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Mathbound · Contest practice

Mixed topics worksheet

Mixed difficulty

Name

Date

1.
Positive integer solutions to $x_1+\cdots+x_{4}=13$ with each $x_i\le 8$ ?
- A.199
- B.204
- C.560
- D.220
- E.205
2.
For a positive integer $n$ , let $f(n)$ be the remainder when $2^{n}$ is divided by $n$ . Call $n$ **good** if $f(n)=2$ . (Every prime is good by Fermat's little theorem, but some composites are good as well.) How many good integers satisfy $n<1000$ ?
- A.148
- B.160
- C.172
- D.184
- E.196
3.
How many positive integers $n\le 250$ satisfy $n\equiv 2\pmod{8}$ ?
- A.32
- B.31
- C.33
- D.34
- E.z4
4.
An item costs $32$ . After a $10\%$ discount (no tax), how much is paid?
- A.28.80
- B.22
- C.32
- D.29.80
- E.3
5.
Lattice paths $(0,0)\to(10,10)$ with steps $(1,0),(0,1)$ avoiding $(5,5)$ . How many?
- A.184756
- B.63504
- C.121504
- D.121252
- E.121242
6.
Regular $16$ -gon vertices labeled. How many triangles formed by three vertices contain the center?
- A.112
- B.107
- C.560
- D.256
- E.128
7.
Lattice points strictly inside the right triangle with vertices $(0,0)$ , $(12,0)$ , $(0,15)$ ?
- A.27
- B.90
- C.77
- D.76
- E.78
8.
Positive integer solutions to $x_1+\cdots+x_{4}=16$ ?
- A.456
- B.455
- C.451
- D.969
- E.1820
9.
How many positive integers $n\le 50$ satisfy $n\equiv 3\pmod{5}$ ?
- A.11
- B.13
- C.9
- D.10
- E.12
10.
How many ordered integer solutions $(x,y,z,w)$ satisfy $x^{2}+y^{2}+z^{2}+w^{2}=25$ ? (Different orderings and sign choices count separately.)
- A.224
- B.232
- C.240
- D.248
- E.256

Answer key · worked solutions

1. B. 204
Inclusion-exclusion after $y_i=x_i-1$ . Upper bound 8. $204$ .
2. C. 172
Primes plus base- $2$ pseudoprimes; cannot test all $999$ values blindly. Sieve primes, then hunt rare composite solutions structurally. $172$ .
3. A. 32
Arithmetic progression. First 2, step 8. $32$ .
4. A. 28.80
Multiply by $(1-d)$ . $32(1-10/100)$ . $28.80$ .
5. D. 121252
$\binom{2n}{n}$ minus paths through the obstacle. Subtract product of binomials. $121252$ .
6. A. 112
Each arc between vertices $<n/2$ . Count valid triples. $112$ .
7. C. 77
$\frac{(A-1)(B-1)}{2}$ for this triangle. Formula. $77$ .
8. B. 455
$\binom{N-1}{k-1}$ . $\binom{15}{3}$ . $455$ .
9. D. 10
Arithmetic progression. First 3, step 5. $10$ .
10. D. 248
Sum of four squares with generating-function / shell enumeration. Residue and parity constraints shrink cases — not $11^4$ naive search. $248$ .

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