each number is no less than 6 and the product of the numbers in each pair is also a number with all its digits being no less than 6.
(4 points)
counting
TOT 2006 Spring - Senior A-Level p2 first decimals of 2^n is 5^k

Source:

2/25/2020
Are there exist some positive integers nn and kk, such that the first decimals of 2n2^n (from left to the right) represent the number 5k5^k while the first decimals of 5n5^n represent the number 2k2^k ? (5)
number theory
Do p and q Exist?

Source: Spring 2006 Tournament of Towns Senior O-Level #2

9/9/2015
Do there exist functions p(x)p(x) and q(x)q(x), such that p(x)p(x) is an even function while p(q(x))p(q(x)) is an odd function (different from 0)?
(3 points)
functions
TOT 2006 Fall - Junior O-Level p2 Knight,Knave,Normal truth or lie

Source:

2/25/2020
A Knight always tells the truth. A Knave always lies. A Normal may either lie or tell the truth. You are allowed to ask questions that can be answered with ''yes" or ''no", such as ''is this person a Normal?" (a) There are three people in front of you. One is a Knight, another one is a Knave, and the third one is a Normal. They all know the identities of one another. How can you too learn the identity of each? (1) (b) There are four people in front of you. One is a Knight, another one is a Knave, and the other two are Normals. They all know the identities of one another. Prove that the Normals may agree in advance to answer your questions in such a way that you will not be able to learn the identity of any of the four people. (3)
combinatorics
TOT 2006 Fall - Senior O-Level p2 lines joining incenters perpendicular

Source:

2/25/2020
The incircle of the quadrilateral ABCDABCD touches AB,BC,CDAB,BC, CD and DADA at E,F,GE, F,G and HH respectively. Prove that the line joining the incentres of triangles HAEHAE and FCGFCG is perpendicular to the line joining the incentres of triangles EBFEBF and GDHGDH. (4)
geometryincenterperpendicular
TOT 2006 Fall - Senior A-Level p2 altitudes wanted

Source:

2/25/2020
Suppose ABCABC is an acute triangle. Points A1,B1A_1, B_1 and C1C_1 are chosen on sides BC,ACBC, AC and ABAB respectively so that the rays A1A,B1BA_1A, B_1B and C1CC_1C are bisectors of triangle A1B1C1A_1B_1C_1. Prove that AA1,BB1AA_1, BB_1 and CC1CC_1 are altitudes of triangle ABCABC. (6)
altitudegeometry
TOT 2006 Fall - Junior A-Level p2 non-acquainted persons to Ann

Source:

2/25/2020
When Ann meets new people, she tries to find out who is acquainted with who. In order to memorize it she draws a circle in which each person is depicted by a chord; moreover, chords corresponding to acquainted persons intersect (possibly at the ends), while the chords corresponding to non-acquainted persons do not. Ann believes that such set of chords exists for any company. Is her judgement correct? (5)
combinatorics