ID stringlengths 6 10 | Year int64 1.98k 2.02k | Problem Number int64 1 15 | Part stringclasses 2
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values | context stringlengths 0 306 | hint1 stringlengths 0 279 | hint2 stringlengths 0 295 | hint3 stringlengths 0 291 | hint4 stringlengths 0 300 | hint5 stringlengths 0 360 |
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2020-I-6 | 2,020 | 6 | I | "A flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the (...TRUNCATED) | 173 | ["Okay, let's try to solve this problem step by step. So, the problem is about two spheres sitting i(...TRUNCATED) | [
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2020-I-7 | 2,020 | 7 | I | "A club consisting of $11$ men and $12$ women needs to choose a committee from among its members so (...TRUNCATED) | 81 | ["</think>\n\nLet's solve the problem step by step.\n\n**Problem Statement:**\nA club consisting of (...TRUNCATED) | ["676455","= no"," something,...( which,,, wait","81","20","676041","3 + 5 + 7 + 11 = 26","1352726",(...TRUNCATED) | [
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] | 0.1875 | Combinatorics | "A combinatorics problem involving counting committee selections from two distinct groups with a fix(...TRUNCATED) | "Define your variables clearly—let m be the number of men chosen so that the number of women is au(...TRUNCATED) | "Carefully determine the range of m by using both the available counts and the requirement that the (...TRUNCATED) | "Write the total count as a sum ∑₍ₘ₌₀₎¹¹ C(11, m)·C(12, m + 1) and consider re-inde(...TRUNCATED) | "Look for a combinatorial identity—specifically, a Vandermonde‐style convolution—that collapse(...TRUNCATED) | "Once you have an expression for N, ensure you factor it carefully to extract its prime divisors bef(...TRUNCATED) |
2020-I-8 | 2,020 | 8 | I | "A bug walks all day and sleeps all night. On the first day, it starts at point $O,$ faces east, and(...TRUNCATED) | 103 | ["Alright, let's tackle this problem step by step. So, we have a bug that starts at point O, faces e(...TRUNCATED) | [
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] | 0.9375 | Geometry | "A geometry problem involving an infinite sequence of scaled, rotated vectors whose sum defines a fi(...TRUNCATED) | "Represent each day's movement as a complex number to encode both magnitude and rotation in a single(...TRUNCATED) | "Recognize that successive displacements form a geometric series; verify convergence by confirming t(...TRUNCATED) | "Apply the infinite geometric series sum formula to determine the overall displacement vector leadin(...TRUNCATED) | "When simplifying complex expressions, multiply by the conjugate to rationalize denominators and ext(...TRUNCATED) | "Separate the computed complex sum into its real and imaginary components to independently verify th(...TRUNCATED) |
2020-I-9 | 2,020 | 9 | I | "Let $S$ be the set of positive integer divisors of $20^9.$ Three numbers are chosen independently a(...TRUNCATED) | 77 | ["</think>\n\nLet's solve the problem step by step.\n\n**Problem Statement:**\nWe need to find the p(...TRUNCATED) | [
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] | 0.4375 | Number Theory | "A Number Theory problem involving the divisors of 20⁹ and finding the probability that three inde(...TRUNCATED) | "Express each divisor in the form 2ᵃ·5ᵇ so that every element of S corresponds to an exponent(...TRUNCATED) | "Translate “a₁ divides a₂” (and similarly for a₂ divides a₃) into the inequalities a₁ (...TRUNCATED) | "Count the total number of divisors using the formula (18+1)(9+1)=190 and compute the total number o(...TRUNCATED) | "For the favorable cases, count the number of non‐decreasing triples of exponents separately for 2(...TRUNCATED) | "After obtaining the counts for both total and favorable cases, simplify the resulting fraction care(...TRUNCATED) |
2020-I-10 | 2,020 | 10 | I | "Let $m$ and $n$ be positive integers satisfying the conditions $\\quad\\bullet\\ \\gcd(m+n,210)=1,$(...TRUNCATED) | 407 | ["</think>\n\nTo solve the problem, let's break down the given conditions and find the least possibl(...TRUNCATED) | ["11"," something is is,,., is. but?,:?,. which is is is ,.","Therefore, the minimal possible value (...TRUNCATED) | [
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] | 0 | Number Theory | "A Number Theory problem involving divisibility and coprimality constraints on expressions built fro(...TRUNCATED) | "Begin by testing small candidate values for n and m—but beware that trivial cases (like n = 1) mu(...TRUNCATED) | "Translate the condition that m^m is a multiple of n^n into constraints on the prime exponents; comp(...TRUNCATED) | "Remember that “m is not a multiple of n” forces m and n to have different “extra” prime fac(...TRUNCATED) | "Analyze the requirement that m+n be coprime with 210 by testing its divisibility modulo each prime (...TRUNCATED) | "Consider structuring your search by splitting into distinct cases (for instance, starting with n = (...TRUNCATED) |
