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Mathbound · Contest practice

Mixed topics worksheet

Mixed difficulty

Name

Date

1.
How many subsets of $\{1,2,3,\ldots,12\}$ (including $\varnothing$ ) have total element-sum divisible by $3$ ?
- A.1360
- B.1376
- C.1392
- D.2048
- E.4096
2.
Lattice points strictly inside the right triangle with vertices $(0,0)$ , $(16,0)$ , $(0,7)$ ?
- A.46
- B.23
- C.56
- D.45
- E.44
3.
Solve for $x$ : $3x+6=21$ .
- A.5
- B.21
- C.6
- D.4
- E.z4
4.
Positive integer solutions to $x_1+\cdots+x_{4}=19$ with each $x_i\le 8$ ?
- A.331
- B.336
- C.1540
- D.816
- E.337
5.
Lattice paths $(0,0)\to(6,6)$ with steps $(1,0),(0,1)$ avoiding $(2,2)$ . How many?
- A.924
- B.420
- C.519
- D.504
- E.494
6.
How many positive divisors does $19$ have?
- A.2
- B.4
- C.3
- D.19
- E.1
7.
If $r$ and $s$ are roots of $x^2+8x+18=0$ , find $r^3+s^3$ (without solving).
- A.88
- B.144
- C.512
- D.80
- E.62
8.
An item costs $28$ . After a $25\%$ discount (no tax), how much is paid?
- A.22.00
- B.7
- C.21
- D.3
- E.28
9.
How many positive integers $n\le 200$ satisfy $n\equiv 6\pmod{8}$ ?
- A.27
- B.26
- C.28
- D.24
- E.25
10.
What is $\operatorname{lcm}(12,60)$ ?
- A.720
- B.12
- C.72
- D.60
- E.0

Answer key · worked solutions

1. B. 1376
Roots-of-unity filter: $\frac{1}{3}\sum_{j=0}^{2}\prod_{k=1}^{12}(1+\omega^{jk})$ . Not exactly $2^{12}/3$ because sums are not uniform. $1376$ .
2. D. 45
$\frac{(A-1)(B-1)}{2}$ for this triangle. Formula. $45$ .
3. A. 5
Isolate $x$ . $3x=21-6$ . $5$ .
4. B. 336
Inclusion-exclusion after $y_i=x_i-1$ . Upper bound 8. $336$ .
5. D. 504
$\binom{2n}{n}$ minus paths through the obstacle. Subtract product of binomials. $504$ .
6. A. 2
Factor and use $\prod(e_i+1)$ . Count divisors of $19$ . $2$ .
7. D. 80
$r^3+s^3=(r+s)((r+s)^2-3rs)$ . Vieta: $r+s=8$ , $rs=18$ . $80$ .
8. C. 21
Multiply by $(1-d)$ . $28(1-25/100)$ . $21$ .
9. E. 25
Arithmetic progression. First 6, step 8. $25$ .
10. D. 60
$\operatorname{lcm}(a,b)=ab/\gcd(a,b)$ . $\gcd=12$ . $60$ .

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