RLAD
Collection
5 items • Updated
completion stringlengths 1.61k 4.45k | query stringlengths 693 3.85k |
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<notes>
<note>
<description>The Chinese Remainder Theorem (CRT) constructs a unique solution modulo the product of pairwise coprime moduli so that each modulus divides its specified residue. It is used to build sequences with prescribed congruence conditions, especially when the sequence must avoid certain residu... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>
When an equation involves floor functions of the form ⌊x/k⌋, observe that for any integer n, x satisfies ⌊x/k⌋=n precisely when x lies in the interval [n·k, (n+1)·k–1]. This transforms a piecewise condition into continuous bounds.
</description>
<example>
Suppose one needs ⌊... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Use indicator functions and linearity of expectation to compute probabilities in geometric settings. Define an indicator I for the desired event, then E[I]=∫ indicator density, which reduces to a volume or area ratio when sampling uniformly.</description>
<example>Suppose points (U... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>When an integer equation involves ⌊x/m⌋, introduce a new variable k=⌊x/m⌋ so that x=m·k and the original congruence becomes k≡a (mod b). This reduces problems with floor functions to simpler congruences on k.</description>
<example>Suppose one needs all x>0 with ⌊x/3⌋≡2 (mod 5).... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>When certain items must be adjacent in a linear arrangement, treat them as a single “block” before counting. This reduces the effective number of objects and ensures the adjacency condition is satisfied. After counting arrangements of these blocks, one can later account for internal or... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>For any four points on a circle, the sum of the measures of one pair of opposite angles equals 180°, and the cosine of one angle can be expressed in terms of the cosines of the adjacent angles. This lets you rewrite one cosine‐of‐angle term as a combination of cosines of two other angl... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Use the Fundamental Theorem of Arithmetic to restrict candidate numbers to those with exactly the desired number of prime factors. For a product of k distinct primes, factor any candidate and check that it has exactly k prime factors; if it has fewer, discard it immediately.</descripti... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>The locus of points equidistant from two intersecting lines consists of their angle bisectors. This principle arises because the angle bisector at the intersection of two lines is the set of points whose perpendicular distances to each line are equal. It is fundamental when aligning tw... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>When you have a rational expression of the form (A + B)/A, rewrite it as 1 + B/A. This separates the integer part (1) from the fractional part (B/A), making it easier to analyze integer and fractional components.</description>
<example>To simplify (x + y)/x, write it as 1 + y/x. Fo... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>When summing a function over a two-dimensional index, one can often reverse the order of summation to simplify the structure. By interchanging the sums, you may convert a complicated double sum into a single sum over one variable, turning the problem into counting or evaluating a one-v... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>When counting outcomes of independent trials, use the product rule: if each trial has m possible results, then k independent trials yield m^k total outcomes. This principle applies whenever choices at each step do not affect the count of others.</description>
<example>For instance,... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>When mixing solutions or alloys with different concentrations, the total amount of pure substance equals the sum of each component’s pure substance. Represent this by setting up an equation ∑(concentrationᵢ·massᵢ) = overall concentration·(∑massᵢ). This holds for any mixture problem.</d... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Use the digit‐sum test to determine whether a number is divisible by 3 or 9: a number is divisible by 3 (resp. 9) exactly when its decimal digits sum to a multiple of 3 (resp. 9). This lets you quickly rule out multiples of 3 or 9 without full division.
