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Mathbound · Contest practice

Mixed topics worksheet

Mixed difficulty

Name

Date

1.
Lattice paths $(0,0)\to(8,8)$ with steps $(1,0),(0,1)$ avoiding $(6,2)$ . How many?
- A.12870
- B.784
- C.12114
- D.12086
- E.12076
2.
How many positive integers $n\le 50$ satisfy $n\equiv 4\pmod{6}$ ?
- A.8
- B.10
- C.9
- D.11
- E.7
3.
Lattice points strictly inside the right triangle with vertices $(0,0)$ , $(13,0)$ , $(0,13)$ ?
- A.73
- B.26
- C.84.5
- D.72
- E.71
4.
What is the sum of the first $26$ positive integers?
- A.351
- B.350
- C.377
- D.52
- E.676
5.
Arithmetic sequence: first term $4$ , common difference $7$ . Sum of first $10$ terms?
- A.355
- B.348
- C.365
- D.40
- E.67
6.
What is the maximum product of positive integers that sum to $21$ ?
- A.2190
- B.2188
- C.1024
- D.2185
- E.2187
7.
Arithmetic sequence: first term $3$ , common difference $7$ . Sum of first $12$ terms?
- A.498
- B.491
- C.510
- D.36
- E.80
8.
Lattice paths $(0,0)\to(7,7)$ with steps $(1,0),(0,1)$ avoiding $(2,5)$ . How many?
- A.3012
- B.2991
- C.2981
- D.3432
- E.441
9.
How many positive integers $k\le 80$ have exactly three positive divisors?
- A.3
- B.8
- C.4
- D.7
- E.5
10.
Arithmetic sequence: first term $2$ , common difference $7$ . Sum of first $10$ terms?
- A.65
- B.335
- C.328
- D.345
- E.20

Answer key · worked solutions

1. D. 12086
$\binom{2n}{n}$ minus paths through the obstacle. Subtract product of binomials. $12086$ .
2. A. 8
Arithmetic progression. First 4, step 6. $8$ .
3. D. 72
$\frac{(A-1)(B-1)}{2}$ for this triangle. Formula. $72$ .
4. A. 351
$n(n+1)/2$ . $26(26+1)/2$ . $351$ .
5. A. 355
$S_n=\frac n2(2a+(n-1)d)$ . Plug values. $355$ .
6. E. 2187
Prefer parts equal to $3$ (use $2$ s when needed; avoid $1$ s). Partition $21$ optimally. $2187$ .
7. A. 498
$S_n=\frac n2(2a+(n-1)d)$ . Plug values. $498$ .
8. B. 2991
$\binom{2n}{n}$ minus paths through the obstacle. Subtract product of binomials. $2991$ .
9. C. 4
Exactly three divisors $\iff k=p^{2}$ for a prime $p$ . Count primes $p\le\sqrt{80}$ . $4$ .
10. B. 335
$S_n=\frac n2(2a+(n-1)d)$ . Plug values. $335$ .

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