id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-014101 | Inequalities: Product Given Sum | 8 | Work this out carefully: Let $x,y>0$ satisfy $x+y=106$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., co... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=106-x$ with $x\\in(0,106)$. Then $P(x)=xy=x(106... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{2809}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=2809$.",
"robustness_analysis": "Sensitivity analysis: AM–GM ... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=53.0$. |
math-014102 | Inequalities: Product Given Sum | 8 | Explain each transformation: Let $x,y>0$ satisfy $x+y=833$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g.... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=833$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\frac{693889}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{693889}{4}$.",
"robustness_a... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=416.5$. (Here the result is $\boxed{\frac{693889}$.) |
math-014103 | Inequalities: Product Given Sum | 8 | Exercise: Let $x,y>0$ satisfy $x+y=33$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., concavity/second d... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=33-x$ with $x\\in(0,33)$. Then $P(x)=xy=x(33-x)... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\frac{1089}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{1089}{4}$.",
"robustness_analysis": "Robustness note:... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Takeaway: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=16.5$. (Here the result is $\boxed{\frac{1089}$.) |
math-014104 | Inequalities: Product Given Sum | 8 | Explain what is being counted/optimized: Let $x,y>0$ satisfy $x+y=15$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus th... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=15-x$ with $x\\in(0,15)$. Then $P(x)=xy=x(15-x)... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\frac{225}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{225}{4}$.",
"robustness_analysi... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=7.5$. (Here the result is $\boxed{\frac{225}$.) |
math-014105 | Optimization: Two Variables — Concavity | 8 | Compute the requested quantity: Let $x,y>0$ satisfy $x+y=812$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=812-x$ with $x\\in(0,812)$. Then $P(x)=xy=x(812... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{164836}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=164836$.",
"robustness_analysis":... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Key idea: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=406.0$. |
math-014106 | Algebra: Extremal Values — Global Bounds | 8 | Find the exact value: Let $x,y>0$ satisfy $x+y=273$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., conca... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=273$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\frac{74529}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{74529}{4}$.",
"robustness_analysis": "Generality not... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Key idea: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=136.5$. (Here the result is $\boxed{\frac{74529}$.) |
math-014107 | Inequalities: AM–GM — Equality Conditions | 8 | Explain why your operations are valid: Let $x,y>0$ satisfy $x+y=471$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus the... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=471-x$ with $x\\in(0,471)$. Then $P(x)=xy=x(471... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\frac{221841}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{221841}{4}$.",
"robustness_a... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=235.5$. (Here the result is $\boxed{\frac{221841}$.) |
math-014108 | Inequalities: Product Given Sum | 8 | Prompt: Let $x,y>0$ satisfy $x+y=337$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., concavity/second de... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=337$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\frac{113569}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{113569}{4}$.",
"robustness_a... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Key idea: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=168.5$. (Here the result is $\boxed{\frac{113569}$.) |
math-014109 | Inequalities: Product Given Sum | 8 | Question: Let $x,y>0$ satisfy $x+y=569$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., concavity/second ... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=569$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\frac{323761}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{323761}{4}$.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=284.5$. (Here the result is $\boxed{\frac{323761}$.) |
math-014110 | Inequalities: Product Given Sum | 8 | Carefully track domains: Let $x,y>0$ satisfy $x+y=150$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., co... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=150$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{5625}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=5625$.",
"robustness_analysis": "Ro... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=75.0$. (Here the result is $\boxed{5625}$.) |
math-014111 | Algebra: Extremal Values — Global Bounds | 8 | Answer with a short justification: Let $x,y>0$ satisfy $x+y=189$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=189-x$ with $x\\in(0,189)$. Then $P(x)=xy=x(189... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\frac{35721}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{35721}{4}$.",
"robustness_analysis": "Sensitivity an... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=94.5$. (Here the result is $\boxed{\frac{35721}$.) |
math-014112 | Algebra: Extremal Values — Global Bounds | 8 | Complete the analysis: Let $x,y>0$ satisfy $x+y=616$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., conc... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=616-x$ with $x\\in(0,616)$. Then $P(x)=xy=x(616... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{94864}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=94864$.",
"robustness_analysis": "... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Key idea: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=308.0$. (Here the result is $\boxed{94864}$.) |
math-014113 | Algebra: Extremal Values — Global Bounds | 8 | Solve and include a self-check: Let $x,y>0$ satisfy $x+y=767$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=767$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\frac{588289}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{588289}{4}$.",
