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Mathbound · Contest practice

Mixed topics worksheet

Mixed difficulty

Name

Date

1.
How many positive integers $n\le 50$ are divisible by $7$ ?
- A.8
- B.43
- C.6
- D.7
- E.z4
2.
A square has side length $33$ . What is its perimeter?
- A.1089
- B.131
- C.66
- D.132
- E.133
3.
Compute $2^{12}\bmod 11$ .
- A.6
- B.7
- C.4
- D.2
- E.5
4.
An item costs $50$ . It is discounted $20\%$ , then $10\%$ tax is applied to the discounted price. How much is paid?
- A.42.50
- B.40
- C.55
- D.46.00
- E.44
5.
Arithmetic sequence: first term $3$ , common difference $3$ . Sum of first $15$ terms?
- A.375
- B.45
- C.360
- D.357
- E.z4
6.
A committee of $2$ is chosen from $11$ people. How many committees?
- A.66
- B.11
- C.45
- D.22
- E.55
7.
How many positive divisors does $12$ have?
- A.5
- B.6
- C.8
- D.7
- E.12
8.
How many positive integers $n\le 200$ satisfy $n\equiv 4\pmod{6}$ ?
- A.36
- B.32
- C.33
- D.35
- E.34
9.
Lattice points strictly inside the right triangle with vertices $(0,0)$ , $(11,0)$ , $(0,15)$ ?
- A.69
- B.71
- C.26
- D.82.5
- E.70
10.
Compute $3^{8}\bmod 11$ .
- A.3
- B.6
- C.7
- D.8
- E.5

Answer key · worked solutions

1. D. 7
Count multiples. $\lfloor 50/7\rfloor$ . $7$ .
2. D. 132
Perimeter $4s$ . $4\cdot33$ . $132$ .
3. C. 4
Successive squaring / cycles. Reduce powers modulo 11. $4$ .
4. E. 44
Discount first, then tax. $50\cdot(1-20/100)\cdot(1+10/100)$ . $44$ .
5. C. 360
$S_n=\frac n2(2a+(n-1)d)$ . Plug values. $360$ .
6. E. 55
$\binom{m}{k}$ . $\binom{11}{2}$ . $55$ .
7. B. 6
Factor and use $\prod(e_i+1)$ . Count divisors of $12$ . $6$ .
8. C. 33
Arithmetic progression. First 4, step 6. $33$ .
9. E. 70
$\frac{(A-1)(B-1)}{2}$ for this triangle. Formula. $70$ .
10. E. 5
Successive squaring / cycles. Reduce powers modulo 11. $5$ .

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