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Mathbound · Contest practice

Mixed topics worksheet

Level 3

Name

Date

1.
Lattice points strictly inside the right triangle with vertices $(0,0)$ , $(15,0)$ , $(0,8)$ ?
- A.60
- B.49
- C.48
- D.50
- E.23
2.
Lattice paths $(0,0)\to(7,7)$ with steps $(1,0),(0,1)$ avoiding $(2,2)$ . How many?
- A.3432
- B.1512
- C.1941
- D.1920
- E.1910
3.
Positive integer solutions to $x_1+\cdots+x_{5}=15$ with each $x_i\le 8$ ?
- A.921
- B.926
- C.3876
- D.1001
- E.927
4.
A bug walks along the edges of a cube. Starting at a vertex, how many walks of length $6$ end at the same vertex?
- A.162
- B.183
- C.192
- D.216
- E.243
5.
If $r$ and $s$ are roots of $x^2+4x+18=0$ , find $r^3+s^3$ (without solving).
- A.-152
- B.-170
- C.-148
- D.72
- E.64
6.
Lattice paths $(0,0)\to(8,8)$ with steps $(1,0),(0,1)$ avoiding $(6,5)$ . How many?
- A.8278
- B.8250
- C.8240
- D.12870
- E.4620
7.
Positive integer solutions to $x_1+\cdots+x_{4}=18$ with each $x_i\le 8$ ?
- A.1330
- B.680
- C.345
- D.339
- E.344
8.
A frog starts at pad $0$ and jumps $+1$ or $+2$ each move. Pad $17$ is unsafe. How many jump sequences reach pad $22$ without landing on pad $17$ ?
- A.7984
- B.7985
- C.7987
- D.7986
- E.28657
9.
Positive integer solutions to $x_1+\cdots+x_{6}=16$ with each $x_i\le 8$ ?
- A.2878
- B.2872
- C.2877
- D.20349
- E.3003
10.
Lattice points strictly inside the right triangle with vertices $(0,0)$ , $(14,0)$ , $(0,11)$ ?
- A.77
- B.65
- C.64
- D.66
- E.25

Answer key · worked solutions

1. B. 49
$\frac{(A-1)(B-1)}{2}$ for this triangle. Formula. $49$ .
2. D. 1920
$\binom{2n}{n}$ minus paths through the obstacle. Subtract product of binomials. $1920$ .
3. B. 926
Inclusion-exclusion after $y_i=x_i-1$ . Upper bound 8. $926$ .
4. B. 183
Cube graph adjacency; distance-type recurrence. Matrix power / state machine on opposite/adj/start. $183$ .
5. A. -152
$r^3+s^3=(r+s)((r+s)^2-3rs)$ . Vieta: $r+s=4$ , $rs=18$ . $-152$ .
6. B. 8250
$\binom{2n}{n}$ minus paths through the obstacle. Subtract product of binomials. $8250$ .
7. E. 344
Inclusion-exclusion after $y_i=x_i-1$ . Upper bound 8. $344$ .
8. B. 7985
Fibonacci with a zeroed forbidden state. DP with $F(17)=0$ . $7985$ .
9. C. 2877
Inclusion-exclusion after $y_i=x_i-1$ . Upper bound 8. $2877$ .
10. B. 65
$\frac{(A-1)(B-1)}{2}$ for this triangle. Formula. $65$ .

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Free AMC Math Worksheet Generator — Mathbound