"robustness_a... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Takeaway: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=383.5$. |
math-014114 | Inequalities: Product Given Sum | 8 | Warm-up: Let $x,y>0$ satisfy $x+y=298$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., concavity/second d... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=298-x$ with $x\\in(0,298)$. Then $P(x)=xy=x(298... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{22201}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=22201$.",
"robustness_analysis": "Robustness note: AM–GM generaliz... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Key idea: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=149.0$. (Here the result is $\boxed{22201}$.) |
math-014115 | Inequalities: AM–GM — Equality Conditions | 8 | Find the exact value: Let $x,y>0$ satisfy $x+y=752$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., conca... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=752$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{141376}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=141376$.",
"robustness_analysis": "Sensiti... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=376.0$. |
math-014116 | Inequalities: AM–GM — Equality Conditions | 8 | Give an answer and a quick verification: Let $x,y>0$ satisfy $x+y=296$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus t... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=296-x$ with $x\\in(0,296)$. Then $P(x)=xy=x(296... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{21904}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=21904$.",
"robustness_analysis": "... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=148.0$. |
math-014117 | Optimization: Two Variables — Concavity | 8 | Determine the requested value: Let $x,y>0$ satisfy $x+y=516$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=516$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{66564}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=66564$.",
"robustness_analysis": "If the pr... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=258.0$. |
math-014118 | Algebra: Extremal Values — Global Bounds | 8 | Explain why your operations are valid: Let $x,y>0$ satisfy $x+y=783$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus the... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=783$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\frac{613089}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{613089}{4}$.",
"robustness_analysis": "If the probl... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Takeaway: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=391.5$. (Here the result is $\boxed{\frac{613089}$.) |
math-014119 | Inequalities: AM–GM — Equality Conditions | 8 | Explain why your operations are valid: Let $x,y>0$ satisfy $x+y=858$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus the... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=858$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{184041}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=184041$.",
"robustness_analysis": "Sensitivity analysis: AM... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Key idea: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=429.0$. (Here the result is $\boxed{184041}$.) |
math-014120 | Inequalities: Product Given Sum | 8 | Answer using clear logical steps: Let $x,y>0$ satisfy $x+y=593$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem ... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=593$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\frac{351649}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{351649}{4}$.",
"robustness_analysis": "If the probl... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Takeaway: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=296.5$. (Here the result is $\boxed{\frac{351649}$.) |
math-014121 | Inequalities: AM–GM — Equality Conditions | 8 | Track quantifiers carefully: Let $x,y>0$ satisfy $x+y=713$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g.... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=713$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\frac{508369}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{508369}{4}$.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Key idea: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=356.5$. (Here the result is $\boxed{\frac{508369}$.) |
math-014122 | Inequalities: Product Given Sum | 8 | Give reasoning, not just computation: Let $x,y>0$ satisfy $x+y=442$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theo... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=442$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{48841}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=48841$.",
"robustness_analysis": "Robustnes... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=221.0$. |
math-014123 | Inequalities: Product Given Sum | 8 | Answer with a short justification: Let $x,y>0$ satisfy $x+y=563$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=563-x$ with $x\\in(0,563)$. Then $P(x)=xy=x(563... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\frac{316969}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{316969}{4}$.",
"robustness_analysis": "If the probl... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=281.5$. |
math-014124 | Algebra: Extremal Values — Global Bounds | 8 | Compute the requested quantity: Let $x,y>0$ satisfy $x+y=771$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=771-x$ with $x\\in(0,771)$. Then $P(x)=xy=x(771... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\frac{594441}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{594441}{4}$.",
"robustness_a... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=385.5$. |
math-014125 | Inequalities: Product Given Sum | 8 | Task: Let $x,y>0$ satisfy $x+y=104$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., concavity/second deri... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=104-x$ with $x\\in(0,104)$. Then $P(x)=xy=x(104... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{2704}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=2704$.",
"robustness_analysis": "Generality note: AM–GM gener... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=52.0$. |
math-014126 | Inequalities: AM–GM — Equality Conditions | 8 | Solve and justify each step: Let $x,y>0$ satisfy $x+y=38$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g.,... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=38-x$ with $x\\in(0,38)$. Then $P(x)=xy=x(38-x)... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{361}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=361$.",
