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Mathbound · Contest practice

Mixed topics worksheet

Mixed difficulty

Name

Date

1.
Circle radius $17$ . Chord length $30$ . Distance from center to chord?
- A.8
- B.2
- C.9
- D.17
- E.15
2.
If $r$ and $s$ are roots of $x^2+7x+22=0$ , find $r^3+s^3$ (without solving).
- A.-141
- B.-112
- C.154
- D.343
- E.-119
3.
Trailing zeros of $50!$ in base $10$ ?
- A.14
- B.10
- C.11
- D.13
- E.12
4.
An item costs $36$ . After a $20\%$ discount (no tax), how much is paid?
- A.16
- B.36
- C.29.80
- D.7
- E.28.80
5.
A frog starts at pad $0$ and jumps $+1$ or $+2$ each move. Pad $13$ is unsafe. How many jump sequences reach pad $18$ without landing on pad $13$ ?
- A.1164
- B.1165
- C.1167
- D.1166
- E.4181
6.
What is the maximum product of positive integers that sum to $30$ ?
- A.59047
- B.59049
- C.59052
- D.59050
- E.32768
7.
What is $\operatorname{lcm}(14,30)$ ?
- A.420
- B.180
- C.2
- D.210
- E.224
8.
Lattice paths $(0,0)\to(10,10)$ with steps $(1,0),(0,1)$ avoiding $(8,8)$ . How many?
- A.107581
- B.107536
- C.107526
- D.184756
- E.77220
9.
Lattice paths $(0,0)\to(10,10)$ with steps $(1,0),(0,1)$ avoiding $(8,6)$ . How many?
- A.139756
- B.139711
- C.139701
- D.184756
- E.45045
10.
What is $\operatorname{lcm}(14,34)$ ?
- A.476
- B.204
- C.2
- D.238
- E.252

Answer key · worked solutions

1. A. 8
Right triangle: half-chord, radius, distance. $\sqrt{17^2-15^2}$ . $8$ .
2. E. -119
$r^3+s^3=(r+s)((r+s)^2-3rs)$ . Vieta: $r+s=7$ , $rs=22$ . $-119$ .
3. E. 12
Exponent of $5$ . $10+2$ . $12$ .
4. E. 28.80
Multiply by $(1-d)$ . $36(1-20/100)$ . $28.80$ .
5. B. 1165
Fibonacci with a zeroed forbidden state. DP with $F(13)=0$ . $1165$ .
6. B. 59049
Prefer parts equal to $3$ (use $2$ s when needed; avoid $1$ s). Partition $30$ optimally. $59049$ .
7. D. 210
$\operatorname{lcm}=ab/\gcd$ . $\gcd=2$ . $210$ .
8. B. 107536
$\binom{2n}{n}$ minus paths through the obstacle. Subtract product of binomials. $107536$ .
9. B. 139711
$\binom{2n}{n}$ minus paths through the obstacle. Subtract product of binomials. $139711$ .
10. D. 238
$\operatorname{lcm}=ab/\gcd$ . $\gcd=2$ . $238$ .

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