answer:Step 1: Calculate Inflation Over Two Years Given the annual inflation rate is (2.5%), we need to calculate the total inflation over two years. We can use the compound interest formula to find this: [ (1 + text{annual rate})^{text{number of years}} ] Substitute the given values: [ left(1 + 0.025right)^2 - 1 ] Calculate: [ = (1.025)^2 - 1 ] [ = 1.050625 - 1 ] [ = 0.050625 ] Convert this to a percentage: [ 0.050625 times 100% = 5.0625% ] Therefore, the total inflation over two years is: [ boxed{5.0625%} ] Step 2: Calculate the Real Yield of the Bank Deposit Given that the nominal rate for the first and second year is (6%). We need to find the real yield on the bank deposit, considering the compounded nominal yield and subtract the inflation calculated above. First, calculate the compounded nominal rate for two years: [ 1.06 times 1.06 ] Calculate it step-by-step: [ = 1.06 times 1.06 ] [ = 1.1236 ] Now, adjust this compounded nominal rate by the two-year inflation rate: [ frac{1.1236}{1 + 0.050625} - 1 ] Calculate the denominator: [ 1 + 0.050625 = 1.050625 ] Thus, [ frac{1.1236}{1.050625} - 1 ] Now, divide and subtract 1: [ = 1.069459 - 1 ] [ = 0.069459 ] Convert this to a percentage: [ 0.069459 times 100% = 6.9459% ] Rounding to two decimal places, we get the real yield: [ boxed{6.95%} ]

answer:1. **Given Conditions:** - ( p ) is a prime number. - ( p < n ). - ( p ) divides ( n + 1 ), i.e., ( n + 1 equiv 0 pmod{p} ). - ( left(leftlfloor frac{n}{p} rightrfloor, (p-1)!right) = 1 ). 2. **Express ( n ) in terms of ( p ):** Since ( p ) divides ( n + 1 ), we can write: [ n + 1 = kp quad text{for some integer } k. ] Therefore, [ n = kp - 1. ] 3. **Calculate ( leftlfloor frac{n}{p} rightrfloor ):** [ leftlfloor frac{n}{p} rightrfloor = leftlfloor frac{kp - 1}{p} rightrfloor = k - 1. ] 4. **Simplify the binomial coefficient ( {n choose p} ):** [ {n choose p} = frac{n(n-1)(n-2)cdots(n-p+1)}{p!}. ] Since ( n = kp - 1 ), we have: [ n - i = kp - 1 - i quad text{for } i = 0, 1, 2, ldots, p-1. ] 5. **Modulo ( p ) simplification:** Since ( n equiv -1 pmod{p} ), we have: [ n - i equiv -1 - i pmod{p}. ] Therefore, the product in the numerator of the binomial coefficient modulo ( p ) is: [ (-1)(-2)(-3)cdots(-(p-1)) equiv (-1)^{p-1}(p-1)! pmod{p}. ] Since ( (-1)^{p-1} = 1 ) for odd ( p ) and ( (p-1)! equiv -1 pmod{p} ) by Wilson's Theorem, we get: [ {n choose p} equiv frac{(-1)^{p-1}(p-1)!}{p!} equiv frac{(p-1)!}{p!} equiv frac{(p-1)!}{p(p-1)!} equiv 0 pmod{p}. ] 6. **Consider the expression ( {n choose p} - leftlfloor frac{n}{p} rightrfloor ):** [ {n choose p} - leftlfloor frac{n}{p} rightrfloor equiv 0 - (k-1) equiv -(k-1) pmod{p}. ] 7. **Check divisibility by ( p cdot leftlfloor frac{n}{p} rightrfloor^2 ):** [ p cdot leftlfloor frac{n}{p} rightrfloor^2 = p cdot (k-1)^2. ] Since ( {n choose p} - leftlfloor frac{n}{p} rightrfloor equiv -(k-1) pmod{p} ), we need to show that: [ p cdot (k-1)^2 text{ divides } -(k-1). ] This is true because ( p ) divides ( -(k-1) ) and ( (k-1)^2 ) is an integer. Therefore, we have shown that ( p cdot leftlfloor frac{n}{p} rightrfloor^2 ) divides ( {n choose p} - leftlfloor frac{n}{p} rightrfloor ). (blacksquare)

answer:1. Recognize that the diagonal of a square divides the square into two 45-45-90 right triangles. Hence, the relation between the side length ( s ) of the square and its diagonal ( d ) is given by: [ d = ssqrt{2} ] 2. Given the diagonal ( d = 12sqrt{2} ) centimeters, solve for ( s ) by rearranging the formula: [ s = frac{d}{sqrt{2}} = frac{12sqrt{2}}{sqrt{2}} = 12 text{ centimeters} ] 3. Calculate the area ( A ) of the square using the side length: [ A = s^2 = 12^2 = 144 text{ square centimeters} ] The area of the square is ( boxed{144} ) square centimeters.

answer:Mary has already put in 2 cups of sugar and needs to add 12 more cups, making a total of 2 + 12 = 14 cups of sugar for her current cake. However, she doubled the amount of flour from 5 cups to 10 cups, which means she also doubled the amount of sugar needed for the recipe. To find out how much sugar the original recipe calls for, we divide the total amount of sugar she is using by 2 (since she doubled the ingredients): 14 cups of sugar ÷ 2 = 7 cups of sugar The recipe calls for boxed{7} cups of sugar.