"robustness_analysis": "Generality no... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=19.0$. |
math-014127 | Inequalities: Product Given Sum | 8 | Explain each transformation: Let $x,y>0$ satisfy $x+y=823$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g.... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=823-x$ with $x\\in(0,823)$. Then $P(x)=xy=x(823... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\frac{677329}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{677329}{4}$.",
"robustness_a... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=411.5$. (Here the result is $\boxed{\frac{677329}$.) |
math-014128 | Optimization: Two Variables — Concavity | 8 | Compute the requested quantity: Let $x,y>0$ satisfy $x+y=80$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=80$ to get $\\sqrt{xy}\\le \\frac{8... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{1600}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=1600$.",
"robustness_analysis": "If the prob... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=40.0$. |
math-014129 | Inequalities: Product Given Sum | 8 | Determine the requested value: Let $x,y>0$ satisfy $x+y=540$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=540$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{72900}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=72900$.",
"robustness_analysis": "... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=270.0$. (Here the result is $\boxed{72900}$.) |
math-014130 | Inequalities: Product Given Sum | 8 | Find the exact value: Let $x,y>0$ satisfy $x+y=751$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., conca... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=751$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\frac{564001}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{564001}{4}$.",
"rob... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=375.5$. |
math-014131 | Inequalities: Product Given Sum | 8 | Checkpoint: Let $x,y>0$ satisfy $x+y=463$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., concavity/secon... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=463$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\frac{214369}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{214369}{4}$.",
"robustness_analysis": "Sensit... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Takeaway: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=231.5$. (Here the result is $\boxed{\frac{214369}$.) |
math-014132 | Optimization: Two Variables — Concavity | 8 | Start by stating any domain restrictions: Let $x,y>0$ satisfy $x+y=26$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus t... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=26$ to get $\\sqrt{xy}\\le \\frac{2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{169}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=169$.",
"robustness_analysis": "If t... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Key idea: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=13.0$. (Here the result is $\boxed{169}$.) |
math-014133 | Inequalities: Product Given Sum | 8 | Be explicit about assumptions: Let $x,y>0$ satisfy $x+y=308$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=308$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{23716}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=23716$.",
"robustness_analysis": "Sensitivity analysis: AM–G... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=154.0$. (Here the result is $\boxed{23716}$.) |
math-014134 | Optimization: Two Variables — Concavity | 8 | Provide a rigorous solution: Let $x,y>0$ satisfy $x+y=391$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g.... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=391$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\frac{152881}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{152881}{4}$.",
"robustness_analysis": "Sensitivity ... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Key idea: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=195.5$. |
math-014135 | Algebra: Extremal Values — Global Bounds | 8 | Give a fully justified solution: Let $x,y>0$ satisfy $x+y=496$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=496-x$ with $x\\in(0,496)$. Then $P(x)=xy=x(496... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{61504}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=61504$.",
"robustness_analysis": "Sensitivity analysis: AM–G... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=248.0$. (Here the result is $\boxed{61504}$.) |
math-014136 | Optimization: Two Variables — Concavity | 8 | Give an answer and a quick verification: Let $x,y>0$ satisfy $x+y=722$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus t... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=722-x$ with $x\\in(0,722)$. Then $P(x)=xy=x(722... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{130321}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=130321$.",
"robustness_analysis": "Robustness note: AM–GM g... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=361.0$. (Here the result is $\boxed{130321}$.) |
math-014137 | Algebra: Extremal Values — Global Bounds | 8 | Explain what is being counted/optimized: Let $x,y>0$ satisfy $x+y=64$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus th... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=64-x$ with $x\\in(0,64)$. Then $P(x)=xy=x(64-x)... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{1024}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=1024$.",
"robustness_analysis": "Sensitivity analysis: AM–GM ... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=32.0$. (Here the result is $\boxed{1024}$.) |
math-014138 | Inequalities: Product Given Sum | 8 | Try to avoid pattern-matching; explain why: Let $x,y>0$ satisfy $x+y=513$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculu... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=513-x$ with $x\\in(0,513)$. Then $P(x)=xy=x(513... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\frac{263169}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{263169}{4}$.",
"rob... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=256.5$. (Here the result is $\boxed{\frac{263169}$.) |
math-014139 | Optimization: Two Variables — Concavity | 8 | Exercise: Let $x,y>0$ satisfy $x+y=838$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., concavity/second ... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=838$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{175561}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=175561$.",
