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"How many ways are there for Nick to travel from \\((0,0)\\) to \\((16,16)\\) in the coordinate plan(...TRUNCATED)
| "This condition is equivalent to the first and last step being in different directions, as if you sw(...TRUNCATED)
| 310,235,040
| 0.625
|
start/end differ;two cases;remove endpoints;binomial count;double by symmetry
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
|
"Let \\( \\triangle ABC \\) be an obtuse triangle with circumcenter \\( O \\) such that \\( \\angle (...TRUNCATED)
| "Let the radius of the circumcircle of \\( \\triangle ABC \\) be \\( r \\).\n\n\\[\n\\begin{align*}\(...TRUNCATED)
| 35
| 0.5
|
Factor with r;Power of D;Isosceles identification;Angle-chase for α;Central–inscribed relation
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "Okay, let's try to tackle this problem step by step. First, let's recall some concepts. The problem(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
|
"Problem: What is the smallest positive integer representable as the sum of the cubes of three posit(...TRUNCATED)
| "Solution: Let this smallest positive integer be represented as \\(a^{3}+b^{3}+c^{3}=d^{3}+e^{3}+f^{(...TRUNCATED)
| 251
| 0.875
| "Assume minimal;Inspect smallest cubes;Place 5th WLOG;Forces double elimination;Analyze sixth unique(...TRUNCATED)
| "Okay, let's try to figure out this problem. The question is asking for the smallest positive intege(...TRUNCATED)
| "Okay, let's try to figure out this problem step by step. The question is asking for the smallest po(...TRUNCATED)
| "Okay, let's try to figure out this problem step by step. The question is asking for the smallest po(...TRUNCATED)
| "Okay, let's try to figure out this problem step by step. The question is asking for the smallest po(...TRUNCATED)
| "Okay, let's try to figure out this problem step by step. The question is asking for the smallest po(...TRUNCATED)
| "Okay, let's try to figure out this problem. The question is asking for the smallest positive intege(...TRUNCATED)
| "Okay, let's try to figure out this problem. The question is asking for the smallest positive intege(...TRUNCATED)
| "Okay, let's try to solve this problem step by step. The question is asking for the smallest positiv(...TRUNCATED)
|
"Let \\( \\triangle ABC \\) be a triangle with \\( BC = 2007 \\), \\( CA = 2008 \\), \\( AB = 2009 \(...TRUNCATED)
| "Let line \\( AD \\) meet \\( \\omega \\) again at \\( H \\). Since \\( AF \\) and \\( AE \\) are ta(...TRUNCATED)
| 2,014,024
| 0.875
|
Harmonic quadruple;Polar through O;Harmonic on BC;Solve harmonic equation
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
|
"Let \\( G, A_{1}, A_{2}, A_{3}, A_{4}, B_{1}, B_{2}, B_{3}, B_{4}, B_{5} \\) be ten points on a cir(...TRUNCATED)
| "Note that \\( G B_{3} \\) is a diameter of the circle. As a result, \\( A_{2}, A_{3} \\) are symmet(...TRUNCATED)
| 12
| 1
| "GB3 is diameter;Swap A2/A3 B1/B5;Chords meet GB3;Angle equals ∠GG2B1;Half difference arcs;Substit(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
|
"At a certain college, there are $10$ clubs and some number of students. For any two different stude(...TRUNCATED)
| "Let \\( C \\) be the set of clubs; each student then corresponds to a subset of \\( C \\) (the club(...TRUNCATED)
| 513
| 1
| "Students as subsets;Define f = S Δ T;Fixed-point-free involution;Partition into pairs;Pairs: canno(...TRUNCATED)
| "Okay, let's try to tackle this problem step by step using the given 7 hints. First, let's parse the(...TRUNCATED)
| "Okay, let's try to tackle this problem step by step. First, let's parse the problem statement caref(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
|
"Let \\( x_{1}, x_{2}, \\ldots, x_{2022} \\) be nonzero real numbers. Suppose that \\( x_{k}+\\frac{(...TRUNCATED)
| "Let the answer be \\( M \\). If \\( M > 1011 \\), there would exist two consecutive positive terms (...TRUNCATED)
| 1,010
| 0.875
| "Pigeonhole: adjacent positives;Alternating sign inequalities;Multiply around cycle;Explicit constru(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
|
"Compute \\(\\arctan \\left(\\tan 65^{\\circ}-2 \\tan 40^{\\circ}\\right)\\). (Express your answer i(...TRUNCATED)
| "First Solution: We have\n\\[\n\\tan 65^{\\circ}-2 \\tan 40^{\\circ}=\\cot 25^{\\circ}-2 \\cot 50^{\(...TRUNCATED)
| 25
| 1
| "convert to cot;cot double-angle;reciprocal → tan;split as 45°+θ;tan double-angle;common denomin(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "Okay, let's try to solve this problem step by step. The problem is to compute \\(\\arctan \\left(\\(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "Okay, let's try to solve this problem step by step using the given hints. The problem is to compute(...TRUNCATED)
| "Okay, let's try to solve this problem step by step. The problem is to compute arctan(tan 65° - 2 t(...TRUNCATED)
| "Okay, let's try to solve this problem step by step. The problem is to compute arctan(tan 65° - 2 t(...TRUNCATED)
| "Okay, let's try to solve this problem step by step. The problem is to compute arctan(tan65° - 2 ta(...TRUNCATED)
|
"Rectangle \\(ABCD\\) has sides in the ratio of \\(\\sqrt{2}\\) to \\(1\\). If \\(\\triangle DEC\\) (...TRUNCATED)
| "Since \\(\\triangle DEC\\) is an isosceles right triangle, we have \\(ED = CD = \\sqrt{2} \\cdot AD(...TRUNCATED)
| 135
| 0.875
|
ED = AD;AED half-supplement;AED = BEC;AEB via full-angle
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "Okay, let's try to solve this problem step by step. First, let's make sure I understand the problem(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "Okay, let's try to solve this problem step by step. First, let's make sure I understand the problem(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
|
"Bassanio has three red coins, four yellow coins, and five blue coins. At any point, he may give Shy(...TRUNCATED)
| "Let \\( r, y, b \\) denote the numbers of red, yellow, and blue coins respectively. Note that each (...TRUNCATED)
| 7
| 1
| "Parity invariant;Even difference determines target;Each exchange reduces total;Balance then convert(...TRUNCATED)
| "Okay, let's try to figure out this problem step by step. So, Bassanio has three red (R), four yello(...TRUNCATED)
| "Okay, let's try to figure out this problem step by step. So, Bassanio has three red (R), four yello(...TRUNCATED)
| "Okay, let's try to figure out this problem step by step. So, Bassanio has 3 red (R), 4 yellow (Y), (...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "Okay, let's try to figure out this problem step by step. So, Bassanio has three red (R), four yello(...TRUNCATED)
| "Okay, let's try to figure out this problem step by step. So, Bassanio has 3 red (R), 4 yellow (Y), (...TRUNCATED)
| "Okay, let's try to figure out this problem step by step. So, Bassanio has three red (R), four yello(...TRUNCATED)
| "Okay, let's try to figure out this problem step by step. So, Bassanio has three red (R), four yello(...TRUNCATED)
|
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