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# Star and Pound Operations

Pro Problems > Math > Number and Quantity > Number Theory > Star Operations

## Star and Pound Operations

If a * b = (a + b)2, and a # b = (a - b)2, find x such that (2 * x) # (x * 1) = 81

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Problem by Mr. Twitchell

## Solution

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## Similar Problems

### Guess the Star

There exists a star (*) operation for which the following equations are true:

2 * 3 = 10
5 * 2 = 35
3 * 6 = 27
8 * 7 = 120

Find a star operation rule which satisfies all of these equations, and then use your rule to calculate the following:

10 * 11 = ?

### Non-Commutative Operation

If the operation a ♦ b is defined as follows: a ♦ b =
a
b
+ a, find the value of x such that:

x  ♦ 2x = 2x ♦ x.

Note that this operation is not commutative; the order of operands cannot be exchanged without altering the result of the operation.

### Diamond Operation

If the operation ♦ is defined as follows: a ♦ b = a2 - b2 , and 18 ♦ (x ♦ 8) = 6 ♦ 1, find all possible values of x.

### h and k

x * y = x2 - y2 + kx - hy. If 5 * 2 = 48 and 4 * 3 = 30, find the ordered pair (h, k).

### Fractional Operation

The operation a » b is defined as follows: a » b =
a
a + 1
+
b
b - 3
. Find the value of a such that a »  a =
12
5
.

### Hashtag Operation

The "hashtag operation" is defined as follows: a # b =
a + b
a
+
a + b
b
.

If x # 7 =
16
3
, find all possible values of x.

### Commutative Operation

The operation * is defined as follows: x * y = (ax + b)(by + a), for some real numbers a and b. Under what conditions is this operation commutative?

### Triangular Operation

The arrangement
a
b
c
means a - (b x c). For example,
20
3
4
means 20 - (3 x 4) = 20 - 12 = 8.

Find the value of
x
y
z
y
x
z
z
x
y
, if x = 10, y = 7, and z = 2.

### Star Operation with Cubes

If the operation (*) is defined as follows: a * b =
a3 + b3
a2 - ab + b2
, find the value of (7 * 3) * 4.

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