"robustness_analysis": "Sensitivity analysis: AM–GM ge... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=419.0$. (Here the result is $\boxed{175561}$.) |
math-014140 | Inequalities: AM–GM — Equality Conditions | 8 | Do not skip justification steps: Let $x,y>0$ satisfy $x+y=561$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=561-x$ with $x\\in(0,561)$. Then $P(x)=xy=x(561... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\frac{314721}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{314721}{4}$.",
"rob... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Key idea: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=280.5$. (Here the result is $\boxed{\frac{314721}$.) |
math-014141 | Inequalities: Product Given Sum | 8 | Make each step logically reversible (or explain if not): Let $x,y>0$ satisfy $x+y=204$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=204-x$ with $x\\in(0,204)$. Then $P(x)=xy=x(204... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{10404}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=10404$.",
"robustness_analysis": "If the problem were pertur... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=102.0$. (Here the result is $\boxed{10404}$.) |
math-014142 | Inequalities: Product Given Sum | 8 | Where appropriate, name the theorem you use: Let $x,y>0$ satisfy $x+y=594$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calcul... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=594-x$ with $x\\in(0,594)$. Then $P(x)=xy=x(594... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{88209}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=88209$.",
"robustness_analysis": "If the problem were pertur... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=297.0$. |
math-014143 | Inequalities: AM–GM — Equality Conditions | 8 | Use two approaches if possible: Let $x,y>0$ satisfy $x+y=136$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=136-x$ with $x\\in(0,136)$. Then $P(x)=xy=x(136... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{4624}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=4624$.",
"robustness_analysis": "Ge... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=68.0$. |
math-014144 | Inequalities: Product Given Sum | 8 | Warm-up: Let $x,y>0$ satisfy $x+y=556$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., concavity/second d... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=556-x$ with $x\\in(0,556)$. Then $P(x)=xy=x(556... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{77284}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=77284$.",
"robustness_analysis": "If the problem were pertur... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Key idea: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=278.0$. (Here the result is $\boxed{77284}$.) |
math-014145 | Algebra: Extremal Values — Global Bounds | 8 | Indicate where a theorem is used: Let $x,y>0$ satisfy $x+y=438$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem ... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=438-x$ with $x\\in(0,438)$. Then $P(x)=xy=x(438... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{47961}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=47961$.",
"robustness_analysis": "Generality note: AM–GM gen... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=219.0$. |
math-014146 | Inequalities: AM–GM — Equality Conditions | 8 | Work carefully and justify each inference: Let $x,y>0$ satisfy $x+y=82$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus ... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=82-x$ with $x\\in(0,82)$. Then $P(x)=xy=x(82-x)... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{1681}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=1681$.",
"robustness_analysis": "Robustness note: AM–GM generalizes... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=41.0$. |
math-014147 | Inequalities: Product Given Sum | 8 | State any required conditions first: Let $x,y>0$ satisfy $x+y=571$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theor... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=571$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\frac{326041}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{326041}{4}$.",
"robustness_a... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Key idea: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=285.5$. (Here the result is $\boxed{\frac{326041}$.) |
math-014148 | Inequalities: Product Given Sum | 8 | Show all reasoning: Let $x,y>0$ satisfy $x+y=704$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., concavi... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=704$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{123904}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=123904$.",
"robustness_analysis": "Robustness note: AM–GM g... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Key idea: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=352.0$. (Here the result is $\boxed{123904}$.) |
math-014149 | Inequalities: AM–GM — Equality Conditions | 8 | Solve and sanity-check: Let $x,y>0$ satisfy $x+y=215$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., con... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=215-x$ with $x\\in(0,215)$. Then $P(x)=xy=x(215... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\frac{46225}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{46225}{4}$.",
"robustness_analysis": "Generali... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Key idea: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=107.5$. (Here the result is $\boxed{\frac{46225}$.) |
math-014150 | Algebra: Extremal Values — Global Bounds | 8 | Solve with verification: Let $x,y>0$ satisfy $x+y=829$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., co... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=829$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\frac{687241}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{687241}{4}$.",
"robustness_a... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=414.5$. |
math-014151 | Optimization: Two Variables — Concavity | 8 | Track units/moduli carefully: Let $x,y>0$ satisfy $x+y=231$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=231$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\frac{53361}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{53361}{4}$.",
"robustness_analysis": "If the problem... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=115.5$. |
math-014152 | Inequalities: Product Given Sum | 8 | Track units/moduli carefully: Let $x,y>0$ satisfy $x+y=288$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=288-x$ with $x\\in(0,288)$. Then $P(x)=xy=x(288... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{20736}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=20736$.",
