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Mathbound · Contest practice

Mixed topics worksheet

Mixed difficulty

Name

Date

1.
Positive integer solutions to $x_1+\cdots+x_{6}=26$ with each $x_i\le 8$ ?
- A.169911
- B.53130
- C.17893
- D.17887
- E.17892
2.
Arithmetic sequence: first term $2$ , common difference $2$ . Sum of first $20$ terms?
- A.440
- B.40
- C.420
- D.418
- E.z4
3.
An item costs $44$ . After a $10\%$ discount (no tax), how much is paid?
- A.44
- B.40.60
- C.4
- D.39.60
- E.34
4.
Solve for $x$ : $3x+5=17$ .
- A.5
- B.3
- C.4
- D.17
- E.z4
5.
How many ordered triples of integers $(x,y,z)$ satisfy both $x^{2}+y^{2}+z^{2}\le 50$ and $x+y+z\equiv 0\pmod 3$ ? (Different orderings count separately, and $(0,0,0)$ is allowed.)
- A.450
- B.468
- C.495
- D.512
- E.540
6.
Arithmetic sequence: first term $2$ , common difference $6$ . Sum of first $20$ terms?
- A.1200
- B.40
- C.116
- D.1180
- E.1174
7.
A frog starts at pad $0$ and jumps $+1$ or $+2$ each move. Pad $15$ is unsafe. How many jump sequences reach pad $24$ without landing on pad $15$ ?
- A.20739
- B.20740
- C.20742
- D.20741
- E.75025
8.
What is $\operatorname{lcm}(24,36)$ ?
- A.36
- B.864
- C.12
- D.96
- E.72
9.
Arithmetic sequence: first term $2$ , common difference $7$ . Sum of first $20$ terms?
- A.1390
- B.40
- C.135
- D.1370
- E.1363
10.
Arithmetic sequence: first term $2$ , common difference $4$ . Sum of first $12$ terms?
- A.288
- B.284
- C.300
- D.24
- E.46

Answer key · worked solutions

1. E. 17892
Inclusion-exclusion after $y_i=x_i-1$ . Upper bound 8. $17892$ .
2. C. 420
$S_n=\frac n2(2a+(n-1)d)$ . Plug values. $420$ .
3. D. 39.60
Multiply by $(1-d)$ . $44(1-10/100)$ . $39.60$ .
4. C. 4
Isolate $x$ . $3x=17-5$ . $4$ .
5. C. 495
Lattice ball + linear congruence; squares mod $3$ are $0$ or $1$ . Couple spherical shell enumeration with residue classes — not $1/3$ of all lattice points. $495$ .
6. D. 1180
$S_n=\frac n2(2a+(n-1)d)$ . Plug values. $1180$ .
7. B. 20740
Fibonacci with a zeroed forbidden state. DP with $F(15)=0$ . $20740$ .
8. E. 72
$\operatorname{lcm}(a,b)=ab/\gcd(a,b)$ . $\gcd=12$ . $72$ .
9. D. 1370
$S_n=\frac n2(2a+(n-1)d)$ . Plug values. $1370$ .
10. A. 288
$S_n=\frac n2(2a+(n-1)d)$ . Plug values. $288$ .

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