</description>
<example>Co... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>In a simple graph, the degree of any vertex lies between 0 and n–1, where n is the number of vertices. This “degree bound” immediately restricts possible degrees you can include in your graph. It’s the first check when trying to build a graph with given degree constraints.</description... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>A function’s range size gives an upper bound on the number of time‐slots needed to schedule its domain elements without overlaps. For a function f with domain D, the pigeonhole principle implies that if you assign each element to a time‐slot equal to f(x), then at most |range(f)| time‐... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>When summing k consecutive integers starting at a, the total is k·a + k(k–1)/2. This comes from pairing terms symmetrically around the central value. Use it to translate any consecutive-sum problem into a quadratic expression in a.</description>
<example>To sum 5 consecutive intege... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Decompose a nonhomogeneous linear recurrence into its homogeneous part and a particular solution. Solve the homogeneous equation to find the complementary solution, then guess (or apply) a method to find a particular solution that matches the nonhomogeneous term’s form. The general sol... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Use the Midpoint Connector Theorem: the segment joining midpoints of two sides of any quadrilateral is parallel to the diagonal connecting the other two vertices and has half its length. This holds regardless of side lengths or angles, providing a direct way to relate smaller shapes to... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Factor out the greatest common monomial from each term to reduce the degree and simplify the polynomial. This works because any common power of the variable divides every term, and extracting it yields a lower-degree polynomial easier to analyze.</description>
<example>Given a poly... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Use the triangle inequalities to translate geometric constraints into algebraic conditions on side lengths. For sides a, b, c, require a + b > c, a + c > b, and b + c > a. This ensures any chosen triple can form a triangle.</description>
<example>Suppose you want triangles... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Legendre’s formula computes the exponent of a prime p in n! by summing the integer parts of n/p^k over k≥1. It works because each multiple of p contributes at least one factor p, each multiple of p^2 contributes an extra, and so on.</description>
<example>To find the exponent of 2 ... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>When all items in a set are distinct, any two can be ordered in exactly two ways. If you record one of these orderings for each unordered pair, you have covered exactly half of all possible orderings. This principle lets you determine how many statements you need to record in order to ... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Translate a “last k digits” condition into a modular congruence. When you require that a product x·n ends in k digits matching n, set up the congruence x·n ≡ n (mod 10^k). This reduces the problem to finding integer solutions in a finite (mod 10^k) sense.</description>
<example>For... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Use the substitution t = x + 1/x to transform a polynomial symmetric in x and 1/x into a lower‐degree polynomial in t. This works because x^k + x^(–k) can be expressed in terms of t, reducing the problem to solving a simpler equation.</description>
<example>For the quartic x^4 + x^... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Place figures in a coordinate system to turn geometric constraints into algebraic equations. By aligning one side with an axis and assigning coordinates to key points, you simplify distance and perpendicularity conditions into linear or quadratic equations.</description>
<example>F... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Use the “sum of squares” norm to detect the trivial solution. If all variables are zero, the sum of their squares is zero, and any multiple of that norm is also zero. This shows that (0,0,…,0) is always a solution for any coefficient.</description>
<example>Consider the equation x₁... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>When an object must include a specified substring, model it as a collection of “blocks”: the substring itself and any characters outside the substring. Counting becomes the number of ways to arrange these blocks in a sequence of fixed length.</description>
<example>To count binary ... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>The sum of the interior angles of any n-sided polygon is given by the formula (n–2)·180°. This follows from triangulating the polygon into n–2 triangles, each contributing 180° to the total.</description>
<example>For a pentagon (n=5), the sum of its interior angles is (5–2)·180°=3... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>When a removal rule applies to groups (e.g. rows or columns), first count the total occurrences of the relevant elements. This gives a lower bound on how many can survive because each group can contribute at most its own size to the final count.</description>
<example>Suppose a gri... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Many integrals over symmetric intervals can be simplified by exploiting symmetry. If a function or its argument is even or odd, split the interval into symmetric halves and apply the corresponding definite‐integral identities: ∫₋a^a g(t) dt = 2∫₀^a g(t) dt for even g, or 0 for odd g.</... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Use the Pythagorean theorem to identify right triangles: for any three side lengths a ≤ b ≤ c, a triangle is right if and only if a² + b² = c². This criterion lets you test every triple of lengths for orthogonality.</description>
<example>Given lengths 2, 3, and √13, check whether ... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Many two-player removal games can be analyzed by partitioning the set of positions into “winning” and “losing” states. A position is losing if every move leads to a winning position; otherwise it is winning. This classification reduces the problem to checking reachable states rather th... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>When a problem requires all substrings of a number to satisfy a property, use the fact that the entire number itself is always a substring. This immediately eliminates any candidate that fails the property on the whole, without needing to check every substring.</description>
<examp... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Any integer composed of k repeated digits d in base b can be written as d·(b^k – 1)/(b – 1). This formula turns digit-repetition patterns into algebraic expressions suitable for divisibility and GCD calculations.</description>