"robustness_analysis": "... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=144.0$. (Here the result is $\boxed{20736}$.) |
math-014153 | Algebra: Extremal Values — Global Bounds | 8 | Determine the requested value: Let $x,y>0$ satisfy $x+y=779$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=779$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\frac{606841}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{606841}{4}$.",
"robustness_a... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=389.5$. |
math-014154 | Inequalities: AM–GM — Equality Conditions | 8 | Answer using clear logical steps: Let $x,y>0$ satisfy $x+y=37$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=37$ to get $\\sqrt{xy}\\le \\frac{3... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\frac{1369}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{1369}{4}$.",
"robustness_analysis": "Generality note:... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Key idea: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=18.5$. |
math-014155 | Algebra: Extremal Values — Global Bounds | 8 | Explain what is being counted/optimized: Let $x,y>0$ satisfy $x+y=130$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus t... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=130-x$ with $x\\in(0,130)$. Then $P(x)=xy=x(130... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{4225}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=4225$.",
"robustness_analysis": "Robustness ... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=65.0$. (Here the result is $\boxed{4225}$.) |
math-014156 | Algebra: Extremal Values — Global Bounds | 8 | Prompt: Let $x,y>0$ satisfy $x+y=549$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., concavity/second de... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=549-x$ with $x\\in(0,549)$. Then $P(x)=xy=x(549... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\frac{301401}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{301401}{4}$.",
"robustness_analysis": "Genera... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Key idea: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=274.5$. (Here the result is $\boxed{\frac{301401}$.) |
math-014157 | Inequalities: AM–GM — Equality Conditions | 8 | Be explicit about assumptions: Let $x,y>0$ satisfy $x+y=57$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=57$ to get $\\sqrt{xy}\\le \\frac{5... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\frac{3249}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{3249}{4}$.",
"robustness_analy... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Takeaway: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=28.5$. (Here the result is $\boxed{\frac{3249}$.) |
math-014158 | Inequalities: Product Given Sum | 8 | Be explicit about assumptions: Let $x,y>0$ satisfy $x+y=67$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=67-x$ with $x\\in(0,67)$. Then $P(x)=xy=x(67-x)... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\frac{4489}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{4489}{4}$.",
"robustn... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Key idea: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=33.5$. |
math-014159 | Inequalities: AM–GM — Equality Conditions | 8 | Indicate where a theorem is used: Let $x,y>0$ satisfy $x+y=520$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem ... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=520$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{67600}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=67600$.",
"robustness_analysis": "... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=260.0$. (Here the result is $\boxed{67600}$.) |
math-014160 | Inequalities: Product Given Sum | 8 | Checkpoint: Let $x,y>0$ satisfy $x+y=534$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., concavity/secon... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=534-x$ with $x\\in(0,534)$. Then $P(x)=xy=x(534... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{71289}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=71289$.",
"robustness_analysis": "If the pr... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Takeaway: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=267.0$. |
math-014161 | Optimization: Two Variables — Concavity | 8 | Track units/moduli carefully: Let $x,y>0$ satisfy $x+y=586$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=586-x$ with $x\\in(0,586)$. Then $P(x)=xy=x(586... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{85849}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=85849$.",
"robustness_analysis": "Robustness note: AM–GM gen... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=293.0$. (Here the result is $\boxed{85849}$.) |
math-014162 | Algebra: Extremal Values — Global Bounds | 8 | Find the exact value: Let $x,y>0$ satisfy $x+y=307$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., conca... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=307-x$ with $x\\in(0,307)$. Then $P(x)=xy=x(307... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\frac{94249}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{94249}{4}$.",
"robustness_ana... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=153.5$. |
math-014163 | Algebra: Extremal Values — Global Bounds | 8 | Explain what is being counted/optimized: Let $x,y>0$ satisfy $x+y=868$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus t... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=868-x$ with $x\\in(0,868)$. Then $P(x)=xy=x(868... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{188356}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=188356$.",
"robustness_analysis": "If the ... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=434.0$. (Here the result is $\boxed{188356}$.) |
math-014164 | Inequalities: AM–GM — Equality Conditions | 8 | Show all reasoning: Let $x,y>0$ satisfy $x+y=899$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., concavi... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=899-x$ with $x\\in(0,899)$. Then $P(x)=xy=x(899... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\frac{808201}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{808201}{4}$.",
"robustness_analysis": "Robustness n... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Takeaway: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=449.5$. |
