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Build a printable AMC-style set by topic and difficulty. Every worksheet includes multiple-choice questions and a worked answer key.

Mathbound · Contest practice

Mixed topics worksheet

Mixed difficulty

Name

Date

1.
Solve for $x$ : $3x+10=28$ .
- A.5
- B.6
- C.28
- D.7
- E.10
2.
Positive integer solutions to $x_1+\cdots+x_{6}=18$ with each $x_i\le 8$ ?
- A.33649
- B.6188
- C.5433
- D.5427
- E.5432
3.
What is $\operatorname{lcm}(24,36)$ ?
- A.36
- B.864
- C.12
- D.96
- E.72
4.
Lattice points strictly inside the right triangle with vertices $(0,0)$ , $(16,0)$ , $(0,7)$ ?
- A.46
- B.23
- C.56
- D.45
- E.44
5.
What is the maximum product of positive integers that sum to $29$ ?
- A.39364
- B.39367
- C.16384
- D.19683
- E.39366
6.
Lattice points strictly inside the right triangle with vertices $(0,0)$ , $(13,0)$ , $(0,9)$ ?
- A.49
- B.22
- C.58.5
- D.48
- E.47
7.
What is the sum of the first $23$ positive integers?
- A.299
- B.46
- C.529
- D.276
- E.275
8.
How many pairs of positive integers $(x,y)$ with $x\le y$ satisfy $\dfrac1x+\dfrac1y=\dfrac1{20}$ ?
- A.8
- B.15
- C.9
- D.20
- E.7
9.
How many positive integers $n\le 250$ satisfy $n\equiv 4\pmod{8}$ ?
- A.32
- B.34
- C.30
- D.31
- E.33
10.
Solve for $x$ : $3x+5=23$ .
- A.5
- B.6
- C.23
- D.7
- E.z4

Answer key · worked solutions

1. B. 6
Isolate $x$ . $3x=28-10$ . $6$ .
2. E. 5432
Inclusion-exclusion after $y_i=x_i-1$ . Upper bound 8. $5432$ .
3. E. 72
$\operatorname{lcm}(a,b)=ab/\gcd(a,b)$ . $\gcd=12$ . $72$ .
4. D. 45
$\frac{(A-1)(B-1)}{2}$ for this triangle. Formula. $45$ .
5. E. 39366
Prefer parts equal to $3$ (use $2$ s when needed; avoid $1$ s). Partition $29$ optimally. $39366$ .
6. D. 48
$\frac{(A-1)(B-1)}{2}$ for this triangle. Formula. $48$ .
7. D. 276
$n(n+1)/2$ . $23(23+1)/2$ . $276$ .
8. A. 8
$(x-t)(y-t)=t^2$ . Divisors $d\le t$ of $t^2$ . $8$ .
9. D. 31
Arithmetic progression. First 4, step 8. $31$ .
10. B. 6
Isolate $x$ . $3x=23-5$ . $6$ .

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