2020-I-11 | 2,020 | 11 | I | "For integers $a,b,c$ and $d,$ let $f(x)=x^2+ax+b$ and $g(x)=x^2+cx+d.$ Find the number of ordered t(...TRUNCATED) | 510 | ["Alright, so I've got this problem here. Let me try to parse it step by step. \n\nWe have two quadr(...TRUNCATED) | [" something. wait is is is","21","100","120","After careful analysis, we find that the number of su(...TRUNCATED) | [
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] | 0 | Algebra | "A quadratic composition problem involving two quadratic functions with bounded integer coefficients(...TRUNCATED) | "Notice that having g(f(2)) = g(f(4)) = 0 forces f(2) and f(4) to be the roots of g, so their expres(...TRUNCATED) | "Calculate f(2) and f(4) in terms of a and b; then observe their difference to decide whether they m(...TRUNCATED) | "When writing g(x) as (x – f(2))(x – f(4)), compare coefficients to derive relationships bet(...TRUNCATED) | "Use the symmetry of a quadratic: if f(2) equals f(4), the axis is fixed and immediately yields a co(...TRUNCATED) | "Always check that any parameters calculated by setting up these relations also satisfy the imposed (...TRUNCATED) |
2020-I-12 | 2,020 | 12 | I | "Let $n$ be the least positive integer for which $149^n-2^n$ is divisible by $3^3\\cdot5^5\\cdot7^7.(...TRUNCATED) | 270 | ["</think>\n\nTo solve the problem, we need to find the least positive integer \\( n \\) such that \(...TRUNCATED) | ["16","315","210","42","18","1200","9 \\times 5^^^77 \\7^5^7 \\ 7^ incorrect^9","9"," is is is,","42(...TRUNCATED) | [
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] | 0 | Number Theory | "It involves finding the smallest positive integer n such that an exponential difference attains div(...TRUNCATED) | "Try rewriting the expression by factoring out common exponents to obtain a congruence like (a/b)ⁿ(...TRUNCATED) | "Determine the multiplicative order of the resulting base modulo each prime power carefully, as erro(...TRUNCATED) | "Use Euler’s theorem and, when applicable, the lifting exponent lemma to handle orders for high pr(...TRUNCATED) | "Once you find the orders for each modulus, combine the conditions by taking their least common mult(...TRUNCATED) | "Factorize the resulting n into its prime components and apply the divisor count formula by incremen(...TRUNCATED) |
2020-I-13 | 2,020 | 13 | I | "Point $D$ lies on side $\\overline{BC}$ of $\\triangle ABC$ so that $\\overline{AD}$ bisects $\\ang(...TRUNCATED) | 36 | ["<think>\nOkay, so I have this problem here about triangle ABC with sides AB=4, BC=5, and CA=6. Poi(...TRUNCATED) | [".","64",",wait??)","36","36","224","\\dfrac{10\\sqrt{3}}{3}","36","50"," is??)","32","36","? wait?(...TRUNCATED) | [
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] | 0.3125 | Geometry | "This geometry problem involves a triangle with given side lengths and constructions of angle and pe(...TRUNCATED) | "Consider placing the triangle in the coordinate plane (for example, with one side along the x‑axi(...TRUNCATED) | "Use the Angle Bisector Theorem to determine how point D divides side BC, as the ratio AB:AC gives B(...TRUNCATED) | "Find the midpoint of segment AD and use the negative reciprocal of AD’s slope to write the equati(...TRUNCATED) | "Determine the equations of the internal bisectors of angles at B and C (by using section formulas o(...TRUNCATED) | "Once the coordinates of A, E, and F are found, apply the shoelace formula (or use the vector cross (...TRUNCATED) |
2020-I-14 | 2,020 | 14 | I | "Let $P(x)$ be a quadratic polynomial with complex coefficients whose $x^2$ coefficient is $1.$ Supp(...TRUNCATED) | 85 | ["</think>\n\nLet's solve the problem step by step.\n\n**Problem Statement:**\nLet \\( P(x) \\) be a(...TRUNCATED) | [
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2020-I-15 | 2,020 | 15 | I | "Let $\\triangle ABC$ be an acute triangle with circumcircle $\\omega,$ and let $H$ be the intersect(...TRUNCATED) | 58 | ["Alright, let me try to work through this problem step by step. It's a bit complex, but I'll take i(...TRUNCATED) | ["18\\sqrt{3}","9","9","7","9",".","9","15","15","perhaps..4. perhaps, but but,.\n\n.\n\n.\n\n5.\n\n(...TRUNCATED) | [
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] | 0 | Geometry | "A geometry problem involving an acute triangle’s orthocenter, its circumcircle(s), and a tangent (...TRUNCATED) | "Review the relationships between an acute triangle’s orthocenter and its circumradius—remember (...TRUNCATED) | "Carefully distinguish between the different circles involved; using the power of a point requires e(...TRUNCATED) | "Construct a precise diagram to track how the tangent to one circle (the circumcircle of HBC) inters(...TRUNCATED) | "Consider using methods from both classical circle theorems (like tangent–secant relationships) an(...TRUNCATED) | "Be cautious of common pitfalls—misinterpreting segment products and tangency properties can lead (...TRUNCATED) |
End of preview. Expand in Data Studio
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