math-014165 | Inequalities: Product Given Sum | 8 | Answer using clear logical steps: Let $x,y>0$ satisfy $x+y=584$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem ... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=584$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{85264}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=85264$.",
"robustness_analysis": "... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=292.0$. (Here the result is $\boxed{85264}$.) |
math-014166 | Inequalities: Product Given Sum | 8 | Challenge: Let $x,y>0$ satisfy $x+y=110$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., concavity/second... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=110$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{3025}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=3025$.",
"robustness_analysis": "Ro... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Takeaway: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=55.0$. (Here the result is $\boxed{3025}$.) |
math-014167 | Optimization: Two Variables — Concavity | 8 | Give a theorem-based solution: Let $x,y>0$ satisfy $x+y=259$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=259-x$ with $x\\in(0,259)$. Then $P(x)=xy=x(259... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\frac{67081}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{67081}{4}$.",
"robustness_ana... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=129.5$. |
math-014168 | Optimization: Two Variables — Concavity | 8 | Derive the result step-by-step: Let $x,y>0$ satisfy $x+y=200$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=200-x$ with $x\\in(0,200)$. Then $P(x)=xy=x(200... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{10000}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=10000$.",
"robustness_analysis": "Generality note: AM–GM generaliz... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=100.0$. |
math-014169 | Inequalities: Product Given Sum | 8 | Solve and include a self-check: Let $x,y>0$ satisfy $x+y=256$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=256$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{16384}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=16384$.",
"robustness_analysis": "Sensitivity analysis: AM–GM gene... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=128.0$. |
math-014170 | Inequalities: Product Given Sum | 8 | Exercise: Let $x,y>0$ satisfy $x+y=749$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., concavity/second ... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=749-x$ with $x\\in(0,749)$. Then $P(x)=xy=x(749... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\frac{561001}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{561001}{4}$.",
"robustness_a... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=374.5$. (Here the result is $\boxed{\frac{561001}$.) |
math-014171 | Optimization: Two Variables — Concavity | 8 | Start by stating any domain restrictions: Let $x,y>0$ satisfy $x+y=345$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus ... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=345-x$ with $x\\in(0,345)$. Then $P(x)=xy=x(345... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\frac{119025}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{119025}{4}$.",
"robustness_analysis": "Sensitivity ... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Takeaway: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=172.5$. (Here the result is $\boxed{\frac{119025}$.) |
math-014172 | Inequalities: AM–GM — Equality Conditions | 8 | Explain what is being counted/optimized: Let $x,y>0$ satisfy $x+y=433$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus t... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=433$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\frac{187489}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{187489}{4}$.",
"robustness_analysis": "Robustness n... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=216.5$. |
math-014173 | Inequalities: Product Given Sum | 8 | Work this out carefully: Let $x,y>0$ satisfy $x+y=891$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., co... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=891$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\frac{793881}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{793881}{4}$.",
"robustness_a... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=445.5$. (Here the result is $\boxed{\frac{793881}$.) |
math-014174 | Optimization: Two Variables — Concavity | 8 | Write the solution set clearly: Let $x,y>0$ satisfy $x+y=53$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=53-x$ with $x\\in(0,53)$. Then $P(x)=xy=x(53-x)... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\frac{2809}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{2809}{4}$.",
"robustn... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Takeaway: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=26.5$. |
math-014175 | Inequalities: AM–GM — Equality Conditions | 8 | Solve and justify each step: Let $x,y>0$ satisfy $x+y=811$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g.... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=811$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\frac{657721}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{657721}{4}$.",
"robustness_analysis": "Robustness n... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Takeaway: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=405.5$. (Here the result is $\boxed{\frac{657721}$.) |
math-014176 | Optimization: Two Variables — Concavity | 8 | Keep the final answer in boxed form: Let $x,y>0$ satisfy $x+y=199$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theor... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=199$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\frac{39601}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{39601}{4}$.",
"robustness_analysis": "Generali... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Takeaway: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=99.5$. (Here the result is $\boxed{\frac{39601}$.) |
math-014177 | Optimization: Two Variables — Concavity | 8 | Make each step logically reversible (or explain if not): Let $x,y>0$ satisfy $x+y=358$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=358$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{32041}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=32041$.",
"robustness_analysis": "Robustnes... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Takeaway: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=179.0$. |
math-014178 | Optimization: Two Variables — Concavity | 8 | Give a fully justified solution: Let $x,y>0$ satisfy $x+y=473$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=473$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\frac{223729}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{223729}{4}$.",
