2025 AMC 12A Problems/Problem 18: Difference between revisions

Revision as of 04:43, 7 November 2025

Problem 19

How many ordered triples $(x, y, z)$ of different positive integers less than or equal to $8$ satisfy $xy > z$ , $xz > y$ , and $yz > x$ ?

$\textbf{(A)}~36 \qquad \textbf{(B)}~84 \qquad \textbf{(C)}~186 \qquad \textbf{(D)}~336 \qquad \textbf{(E)}~486$

Solution 1

let 0<=x<y<z<=8; x cannot be 0 because it makes xy>z --> 0>z; x cannot be 1 because it makes xy>z ---> y>z;

x=2, y=3, z can be 4, 5 but not others; x=2, y=4, z can be 5, 6, 7; x=2, y=5, z can be 6, 7, 8; x=2, y=6, z can be 7, 8; x=2, y=7, z can be 8; for x=2, total 11 cases;

similarly, for x=3,y=4, 5, 6, 7, total 10 cases; for x=4, y =5, 6, 7, total 6 cases; x=5, y=6, 7, 3 cases; x=6, y=7, z=8, 1 cases;

total = 11 + 10 +6 +3 +1 = 31. permutate x, y, z for ordered triple, it is 31*6=186, C.

~imagination

Solution 2

For now, assume $x\leq y\leq z$ .

First note that no number can be 0, as it would imply $0y>z$ . Similarly, no number can be 1, as it would imply $1y>z$ . So we only need to consider numbers between 2 and 8, inclusive.

We may use complementary counting:

Consider when $xy\leq z$ . This implies $xy\leq 8$ . Some quick calculations gives us the products $2\cdot2,2\cdot3,2\cdot4$ . We may now calculate the number of times each happens (we are no longer assuming $x\leq y\leq z$ ):

$2\cdot2$ : This case is invalid as it asks for distinct integers.
$2\cdot3$ : $2,3,(n\geq6)$ . Then we have $6\cdot3=18$ cases.
$2\cdot4$ : $2,4,(n\geq8)$ . Then we have $6\cdot1=6$ cases.

In total, there are $7\cdot6\cdot5=210$ total cases, so our final answer is $210 - (18 + 6) = \boxed{186}$ .

~SilverRush

Solution 3

Note that if $x,y,z \ge 3$ , all pairs work - hence, we have $\binom{6}{3} \cdot 6 = 120$ pairs. Now, note that if $x=1$ , we get $y>z$ and $z>y$ , contradiction. Therefore, assume $x=2$ (since we said $x < 3, x \neq 1$ ). Note that we need $2y>z>\frac{y}{2}$ . WLOG assume $y>z$ , we get there are $11$ pairs that work (we just need $2z>y$ ): $(y,z) = (8,5), (8,6), (8,7), (7,4), (7,5), (7,6), (6,4), (6,5), (5,3), (5,4), (4,3)$ . With $x=2$ , we can re-arrange these in $3!=6$ ways each, hence the answer is just $120 + 6 \cdot 11 = \boxed{186}$ .

~ScoutViolet

Video Solution 1 by OmegaLearn

https://youtu.be/OPQbAyYZBtA

@@ Line 40: / Line 40: @@
 ==Solution 3==
 Note that if <imath>x,y,z \ge 3</imath>, all pairs work - hence, we have <imath>\binom{6}{3} \cdot 6 = 120</imath> pairs.
-Now, note that if <imath>x=1</imath>, we get <imath>y>z</imath> and <imath>z>y</imath>, contradiction. Therefore, assume <imath>x=2</imath> (since we said <imath>x < 3, x \neq 1</imath>). Note that we need <imath>2y>z>\frac{y}{2}</imath>. WLOG assume <imath>y>z</imath>, we get there are <imath>11</imath> pairs that work (we just need <imath>2z>y</imath>) - <imath>(y,z) = (8,5), (8,6), (8,7), (7,4), (7,5), (7,6), (6,4), (6,5), (5,3), (5,4), (4,3)</imath>. With <imath>x=2</imath>, we can re-arrange these in <imath>3!=6</imath> ways each, hence the answer is just <imath>120 + 6 \cdot 11 = \boxed{186}</imath>.
+Now, note that if <imath>x=1</imath>, we get <imath>y>z</imath> and <imath>z>y</imath>, contradiction. Therefore, assume <imath>x=2</imath> (since we said <imath>x < 3, x \neq 1</imath>). Note that we need <imath>2y>z>\frac{y}{2}</imath>. WLOG assume <imath>y>z</imath>, we get there are <imath>11</imath> pairs that work (we just need <imath>2z>y</imath>): <imath>(y,z) = (8,5), (8,6), (8,7), (7,4), (7,5), (7,6), (6,4), (6,5), (5,3), (5,4), (4,3)</imath>. With <imath>x=2</imath>, we can re-arrange these in <imath>3!=6</imath> ways each, hence the answer is just <imath>120 + 6 \cdot 11 = \boxed{186}</imath>.
 ~ScoutViolet

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