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Mathbound · Contest practice

Mixed topics worksheet

Mixed difficulty

Name

Date

1.
What is the sum of the first $14$ positive integers?
- A.196
- B.105
- C.104
- D.119
- E.28
2.
Positive integer solutions to $x_1+\cdots+x_{6}=25$ with each $x_i\le 8$ ?
- A.142506
- B.42504
- C.17137
- D.17131
- E.17136
3.
Arithmetic sequence: first term $3$ , common difference $7$ . Sum of first $25$ terms?
- A.2200
- B.75
- C.171
- D.2175
- E.2168
4.
Positive integer solutions to $x_1+\cdots+x_{6}=23$ with each $x_i\le 8$ ?
- A.14413
- B.14407
- C.14412
- D.98280
- E.26334
5.
Arithmetic sequence: first term $3$ , common difference $5$ . Sum of first $25$ terms?
- A.1570
- B.1600
- C.75
- D.123
- E.1575
6.
Equilateral triangle side $12$ . Area is $a\sqrt{3}$ . What is $a$ ?
- A.107
- B.109
- C.144
- D.72
- E.108
7.
A square has side length $38$ . What is its perimeter?
- A.152
- B.153
- C.1444
- D.151
- E.76
8.
How many positive divisors does $29$ have?
- A.3
- B.4
- C.1
- D.2
- E.5
9.
Lattice paths $(0,0)\to(9,9)$ with steps $(1,0),(0,1)$ avoiding $(5,7)$ . How many?
- A.48620
- B.11880
- C.36866
- D.36740
- E.36730
10.
How many positive integers $n\le 200$ satisfy $n\equiv 1\pmod{5}$ ?
- A.41
- B.43
- C.39
- D.40
- E.42

Answer key · worked solutions

1. B. 105
$n(n+1)/2$ . $14(14+1)/2$ . $105$ .
2. E. 17136
Inclusion-exclusion after $y_i=x_i-1$ . Upper bound 8. $17136$ .
3. D. 2175
$S_n=\frac n2(2a+(n-1)d)$ . Plug values. $2175$ .
4. C. 14412
Inclusion-exclusion after $y_i=x_i-1$ . Upper bound 8. $14412$ .
5. E. 1575
$S_n=\frac n2(2a+(n-1)d)$ . Plug values. $1575$ .
6. E. 108
$\frac{\sqrt{3}}{4}s^2$ . $a=s^2/4$ . $108$ .
7. A. 152
Perimeter $4s$ . $4\cdot38$ . $152$ .
8. D. 2
Factor and use $\prod(e_i+1)$ . Count divisors of $29$ . $2$ .
9. D. 36740
$\binom{2n}{n}$ minus paths through the obstacle. Subtract product of binomials. $36740$ .
10. D. 40
Arithmetic progression. First 1, step 5. $40$ .

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