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Mathbound · Contest practice

Mixed topics worksheet

Mixed difficulty

Name

Date

1.
How many pairs of positive integers $(x,y)$ with $x\le y$ satisfy $\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{1}{12}$ ?
- A.6
- B.7
- C.8
- D.9
- E.12
2.
A circle has radius $5$ . Chord of length $8$ — distance from center to chord?
- A.\sqrt{41}
- B.3
- C.\sqrt{39}
- D.2
- E.4
3.
Coefficient of $x^4$ in $(1+x+x^2)^5$ ?
- A.70
- B.45
- C.50
- D.55
- E.87
4.
How many positive integers $n\le 150$ satisfy $n\equiv 4\pmod{6}$ ?
- A.27
- B.26
- C.28
- D.24
- E.25
5.
An item costs $45$ . It is discounted $15\%$ , then $10\%$ tax is applied to the discounted price. How much is paid?
- A.40.58
- B.38
- C.50
- D.44.08
- E.42.08
6.
An item costs $50$ . It is discounted $10\%$ , then $5\%$ tax is applied to the discounted price. How much is paid?
- A.49.25
- B.47.25
- C.45.75
- D.45
- E.53
7.
Trailing zeros of $50!$ in base $10$ ?
- A.14
- B.10
- C.11
- D.13
- E.12
8.
What is the distance from the origin to the line $3x+4y-10=0$ ?
- A.3
- B.1
- C.3/2
- D.5/2
- E.2
9.
How many prime numbers satisfy $20 < p < 40$ ?
- A.6
- B.4
- C.7
- D.3
- E.5
10.
How many ordered pairs of positive integers $(m,n)$ satisfy $\operatorname{lcm}(m,n)=2^{5}\cdot3^{4}\cdot5^{2}$ ?
- A.405
- B.450
- C.495
- D.540
- E.729

Answer key · worked solutions

1. C. 8
$(x-12)(y-12)=144$ after clearing denominators. Count factor pairs $d\le12$ of $144$ with $d\le144/d$ . $8$ .
2. B. 3
Right triangle half-chord. $\sqrt{25-16}=3$ . $3$ .
3. B. 45
Generating functions. Exponents in $\{0,1,2\}$ sum to $4$ . $45$ .
4. E. 25
Arithmetic progression. First 4, step 6. $25$ .
5. E. 42.08
Discount first, then tax. $45\cdot(1-15/100)\cdot(1+10/100)$ . $42.08$ .
6. B. 47.25
Discount first, then tax. $50\cdot(1-10/100)\cdot(1+5/100)$ . $47.25$ .
7. E. 12
Exponent of $5$ . $10+2$ . $12$ .
8. E. 2
Point-line distance. $10/5=2$ . $2$ .
9. B. 4
List. $23,29,31,37$ . $4$ .
10. C. 495
Per prime, $\max(a,b)=e$ has $2e+1$ pairs of exponents. $(11)(9)(5)$ . $495$ .

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