"robustness_a... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Takeaway: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=236.5$. (Here the result is $\boxed{\frac{223729}$.) |
math-014179 | Inequalities: AM–GM — Equality Conditions | 8 | Make each step logically reversible (or explain if not): Let $x,y>0$ satisfy $x+y=702$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=702-x$ with $x\\in(0,702)$. Then $P(x)=xy=x(702... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{123201}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=123201$.",
"robustness_analysis": "Sensitivity analysis: AM–GM ge... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=351.0$. (Here the result is $\boxed{123201}$.) |
math-014180 | Optimization: Two Variables — Concavity | 8 | Do not skip justification steps: Let $x,y>0$ satisfy $x+y=578$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=578-x$ with $x\\in(0,578)$. Then $P(x)=xy=x(578... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{83521}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=83521$.",
"robustness_analysis": "If the problem were perturbed: A... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=289.0$. |
math-014181 | Algebra: Extremal Values — Global Bounds | 8 | Give an answer and a quick verification: Let $x,y>0$ satisfy $x+y=669$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus t... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=669-x$ with $x\\in(0,669)$. Then $P(x)=xy=x(669... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\frac{447561}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{447561}{4}$.",
"robustness_analysis": "If the... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=334.5$. (Here the result is $\boxed{\frac{447561}$.) |
math-014182 | Algebra: Extremal Values — Global Bounds | 8 | Try to avoid pattern-matching; explain why: Let $x,y>0$ satisfy $x+y=392$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculu... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=392-x$ with $x\\in(0,392)$. Then $P(x)=xy=x(392... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{38416}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=38416$.",
"robustness_analysis": "If the problem were pertur... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Takeaway: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=196.0$. (Here the result is $\boxed{38416}$.) |
math-014183 | Algebra: Extremal Values — Global Bounds | 8 | Complete the analysis: Let $x,y>0$ satisfy $x+y=682$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., conc... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=682$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{116281}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=116281$.",
"robustness_analysis": "Robustness note: AM–GM g... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=341.0$. |
math-014184 | Optimization: Two Variables — Concavity | 8 | Be explicit about assumptions: Let $x,y>0$ satisfy $x+y=873$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=873$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\frac{762129}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{762129}{4}$.",
"robustness_analysis": "Robust... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=436.5$. (Here the result is $\boxed{\frac{762129}$.) |
math-014185 | Optimization: Two Variables — Concavity | 8 | Give reasoning, not just computation: Let $x,y>0$ satisfy $x+y=505$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theo... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=505$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\frac{255025}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{255025}{4}$.",
"robustness_a... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=252.5$. (Here the result is $\boxed{\frac{255025}$.) |
math-014186 | Optimization: Two Variables — Concavity | 8 | Indicate where a theorem is used: Let $x,y>0$ satisfy $x+y=252$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem ... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=252-x$ with $x\\in(0,252)$. Then $P(x)=xy=x(252... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{15876}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=15876$.",
"robustness_analysis": "If the pr... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=126.0$. |
math-014187 | Optimization: Two Variables — Concavity | 8 | Give an answer and a quick verification: Let $x,y>0$ satisfy $x+y=257$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus t... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=257-x$ with $x\\in(0,257)$. Then $P(x)=xy=x(257... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\frac{66049}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{66049}{4}$.",
"robustness_analysis": "Sensitiv... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Remember: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=128.5$. |
math-014188 | Optimization: Two Variables — Concavity | 8 | Where appropriate, name the theorem you use: Let $x,y>0$ satisfy $x+y=340$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calcul... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=340-x$ with $x\\in(0,340)$. Then $P(x)=xy=x(340... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{28900}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=28900$.",
"robustness_analysis": "Robustness note: AM–GM generaliz... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=170.0$. (Here the result is $\boxed{28900}$.) |
math-014189 | Inequalities: AM–GM — Equality Conditions | 8 | Use two approaches if possible: Let $x,y>0$ satisfy $x+y=25$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=25-x$ with $x\\in(0,25)$. Then $P(x)=xy=x(25-x)... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\frac{625}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{625}{4}$.",
"robustness_analysis": "Sensitivity ... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Key idea: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=12.5$. (Here the result is $\boxed{\frac{625}$.) |
math-014190 | Algebra: Extremal Values — Global Bounds | 8 | Checkpoint: Let $x,y>0$ satisfy $x+y=855$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., concavity/secon... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=855-x$ with $x\\in(0,855)$. Then $P(x)=xy=x(855... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\frac{731025}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{731025}{4}$.",
