For any real number x, let l x J be the largest integer less than or equal to x and x = x - l x J. Which of the following statements is incorrect. A does not teach Mathematics and B does not work in school Z;. ii) The teacher in school Z teaches Music;. iii). It is known that the largest root of the equation is -k times the smallest root.
When both the largest and smallest numbers are removed, the average of the remaining four numbers is 20. Given that the area of triangle CGF is 351, calculate the area of rectangle DEFG. On each occasion, you may draw at most one red marble, at most two blue marbles and a total of at most five marbles from the pack.
Find the minimum number of shots needed to remove all the marbles from the pack. The number of ways in which all k people can sit so that the couple sits together is equal to the number of ways in which the (k - 2) people, without the couple present, can sit.
Answer: 4
Answer: 25
Answer: 9
Answer: 110
Answer: 50
Answer: 35
Answer: 1818
Answer: 4
Answer: 3150
Answer: 840
Answer: 27
Answer: 1
Answer: 12
Answer: 1976
Answer: 12
Let M, N be the bisectors of AP, BP and Q the intersection of M C and N D respectively. Prove that 0, P and Q are collinear. Is there an integer A such that each of the ten digits is 0, 1, . exactly once as a digit in exactly one of the numbers A, A2, A 3. In L.ABC, the external bisectors LA and LB meet at point D. Prove that the center of the circle is L.ABD and the points C, D lie on the same line.
Determine the values of the positive integer n for which the following system of equations has a solution in positive integers x1, x2, •. Find all solutions for each of these n. Prove that S contains a prime number. Please note". For the multiple choice questions, fill in your answer on the answer sheet by shading the balloon with the letter (A, B, C, D or E) that corresponds to the correct answer. If the last three digits of 2012m and 2012n are identical, find the smallest possible value of m + n.
Find the number of ways to arrange the letters A, A, B, B, C, C, D and E in a line, so that there are no consecutive identical letters. Find the smallest value of the expression (x + y) (y + z), given that x, y, z are positive real numbers that satisfy the equation. Find the number of selections where the sum of the three integers is divisible by 3.
The losing player is eliminated while the winning player continues to play with the next player on the opposite team. Let L denote the minimum value of the quotient of a 3-digit number formed by three different digits divided by the sum of its digits. There are � ways to arrange the letters so that both As and B are consecutive.
Answer: 384
Answer: 3432
Answer: 105
A circle w through the center I of a triangle ABC and tangent to AB at A, intersects the segment BC at D and the extension of BC at E. Determine all positive integers n such that n is equal to the square of the sum of the digits i n. If 46 squares are colored red in a 9 x 9 board, show that there is a 2 x 2 block on the board in which at least 3 of the squares are colored red.
In the same way we can show that ak 2: -bk for all k and therefore ak :::; bk as required. Write your answers on the provided answer sheet and color the correct bubbles below your answers. Find the total number of sets of positive integers (x, y, z), where x, y, and z are positive integers, with x < y < z such that.
Determine the largest even positive integer that cannot be expressed as the sum of two composite odd positive integers. In the diagram below, the point and D lie inside the triangle ABC, so LEAD= LBCD LBDC = 90°. Given that f is a real valued function on the set of all real numbers, such that for all real numbers a and b,.
Determine the maximum number of different sets consisting of three terms that form an arithmetic progression that can be chosen from a set of real numbers a1 , a2, · · · , a101 , where.
Answer: 121
Answer: 7986
Answer: 4
Answer: 3333
Answer: 62
Answer: 3015
Answer: 38
Answer: 76
Answer: 4
Answer: 2015
Answer: 86
Answer: 337
- Answer: 21
- Answer: 2500
- Answer: 20121
We will show that this is its only complete solution by proving that if x,y,z is a solution of this equation and whenever x,y,z are divisible by 2k, they are also divisible by 2k+ 1 for every k 2: : 0. We now show that for any given sequence of real numbers a1 < a2 < · · · < aw1 , there are at most 2500 different three-term arithmetic progressions that can be chosen from this sequence. So the number of different three-term arithmetic progressions that can be chosen from this sequence is at most.
The inscribed circle with center I of triangle ABC touches sides BC, CA and AB at points D, E and F respectively. Show that we can find integers x, y, z such that xai + ybi + zci is odd by at least 4N /7 of different values i. There are 2012 different points on the plane, each of which must be painted with one of n colors, so that the number of points of each color is different.
We can check directly that any function of this form (for some a, b E JR.) satisfies the given equation. There are thus choices of (x, y, z) for which at least 4N/7 of the corresponding sums are odd. There are thus 4 odd numbers in each row, which gives a total of 4N odd sums in the table.
The last summation counts the even-sized nonempty subsets of a p-element set, of which there are 2P-1 - 1.
