Contradiction. Instead, assume reported rate is approximate, but in math olympiad context, expect exact fraction:

["Understanding Contradiction in Mathematical Olympiads: The Exact Fraction Is Key", "In the high-stakes world of mathematical olympiads, precision is paramount. Contestants face problems that demand sharp logic, creative insight, and strict adherence to mathematical rigor—especially when dealing with conditions that hinge on clear logical outcomes. Among the central themes in competition problems is the concept of contradiction—a powerful tool used to disprove assumptions, validate statements, and guide problem-solving strategies.", "Though often talked about in passing, contradiction is far more than just "saying something is false." In the context of math olympiads, contradiction is a structured method: assume the opposite of what you aim to prove and show that this assumption leads to an inconsistency—often a violation of a known rule, a well-defined definition, or an impossible mathematical fact.", "### Why Contradiction is Integral to Olympiad Problems", "At first glance, contradiction may seem indirect, but in olympiad problems, especially those involving inequalities, divisibility, ratios, or equivalence, contradiction serves as a sharp pencil to eliminate impossible cases swiftly. The reported rate in many competition problems—even when ostensibly presented as approximate—is actually expected to be treated as an exact fraction, reinforcing the necessity of precision.", "For instance, consider a problem stating: The ratio of two numbers is 3/5. Show that their sum exceeds 16. Rather than evaluating the ratio in decimal form, contestants use exact fractions: let the numbers be ( \frac{3k}{5k} ), then ( \frac{3k + 5k}{k} = \frac{8k}{k} = \frac{8}{1} = 8 ). So the sum is exactly 8—but the condition may require exceeding 16, triggering contradiction if constraints are misapplied. The exact fractional form is critical for correct reasoning.", "### When Does Contradiction Work in Olympiads?", "1. Disproving Invalid Assumptions\n If a problem asks to prove a statement is always true, suppose a contestant assumes it’s false and derives a result such as ( \frac{a}{b} < 0 ) given ( a, b > 0 )—a contradiction with the premise. This forces reevaluation and strengthens proof.", "2. Confirming Equality Through Contradiction\n Problems often ask to show two expressions are equal. Assume they differ (e.g., ( \frac{p}{q} <br/>\neq \frac{r}{s} )), then derive an impossible equation—such as ( p \cdot s <br/>\neq r \cdot q )—contradicting fundamental fraction rules.", "3. Eliminating Impossible Cases\n In optimization or ratio-based problems, suppose a candidate assumes a certain value satisfies multiple impossible constraints (e.g., a fraction being both greater than 1 and less than 0). The contradiction invalidates that assumption, sharpening focus on viable solutions.", "### The Role of Exact Fractions Over Approximations", "Unlike casual math, olympiad contexts demand exactness. Reporting a “reported rate” as approximate is misleading—true olympiad problems expect precise fractional values. For example, stating “the ratio is approximately 2/3” is insufficient unless tied to an exact relationship. Contestants must work with fractions like ( \frac{2}{3} ), ( \frac{5}{8} ), or ( \frac{a}{b} ) where ( a, b ) are integers, reinforcing the necessity of exact reasoning.", "Misinterpreting approximate values can lead to incorrect conclusions, making the exact fraction a cornerstone of correct strategy. Always verify decimal approximations against rational representations before proceeding with proof.", "### Mastery Through Practice", "To excel in olympiads, students must internalize contradiction as a core technique—not a fallback. Practice problems where contradiction is central, such as proving uniqueness, ruling out values, or validating identities using fractional exactness. Remember: contradiction works when the assumption invariably contradicts an objectively true proposition.", "### Final Thoughts", "Contradiction, when applied with rigor and exactness in fractional form, is indispensable in mathematical olympiad problem-solving. It transforms vague suspicions into mathematical certainty and turns complex problems into clear, solvable puzzles. Embrace contradiction not just as a trick, but as a foundational lens through which contradiction reveals truths with absolute clarity—especially when numbers are fractions, not approximations.", "In math olympiads, the exact fraction is your ally. Respect the contradiction, honor the ratio, and let precision guide your proof."]