"robustness_analysis": "If the probl... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Takeaway: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=427.5$. (Here the result is $\boxed{\frac{731025}$.) |
math-014191 | Algebra: Extremal Values — Global Bounds | 8 | Track quantifiers carefully: Let $x,y>0$ satisfy $x+y=378$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g.... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=378$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{35721}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=35721$.",
"robustness_analysis": "... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=189.0$. (Here the result is $\boxed{35721}$.) |
math-014192 | Inequalities: Product Given Sum | 8 | Give a theorem-based solution: Let $x,y>0$ satisfy $x+y=551$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=551$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\frac{303601}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{303601}{4}$.",
"robustness_a... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=275.5$. (Here the result is $\boxed{\frac{303601}$.) |
math-014193 | Inequalities: Product Given Sum | 8 | Indicate where a theorem is used: Let $x,y>0$ satisfy $x+y=192$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem ... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=192-x$ with $x\\in(0,192)$. Then $P(x)=xy=x(192... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{9216}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=9216$.",
"robustness_analysis": "Generality ... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=96.0$. (Here the result is $\boxed{9216}$.) |
math-014194 | Algebra: Extremal Values — Global Bounds | 8 | Carefully track domains: Let $x,y>0$ satisfy $x+y=866$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., co... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=866-x$ with $x\\in(0,866)$. Then $P(x)=xy=x(866... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{187489}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=187489$.",
"robustness_analysis": "Generality note: AM–GM general... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Takeaway: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=433.0$. (Here the result is $\boxed{187489}$.) |
math-014195 | Algebra: Extremal Values — Global Bounds | 8 | Find the exact value: Let $x,y>0$ satisfy $x+y=604$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., conca... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=604-x$ with $x\\in(0,604)$. Then $P(x)=xy=x(604... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{91204}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=91204$.",
"robustness_analysis": "Robustness note: AM–GM generaliz... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Takeaway: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=302.0$. (Here the result is $\boxed{91204}$.) |
math-014196 | Optimization: Two Variables — Concavity | 8 | Solve and justify each step: Let $x,y>0$ satisfy $x+y=383$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g.... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=383$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\frac{146689}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{146689}{4}$.",
"rob... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=191.5$. |
math-014197 | Inequalities: Product Given Sum | 8 | Challenge: Let $x,y>0$ satisfy $x+y=697$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., concavity/second... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=697$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\frac{485809}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{485809}{4}$.",
"rob... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Core principle: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=348.5$. (Here the result is $\boxed{\frac{485809}$.) |
math-014198 | Inequalities: AM–GM — Equality Conditions | 8 | Challenge: Let $x,y>0$ satisfy $x+y=281$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., concavity/second... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=281-x$ with $x\\in(0,281)$. Then $P(x)=xy=x(281... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\frac{78961}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{78961}{4}$.",
"robustness_analysis": "Sensitivity an... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Takeaway: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=140.5$. |
math-014199 | Optimization: Two Variables — Concavity | 8 | Question: Let $x,y>0$ satisfy $x+y=698$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem (e.g., concavity/second ... | [
{
"method_name": "AM–GM",
"approach": "Apply AM–GM to $x$ and $y$ to bound $\\sqrt{xy}$ in terms of $x+y$; equality characterizes the maximizer.",
"steps": [
"Step 1: By AM–GM, for $x,y>0$ we have $\\frac{x+y}{2}\\ge \\sqrt{xy}$.",
"Step 2: Substitute $x+y=698$ to get $\\sqrt{xy}\\le \\frac{... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{121801}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=121801$.",
"robustness_analysis":... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Takeaway: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=349.0$. (Here the result is $\boxed{121801}$.) |
math-014200 | Inequalities: AM–GM — Equality Conditions | 8 | Answer using clear logical steps: Let $x,y>0$ satisfy $x+y=327$.
(a) Find the maximum possible value of $xy$.
(b) State precisely when equality (the maximum) occurs.
(c) Explain why your argument guarantees a global maximum (not just a local one).
Your solution must explicitly cite either AM–GM or a calculus theorem ... | [
{
"method_name": "Calculus (Concave Quadratic)",
"approach": "Use the constraint to write $xy$ as a concave quadratic in one variable; a concave function on an interval has a unique global maximum at its critical point.",
"steps": [
"Step 1: Write $y=327-x$ with $x\\in(0,327)$. Then $P(x)=xy=x(327... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\frac{106929}$.\nAM–GM gives $xy\\le (S/2)^2$ and calculus finds the maximizer $x=y=S/2$ for the concave quadratic $x(S-x)$. Both yield the same maximum value $S^2/4=\\frac{106929}{4}$.",
"rob... | [
{
"error_description": "Squared an inequality without stating nonnegativity.",
"why_plausible": "Squaring is common and seems automatically valid.",
"why_wrong": "Squaring reverses inequalities when negative quantities are involved; the justification is that $\\sqrt{xy}\\ge 0$ and $S/2>0$.",
"which_... | Key idea: With a fixed positive sum, the product is maximized when the numbers are equal; AM–GM gives a global bound and identifies equality at $x=y=163.5$. |
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