Received: 5/5/20, Revised: 12/28/20, Accepted: 5/10/21, Published: 5/17/21 Abstract Let Sbe a set of points in R2 contained in a circle and P an unrestricted point set in R2 We prove that the number of distinct distances between points in Sand points in Pis at least the minimum of jSjjPj1 =4 ";jSj2=3jPj2 3;jSj2; and jPj2 This
Jun 22, 2003 · De nition 1 2 3 (Congruence) Let a;b;mbe integers We say that a b(mod m) (\ais congruent to bmodulo m") if mja b Examples: 11 10 (mod 7) because 7 j11 ( 10) = 21:Two integers are congruent modulo 2 exactly if they have the same parity (both are even or both are odd) Exercise 1 2 4 Prove the following statements The universe is Z
produces a matrix whose elements are integers between 1 and n2 and which has equal row, column and diagonal sums What is not obvious, but is true, is that there are no duplicates So M must contain all of the integers between 1 and n2 and consequently is a magic square M = 17 24 1 8 15 23 5 7 14 16 4 6 13 20 22 10 12 19 21 3 11 18 25 2 9
24 If y= 2x+ 1 is tangent to the circle centered at the origin in an xy-plane Find the radius of the circle 1(a) p 3 (b) 3 p 5 (c) 5 p 2(d) p 2 (e) 2 25 Find the area of the given gure (Note: Figure not drawn to scale) 1 2 1 2 p 3 2 p 3 2 p 3 (a) 4 p 3 (b) 2 p 3+1 (c) 4 p 3+1 (d)None of these 2 (e) 26 How many integers between 1 and 1500
6 How many positive integers between 0 and 100, inclusive, have no prime factor greater than 5? 7 A positive integer leaves a remainder of 2 when divided by 5 and a remainder of 7 when divided by 9 If is less than 100, what is the maximum possible value of ? 1 2 $ 3 faces 4 square inches 5 6 positive integers 7
How many integers from 1 to 1000 are divisible by 3 or 5? Solution: This is an application of the inclusion/exclusion principle There are 333 integers divisible by 3, 200 divisible by 5 and 66 divisible by both This gives a total of 333 + 200 - 66 = 467 Answer:
Problem 1: (Section 6 1 Exercise 14) How many integers between 1 and 21 ⌋ = 47 C ∩ D = ⌊1000 35 ⌋ = 28 A ∩ B ∩ C = ⌊1000 30 ⌋ = 33 Problem 6: (section 6 2 Exercise 10) How many possible telephone numbers consist
How many integers between 2004 and 4002 are perfect squares? 5 Let the operation 9 What is the area of the region of the plane satisfying 1 ≤ x+y ≤ 2? 10 + 1 ac + 1 bc ? 21 How many integers n satisfy n4 + 6n < 6n3 + n2? 22
You randomly select two distinct integers from 1 to 10 What is the Triangle ABC has tan(A)=3/4 and tan(B) = 21/20 What is AC/BC? How many integers between 1 and 1000 cannot be expressed as the difference of squares of integers?
Homework 12: 1,7,8,10,18,19,25,28,39,41,42 p 310 a) If we pick two representatives, one from each major, this is the scenario of the We consider positive integers less than 1000 and count how many satisfy certain properties b) no letters can be repeated: 26 x 25 x 24 x 23 x 22 x 21 x 20 x 19 = 62 990 928000
October 24, 2015 1 For how many integers n in the set {1,2,3, ,150} is n3 − n2 the Thus, there are 7 integers between 1 and 100 divisible by 14 21 Let f1 be the function defined by f1(x) = 1 - 1 x Let f2 be defined by f2(x) = f1(f1(x))
The Junior Mathematical Challenge (JMC) is a multiple-choice paper correct answers by working backwards from the given alternatives, or by showing that four of 9 D 10 E 11 B 12 D 13 C 14 B 15 A 16 D 17 B 18 A 19 A 20 B 21 C 22 B the 4 × 4 grid shown, with one integer in each small square, so that