<example>For instance, a 5-digit number in base 10 made... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Vieta’s formulas connect the coefficients of a polynomial to the elementary symmetric sums of its roots. In particular, for a polynomial of degree n, the sum of the roots, sum of pairwise products, and so on up to the product of all roots can be read off directly from its coefficients.... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Impose the necessary divisibility condition on linear combinations: for integers m and n, any integer combination m·x + n·y is divisible by g = gcd(m,n). Thus a positive integer n cannot be expressed as m·x + n·y unless g divides n.</description>
<example>Let m=6 and n=9. Their gcd... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Represent relationships by a graph whose vertices are the participants and whose edges encode the handshake (or any binary relation). Counting ordered triples with certain adjacency or non-adjacency patterns reduces to counting ordered triples of vertices in that graph. This turns a co... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Use the Euler totient theorem to reduce large exponents modulo φ(m) when gcd(a,m)=1. Since a^φ(m)≡1 (mod m), any exponent k can be reduced by taking k mod φ(m), simplifying computations for high powers.</description>
<example>To compute 7^100 mod 10, note φ(10)=4 and gcd(7,10)=1, s... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>In any triangle, the internal angle bisector from a vertex divides the opposite side into segments proportional to the adjacent sides. This follows from the Angle Bisector Theorem, which states that if a ray from vertex A meets side BC at D, then BD/DC = AB/AC. It is useful whenever a ... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>When a function is a polynomial of odd degree, it is automatically bijective over the real line because it is strictly monotonic (its derivative never changes sign). This ensures the inverse exists and is unique. It simplifies the search for inverse‐related constructions.</description>... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Use base-10 expansions to express numbers with repeated digits as sums of powers of ten times a single digit. This turns digit-based questions into algebraic sums, making it easier to manipulate and compare numbers. It works because each digit’s contribution grows by a factor of ten in... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>When summing over all subsets of a set, partition the subsets by their size. For each nonnegative integer k, count how many k-element subsets there are (via binomial coefficients) and then sum contributions of each size separately. This reduces a global sum to sums over simpler “size-k... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Use a strategic substitution to turn a complicated summand into a simpler, more tractable form. By choosing a parameter or index that matches a pattern inside the sum, you can often rewrite the expression in terms of well-known functions or simpler algebraic terms.</description>
<e... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>In a right triangle, the radius r of the inscribed circle relates to the legs a, b and hypotenuse c by the formula r = (a + b – c)/2. This comes from the area A = rs and the fact that for a right triangle s = (a + b + c)/2. Use this whenever you know both the inradius and one side (or ... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Model a sorted selection of k items from a line of N as a k-tuple of nonnegative integers (d₁,…,dₖ) with d₁+d₂+⋯+dₖ=N−k. This bijection turns selection problems into counting integer compositions subject to bounds, making combinatorial tools directly applicable.</description>
<exam... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Use modular invariants to limit possible values: by computing the target expression modulo a small integer, one can derive a finite list of residue classes that terms can occupy. Any candidate sequence must lie entirely within that residue class, drastically reducing the search space.<... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>When a region is tiled by congruent shapes, each shape covers exactly one unit of a chosen additive measure (area, perimeter, etc.). Thus the total measure of the region equals the number of shapes times the measure of one shape. This lets you compute the minimum number of shapes by di... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Use the side–side–angle (SSA) criterion to show two triangles are congruent. If two triangles share two equal sides and the angle opposite one of those sides is equal, then the triangles are congruent. This works in the plane when the included angle between the two known sides is acute... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Use the perpendicularity condition to relate side lengths: if a diagonal is perpendicular to one side of a parallelogram, the triangle formed by that diagonal and the side is right-angled. In such a triangle, the Pythagorean theorem gives the length of the diagonal in terms of the side... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Use modular arithmetic to test whether a target value can ever appear as a difference of two exponential bases. By reducing each base to a residue class modulo a chosen integer, one obtains a finite set of possible residues and can rule out the target if it never appears in that class.... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Use the Euclidean algorithm to compute the greatest common divisor (gcd) of the coefficients. This reveals whether a linear combination of variables can exactly match a given total, and if so, simplifies the system by dividing through by the gcd.</description>
<example>Suppose you ... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
<notes>
<note>
<description>Use Fermat’s little theorem to test divisibility by a prime: for any integer a and any prime p, a^p ≡ a (mod p). This holds because the multiplicative order of a modulo p divides p–1, so a^(p–1) ≡ 1 (mod p) if a is not divisible by p. It is especially useful when you need to show a^p–1... | <|im_start|>user
Given the following math problem, generate a list of insightful hints that help guide a student toward solving the problem. Each hint should be wrapped in a <note> block with the following structure:
<note>
<description>[Brief explanation of a key idea or technique relevant to the problem]</descriptio... |
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RLAD-HintGen-warmup
Warm-start SFT data for the RLAD hint/abstraction generator.
Mirror of Asap7772/dapo5k-offlinedata-hintgen-qwen3-4b-lr1e6-warmstartfinal
(revision b4cf670b04d2336d3d87d52629b0d1171c552fa3), copied unmodified: 4,500 train / 500 test
(query, completion) pairs.
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