7. If [x] means the largest integer n =< x. Example: [17/2]= 8. Then what is {Squareroot(1)]+[Squareroot(2)]+[Squareroot(3)]+...+[Squareroot(16)]
10. Each old license plate consisted of a letter(English Alphabet) followed by 4 digits. Each new license plate consists of 3 letters followed by 3 digits. By how many times is the number of possible license plates increased?
12. 3 People: A,B,C split $1000 among them to be invested in different ways. Each begins with a different amount. At the end of one year they have a total of $1500. B and C have doubled their money, while A lost $100. How much did A have to start with?
13. Let &(x) denote the sum of digits of a positive integer x. Ex. &(342)=2+3+4=9. For how many 2-digit values of x is &(&(x))=3?
17. An ice-cream cone consists of a sphere of ice cream and a right circular cone that has the same diameter as the sphere. If the ice-cream melts, it will exactly fill the cone. Assume that the melted ice-cream occupies 75% of the volume of the frozen ice-cream. What is the ratio of the cone's height to its radius? ( A cone with radius r and height h has volume Pi(r^2h)/3 and sphere with radius r has volume Pi(4r^3)/3)
18. What is the largest integer that is a divisor of s where
s= (n+1)(n+3)(n+5)(n+7)(n+9)
for all positive EVEN integers n?
25. How many distinct 4-digit numbers are divisible by 3 and have 23 as their last 2 digits?
Solutions
7. Until [Squareroot(4)] all [ ]s are equal to 1 because squareroot(4) is 2. All [ ]s betweent [sqrt(4)] and [sqrt(9) are equal to 2. All [ ]s between [sqrt(9)] and [sqrt(16)] are equal to 3. And [sqrt(16)]=4. So add them all up:
1+1+1+2+2+2+2+2+3+3+3+3+3+3+3+4=38
10. Number of all possible license plates for 1 letter and 4 digits is 26(10^4) and number of all possible license plates for 3 letters and 3 digits is 26^3(10^3). So 26^3(10^3)/26(10^4)=26^2/10
12. A+B+C=1000 and after one year 2B+2C+A-100=1500. Then 2B+2C+A=1600.Subtracting the previous equation from this we get B+C=600. If A+B+C=1000 and B+C=600 then A=400
13. First number that works is 12, &(&(12))=3 The next is 21.. We see that these numbers appear every 9 numbers. So 12, 21, 30, 39, 48, 57, 66, 75, 84, 93 are the only ones that work. When we count we see there are 10. Another way to solve this is to subtact 12 from 99 to get 87. When we divide this by 9 we get 9.66666...This shows there are 9 more numbers that work after 12. If we count also count 12 there will be 10 such numbers.
17. When the ice-cream melts or loses 25% of its volume, its volume equals the cones volume. So 75% of the ice-cream equals 100% of the cone. 3(Pi(4r^3)/4(3) =Pi(hr^2)/3.. First of all we simplify 3(Pi(4r^3)/4(3) and we get Pi(r^3)= Pi(hr^2)/3. So 3Pi(r^3)=Pi(hr^2). Pi's cancel out to leave 3r^3=hr^2. If we divide both sides by r^2 we get 3r=h. So the ratio is 3 : 1.
18. For all even integers n, all factors of n are going to be odd numbers. At least 1 of them is going to be a multiple of 5 because there is one multiple of 5 in each 5 consecutive ODD integers.This doesn't work for 9 because there is 1 multiple of 9 in every 9 odd consecutive integers and there is one multiple of 7 in every 7 consecuitve odd integers. But here were dealing with 5 consecutive odd integers. There will be a multiple of 3 in every 3 consecutive odd integers. So at least one of the factors is a multiple of 3. So 3*5=15 will be the largest common factor of s for ANY even integer n.
25. Since the rule of being divisible by 3 is if the sum of the digits are divisible by 3 and since 23 adds up to 5 then the rest must complete it by adding 1+any number that is 0,3,6,9. We find all 2 digit numbers that are in the form 3n+1. 10 being the first number of the form 3n+1, where n equals to 3. We find the largest 2 digit number of the form 3n+1. 3(33)+1 is 100 so the largest one should be 3(32)+1. So 3n+1 is a 2 digit number for n=3,4,5,6...32. Counting these we find that there are 30 such 4 digit integers which the last 2 digits are 23 which are divisible by 3.