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Mathbound · Contest practice

Mixed topics worksheet

Mixed difficulty

Name

Date

1.
An item costs $60$ . After a $10\%$ discount (no tax), how much is paid?
- A.50
- B.60
- C.55.00
- D.6
- E.54
2.
How many positive integers $n\le 50$ are divisible by $6$ ?
- A.8
- B.7
- C.9
- D.44
- E.z4
3.
How many positive integers $n\le 250$ satisfy $n\equiv 7\pmod{8}$ ?
- A.32
- B.34
- C.30
- D.31
- E.33
4.
A square has side length $13$ . What is its perimeter?
- A.26
- B.52
- C.53
- D.169
- E.51
5.
What is $\operatorname{lcm}(24,42)$ ?
- A.192
- B.168
- C.126
- D.1008
- E.6
6.
What is $\operatorname{lcm}(18,54)$ ?
- A.0
- B.972
- C.18
- D.72
- E.54
7.
What is $\operatorname{lcm}(18,70)$ ?
- A.560
- B.2
- C.630
- D.648
- E.1260
8.
If $r$ and $s$ are roots of $x^2+8x+6=0$ , find $r^3+s^3$ (without solving).
- A.512
- B.368
- C.362
- D.376
- E.48
9.
What is $\operatorname{lcm}(12,44)$ ?
- A.132
- B.144
- C.528
- D.88
- E.4
10.
Lattice paths $(0,0)\to(9,9)$ with steps $(1,0),(0,1)$ avoiding $(5,6)$ . How many?
- A.48620
- B.16170
- C.32576
- D.32450
- E.32440

Answer key · worked solutions

1. E. 54
Multiply by $(1-d)$ . $60(1-10/100)$ . $54$ .
2. A. 8
Count multiples. $\lfloor 50/6\rfloor$ . $8$ .
3. D. 31
Arithmetic progression. First 7, step 8. $31$ .
4. B. 52
Perimeter $4s$ . $4\cdot13$ . $52$ .
5. B. 168
$\operatorname{lcm}(a,b)=ab/\gcd(a,b)$ . $\gcd=6$ . $168$ .
6. E. 54
$\operatorname{lcm}(a,b)=ab/\gcd(a,b)$ . $\gcd=18$ . $54$ .
7. C. 630
$\operatorname{lcm}=ab/\gcd$ . $\gcd=2$ . $630$ .
8. B. 368
$r^3+s^3=(r+s)((r+s)^2-3rs)$ . Vieta: $r+s=8$ , $rs=6$ . $368$ .
9. A. 132
$\operatorname{lcm}=ab/\gcd$ . $\gcd=4$ . $132$ .
10. D. 32450
$\binom{2n}{n}$ minus paths through the obstacle. Subtract product of binomials. $32450$ .

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