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Mathbound · Contest practice

Mixed topics worksheet

Mixed difficulty

Name

Date

1.
How many involutions (bijections $f$ with $f\circ f=\mathrm{id}$ ) exist on a set of $11$ elements?
- A.39266
- B.35696
- C.2048
- D.35707
- E.35685
2.
A committee of $3$ is chosen from $13$ people. How many committees?
- A.220
- B.39
- C.286
- D.299
- E.78
3.
Right triangle with legs $6$ and $8$ . Hypotenuse?
- A.14
- B.10
- C.2
- D.11
- E.9
4.
A circle has radius $10$ . Two parallel chords in this circle have lengths $12$ and $16$ , and both lie on the same side of the center. What is the distance between these two chords?
- A.1
- B.2
- C.3
- D.4
- E.5
5.
An arithmetic sequence has first term $5$ and common difference $4$ . What is the sum of its first $20$ terms?
- A.840
- B.860
- C.880
- D.800
- E.820
6.
How many positive integers $n\le 250$ satisfy $n\equiv 2\pmod{7}$ ?
- A.37
- B.38
- C.35
- D.36
- E.z4
7.
How many positive integers $k\le 225$ have exactly three positive divisors?
- A.5
- B.6
- C.15
- D.7
- E.9
8.
Let $S$ be the set of integers from $1$ through $300$ inclusive. How many elements of $S$ are divisible by $2$ or by $3$ , but are **not** divisible by $5$ ?
- A.140
- B.148
- C.152
- D.160
- E.168
9.
Solve for $x$ : $3x+2=26$ .
- A.8
- B.26
- C.9
- D.2
- E.7
10.
Arithmetic sequence: first term $4$ , common difference $6$ . Sum of first $10$ terms?
- A.320
- B.40
- C.58
- D.310
- E.304

Answer key · worked solutions

1. B. 35696
$I_n=I_{n-1}+(n-1)I_{n-2}$ — not a $4$ -element listing drill. Compute the recurrence up to $n=11$ . $35696$ .
2. C. 286
$\binom{m}{k}$ . $\binom{13}{3}$ . $286$ .
3. B. 10
Pythagorean theorem. $\sqrt{6^2+8^2}$ . $10$ .
4. B. 2
Half-chord distances from center: $\sqrt{100-36}$ and $\sqrt{100-64}$ . Subtract perpendicular distances from center to each chord. $8-6=2$ .
5. B. 860
$S_n=\frac n2(2a+(n-1)d)$ . $10(10+76)=860$ . $860$ .
6. D. 36
Arithmetic progression. First 2, step 7. $36$ .
7. B. 6
Exactly three divisors $\iff k=p^2$ . Primes $p\le\sqrt{225}$ . $6$ .
8. B. 148
Inclusion–exclusion on $(2\mathbb Z\cup3\mathbb Z)\setminus5\mathbb Z$ . Count $|A\cup B|$ then subtract those also divisible by $5$ . $148$ .
9. A. 8
Isolate $x$ . $3x=26-2$ . $8$ .
10. D. 310
$S_n=\frac n2(2a+(n-1)d)$ . Plug values. $310$ .

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