Free AMC Math Worksheet Generator

Mathbound · Contest practice

Mixed topics worksheet

Level 4

Name

Date

1.
Call a positive integer $n$ **square-linked** if $n$ is a perfect square and the integer $\lfloor n\sqrt{2}\rfloor$ is also a perfect square. (For example, $1$ is square-linked because both $1$ and $\lfloor 1\cdot\sqrt{2}\rfloor=1$ are squares.) A contestant initially believes there might be hundreds of such integers below $10^{6}$ and plans to test every perfect square $n\le10^{6}$ by brute force. How many square-linked integers actually satisfy $n\le 10^{6}$ ?
- A.3
- B.4
- C.5
- D.6
- E.7
2.
Regular $18$ -gon vertices labeled. How many triangles formed by three vertices contain the center?
- A.816
- B.324
- C.186
- D.168
- E.163
3.
A particle moves on the integer lattice from $(0,0)$ to $(8,8)$ using only steps $(1,0)$ and $(0,1)$ , never leaving the first quadrant. The particle must avoid the obstacle point $(3,5)$ (it may not visit that point, though it may pass through other lattice points). How many admissible lattice paths exist?
- A.8460
- B.9120
- C.9734
- D.10240
- E.12870
4.
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
5.
How many ordered quadruples of **positive** integers $(a,b,c,d)$ satisfy $a+b+c+d=20$ and have the property that $abcd$ is a perfect square?
- A.72
- B.80
- C.85
- D.90
- E.96
6.
How many ordered pairs of integers $(a,b)$ with $1\le a<b\le 100$ satisfy that $a^{2}+b^{2}+ab$ is a perfect square?
- A.55
- B.58
- C.61
- D.64
- E.67
7.
Consider all nonempty subsets of $\{1,2,3,\ldots,10\}$ . How many of these subsets have the property that the product of their elements is divisible by $60$ ?
- A.560
- B.584
- C.608
- D.632
- E.656
8.
How many ordered pairs of positive integers $(m,n)$ satisfy $\operatorname{lcm}(m,n)=2^{4}\cdot3^{3}\cdot5^{1}$ ?
- A.189
- B.378
- C.199
- D.16\cdot27\cdot5
- E.184
9.
Regular $10$ -gon vertices labeled. How many triangles formed by three vertices contain the center?
- A.20
- B.15
- C.120
- D.100
- E.30
10.
A researcher lists all triples of positive integers $(x,y,z)$ with $x\le y\le z$ such that $\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=\dfrac{1}{2}$ . (This equation describes how three identical pint containers can be filled from a half-pint scoop if each container must receive an integer number of scoopfuls.) How many triples appear on the list?
- A.8
- B.9
- C.10
- D.11
- E.12

Answer key · worked solutions

1. C. 5
Pell-type recurrence from $\sqrt{2}$ ; squares $a^2$ with $\lfloor a^2\sqrt{2}\rfloor$ square. Brute listing fails — discover the continued-fraction / recurrence structure of convergents. $5$ : $1,1369,5476,12321,21904$ .
2. D. 168
Each arc between vertices $<n/2$ . Count valid triples. $168$ .
3. C. 9734
Total $\binom{16}{8}$ minus paths through $(3,5)$ . Subtract $\binom{8}{3}\binom{8}{5}$ from $\binom{16}{8}$ . $9734$ .
4. A. 112
Each arc between vertices $<n/2$ . Count valid triples. $112$ .
5. C. 85
Parity of exponents in prime factorization of $abcd$ must all be even. Looks like $\binom{19}{3}$ partitions — square condition is the hidden filter. $85$ .
6. C. 61
$a^2+b^2+ab=N(a-b\omega)$ in Eisenstein integers; classify norms. Camouflaged as a double loop — algebraic structure collapses the search. $61$ .
7. C. 608
Need factors $2^2$ , $3$ , and $5$ in the subset product. Use inclusion–exclusion on subsets missing a required prime power — not $2^{10}-1$ brute. $608$ .
8. A. 189
Per prime exponent $e$ : $2e+1$ pairs. Multiply across primes. $189$ .
9. A. 20
Each arc between vertices $<n/2$ . Count valid triples. $20$ .
10. C. 10
Egyptian fraction: bound $3\le x\le6$ , clear denominators for integrality of $z$ . Hidden factorization after fixing $x$ . $10$ triples.

Teaching a class or club? Find classroom-ready starting sets on the teacher resources page.

Math worksheet generator