Finding the Number of Ordered Integer Triplets

In this article, we will explore the method to find the number of ordered integer triplets (x, y, z) that satisfy the given inequality. The process involves completing the square, changing variables, and considering different cases based on the values of the transformed variables. Let's begin with the given inequality:

Step-by-Step Solution

Inequality: 2x^2 3y^2 3z^2 2xy 2xz 4yz - 5 leq 0

First, rearrange the inequality to isolate the constant term:

x^2 - 2xy y^2 x^2 - 2xz z^2 2y^2 4yz - 2xz - 5 leq 0

Simplify by combining terms:

x^2 y^2 x^2 z^2 2y^2 - 4yz 2z^2 - 2xz - 5 leq 0

Further simplify:

x^2 - 2xy y^2 x^2 - 2xz z^2 2y^2 - 4yz 2z^2 - 2xz - 5 2x^2 3y^2 3z^2 2xy 2xz 4yz - 5

Complete the square:

x^2 - 2xy y^2 x^2 - 2xz z^2 2y^2 - 4yz 2z^2 - 2xz - 5 (x - y)^2 (x - z)^2 2y^2 - 4yz 2z^2 - 5 leq 3u - 2v^2 5v^2 - 15 leq 5

Define new variables:

u x - y v x - z

Substitute:

u^2 v^2 leq 15

Express the inequality in terms of squares:

3u - 2v^2 5v^2 - 15 leq 5

This yields the following solutions:

uv 00 pm 10 pm 1pm 1pm 1

Substitute back to the original variables:

x t, y u - t, z v - t text{ where } t leq 100

Analysis of Solutions

uv 00 Rightarrow xyz t - t - t uv 10 Rightarrow xyz t, 1-t - t uv -10 Rightarrow xyz t, -1-t - t uv 040 Rightarrow xyz t, 4-t - t uv 041 Rightarrow xyz t, 4-t - t 1 uv 100 Rightarrow xyz t, 1-t - t uv 110 Rightarrow xyz t, 1-t - t 1 uv 101 Rightarrow xyz t, 1-t - t - 1 uv 140 Rightarrow xyz t, 1-t - t 2

Count the solutions for each case:

uv 00 Rightarrow N_1 201 uv 10 Rightarrow N_2 200 uv -10 Rightarrow N_3 200 uv 040 Rightarrow N_4 400 uv 100 Rightarrow N_5 200 uv 110 Rightarrow N_6 200 uv 101 Rightarrow N_7 200 uv 140 Rightarrow N_8 400 uv 041 Rightarrow N_9 200

Summing all solutions:

N_{text{total}} N_1 N_2 N_3 N_4 N_5 N_6 N_7 N_8 N_9 201 200 200 400 200 200 200 400 200 1401

Thus, the total number of ordered integer triplets satisfying the inequality is 1401.