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Mathbound · Contest practice

Mixed topics worksheet

Mixed difficulty

Name

Date

1.
Arithmetic sequence: first term $4$ , common difference $4$ . Sum of first $12$ terms?
- A.312
- B.308
- C.324
- D.48
- E.z4
2.
How many positive divisors does $35$ have?
- A.6
- B.3
- C.4
- D.7
- E.5
3.
What is $\operatorname{lcm}(30,48)$ ?
- A.6
- B.270
- C.240
- D.192
- E.1440
4.
How many positive integers $n\le 150$ satisfy $n\equiv 1\pmod{5}$ ?
- A.29
- B.30
- C.32
- D.31
- E.33
5.
Lattice paths $(0,0)\to(6,6)$ with steps $(1,0),(0,1)$ avoiding $(4,2)$ . How many?
- A.714
- B.699
- C.689
- D.924
- E.225
6.
What is $\operatorname{lcm}(20,48)$ ?
- A.240
- B.260
- C.960
- D.192
- E.4
7.
How many positive integers $n\le 200$ satisfy $n\equiv 2\pmod{6}$ ?
- A.33
- B.34
- C.36
- D.35
- E.alt4
8.
How many pairs of positive integers $(x,y)$ with $x\le y$ satisfy $\dfrac1x+\dfrac1y=\dfrac1{12}$ ?
- A.8
- B.15
- C.9
- D.12
- E.7
9.
A committee of $4$ is chosen from $13$ people. How many committees?
- A.495
- B.52
- C.715
- D.728
- E.286
10.
An item costs $65$ . It is discounted $25\%$ , then $5\%$ tax is applied to the discounted price. How much is paid?
- A.68
- B.53.19
- C.51.19
- D.49.69
- E.49

Answer key · worked solutions

1. A. 312
$S_n=\frac n2(2a+(n-1)d)$ . Plug values. $312$ .
2. C. 4
Factor and use $\prod(e_i+1)$ . Count divisors of $35$ . $4$ .
3. C. 240
$\operatorname{lcm}(a,b)=ab/\gcd(a,b)$ . $\gcd=6$ . $240$ .
4. B. 30
Arithmetic progression. First 1, step 5. $30$ .
5. B. 699
$\binom{2n}{n}$ minus paths through the obstacle. Subtract product of binomials. $699$ .
6. A. 240
$\operatorname{lcm}=ab/\gcd$ . $\gcd=4$ . $240$ .
7. B. 34
Count an arithmetic progression. First 2, common difference 6. $34$ .
8. A. 8
$(x-t)(y-t)=t^2$ . Divisors $d\le t$ of $t^2$ . $8$ .
9. C. 715
$\binom{m}{k}$ . $\binom{13}{4}$ . $715$ .
10. C. 51.19
Discount first, then tax. $65\cdot(1-25/100)\cdot(1+5/100)$ . $51.19$ .

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