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Mathbound · Contest practice

Mixed topics worksheet

Mixed difficulty

Name

Date

1.
Lattice points strictly inside the right triangle with vertices $(0,0)$ , $(17,0)$ , $(0,11)$ ?
- A.93.5
- B.80
- C.79
- D.81
- E.28
2.
Arithmetic sequence: first term $4$ , common difference $5$ . Sum of first $15$ terms?
- A.600
- B.60
- C.74
- D.585
- E.580
3.
If $x+\dfrac1x=5$ , then $x^2+\dfrac1{x^2}=$
- A.25
- B.23
- C.27
- D.21
- E.24
4.
Compute $3^{8}\bmod 13$ .
- A.3
- B.10
- C.11
- D.12
- E.9
5.
What is the maximum product of positive integers that sum to $20$ ?
- A.1456
- B.1459
- C.1024
- D.729
- E.1458
6.
Lattice paths $(0,0)\to(10,10)$ with steps $(1,0),(0,1)$ avoiding $(5,6)$ . How many?
- A.126796
- B.126544
- C.126534
- D.184756
- E.58212
7.
How many ordered pairs of positive integers $(m,n)$ satisfy $\operatorname{lcm}(m,n)=2^{5}\cdot3^{4}\cdot5^{2}$ ?
- A.490
- B.8100
- C.495
- D.90
- E.505
8.
Two fair dice are rolled. Probability the sum equals $11$ ?
- A.2/36
- B.1/6
- C.3/36
- D.2/12
- E.2/18
9.
What is $\operatorname{lcm}(16,72)$ ?
- A.72
- B.8
- C.144
- D.160
- E.1152
10.
Solve for $x$ : $3x+6=27$ .
- A.8
- B.6
- C.7
- D.27
- E.z4

Answer key · worked solutions

1. B. 80
$\frac{(A-1)(B-1)}{2}$ for this triangle. Formula. $80$ .
2. D. 585
$S_n=\frac n2(2a+(n-1)d)$ . Plug values. $585$ .
3. B. 23
Square both sides. $25-2=23$ . $23$ .
4. E. 9
Successive squaring / cycles. Reduce powers modulo 13. $9$ .
5. E. 1458
Prefer parts equal to $3$ (use $2$ s when needed; avoid $1$ s). Partition $20$ optimally. $1458$ .
6. B. 126544
$\binom{2n}{n}$ minus paths through the obstacle. Subtract product of binomials. $126544$ .
7. C. 495
For each prime with LCM exponent $e$ , there are $2e+1$ exponent pairs. Multiply over primes. $495$ .
8. A. 2/36
Count ordered pairs out of $36$ . $2$ favorable. $2/36$ .
9. C. 144
$\operatorname{lcm}=ab/\gcd$ . $\gcd=8$ . $144$ .
10. C. 7
Isolate $x$ . $3x=27-6$ . $7$ .

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