1

Algebra I Extra Credit Problems

#1. If

(a + b)2

(a − b)2= 1 , then which of the following

statements must be true?

I. a = 0

II. b = 0

III. a = –b

(A) None

(B) I only

(C) II only

(D) III only

(E) I, II, and III

#2. Is 2( 4) 4 ?y y− = −− = −− = −− = −

(1) |y – 3| ≤ 1

(2) y · |y| > 0

#3. The sequence an is defined so that, for all

n ≥ 3, an is the greater of (an – 2 + 1) and

(an – 1). (If the two quantities are the same,

then an is equal to either of them.) Which of

the following values of a1 and a2 will produce

a sequence in which no value is repeated?

(A) a1 = –1, a2 = –1.5

(B) a1 = –1, a2 = 1

(C) a1 = 1, a2 = –1

(D) a1 = 1, a2 = 1.5

(E) a1 = 1.5, a2 = 1

#4. If x and y are integers such that x < y < 0,

what is x – y?

(1) (x + y)(x – y) = 5

(2) xy = 6

2

#1. Expand out the squared binomials in the numerator and denominator:

a2

+ 2ab + b2

a2

− 2ab + b2

= 1

Multiply by the denominator, thereby getting rid of fractions: a2 + 2ab + b

2 = a

2 – 2ab + b

2.

Cancel terms: 2ab = –2ab

4ab = 0

ab = 0

Therefore, a = 0 OR b = 0 OR a and b both equal 0, but neither of the two must be 0. In addition, as long

as one of a and b is 0, the other can be any number. Therefore, none of the statements is necessarily true.

The correct answer is A.

#2. The complicated expression in the question stem leads to a disguised Positive/Negative problem. In

general, 2

x x= . Think about this relationship with a real example:

32= 9 = 3 (−3)2

= 9 = 3

In both cases (positive or negative 3), the end result is 3. Thus in general, 2

x will always result in a

positive value, or |x|. Therefore, ( )2

4 4− = −y y . We can rephrase the original question:

Is |y – 4| = 4 – y? → Is |y – 4| = –(y – 4)?

Since the absolute value of y – 4 must be positive or zero, we can rephrase the question further:

Is –(y – 4) ≥ 0? Is (y – 4) ≤ 0? Is y ≤ 4?

(1): SUFFICIENT: Solve for the range of y. To evaluate the absolute value, set up two equations.

+ (y – 3) ≤ 1 – (y – 3) ≤ 1

y – 3 ≤ 1 –y + 3 ≤ 1

y ≤ 4 –y ≤ –2

y ≥ 2

Taking these two equations together, we find that 2 ≤ y ≤ 4. Therefore, y is definitely less than or equal to

4, and the statement is sufficient.

(2): INSUFFICIENT: If y · |y| > 0, then y · |y| is positive. The term |y| is non-negative. Therefore, y must

be positive. However, knowing that y is positive doesn’t tells us whether y ≤ 4.

The correct answer is A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

3

#3. Work out the sequence for each of the given pairs of initial values until either (a) a repeated value

eliminates that answer choice from contention, or (b) a pattern emerges in which no values are repeated.

Choice (A) gives –1, –1.5, 0, 0, … Since zero is repeated, this choice is incorrect.

Choice (B) gives –1, 1, 1, … Since 1 is repeated, this choice is incorrect.

Choice (C) gives 1, –1, 2, 2, … Since 2 is repeated, this choice is incorrect.

Choice (D) gives 1, 1.5, 2, 2.5, 3, 3.5, … The pattern here is clear: each value in the sequence is 0.5

greater than the preceding value. Therefore, none of the values will repeat.

Choice (E) gives 1.5, 1, 2.5, 2.5, … Since 2.5 is repeated, this choice is incorrect.

The correct answer is D.

#4.

(1): SUFFICIENT. This statement is tricky. One approach is to distribute the left side to get the

difference of squares:

x2 – y

2 = 5 → (x + y)(x – y) = 5

Note that both x + y and x – y are themselves integers. Looking at the statement, we have

int × int = 5

The only possible integer pairs that give 5 as a product are (5, 1) and (-5, -1), since 5 is prime. Now,

because both x and y are negative, the (5, 1) pair won’t work either way (either with x + y = 5 or with x +

y = 1). So let’s try (-5, -1):

x + y = –5

x – y = –1

Adding these two equations, we get 2x = –6, or x = –3. Substituting back into

x + y = –5, we get y = –2. (If we had assigned x + y = –1 and x – y = –5, we would have gotten y = 2,

which doesn’t fit the problem constraints.)

(2): INSUFFICIENT. There are two pairs of integers that fit the constraint x < y < 0 and the statement xy

= 6: (–3)(–2) = 6 AND (–6)(–1) = 6.

The correct answer is A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.