aops_id int64 3 3.62M | problem stringlengths 17 4.71k | best_solution stringlengths 7 38.7k | problem_vector listlengths 4.1k 4.1k | best_solution_vector listlengths 4.1k 4.1k | last_modified stringdate 2025-08-25 00:00:00 2025-08-25 00:00:00 |
|---|---|---|---|---|---|
231,038 | Let \(a_1\) be a natural number not divisible by \(5\). The sequence \(a_1,a_2,a_3,\dots\) is defined by
\[
a_{n+1}=a_n+b_n,
\]
where \(b_n\) is the last digit of \(a_n\). Prove that the sequence contains infinitely many powers of two. | [quote="outback"]Let $ a_1$ be a natural number not divisible by $ 5$. The sequence $ a_1, a_2, a_3, . . .$ is defined by $ a_{n \plus{} 1} \equal{} a_n \plus{} b_n$, where $ b_n$ is the last digit of $ a_n$. Prove that the sequence contains infinitely many powers of two.[/quote]
[hide="Solution"]
The second number of... | [
0.2541322708129883,
0.12177586555480957,
2.8280839920043945,
-0.09444086998701096,
0.293412983417511,
-2.9982411861419678,
-2.3400046825408936,
1.958577275276184,
-0.7243392467498779,
0.5812130570411682,
2.799819231033325,
2.5093653202056885,
2.4586410522460938,
1.320928931236267,
0.3369... | [
0.5862583518028259,
1.3891080617904663,
3.1553313732147217,
-0.012347571551799774,
-0.0019843271002173424,
-2.3261184692382812,
-2.729907512664795,
2.0136704444885254,
-0.6429914236068726,
0.7785053253173828,
3.133678674697876,
1.9021496772766113,
-0.3365040123462677,
0.8972617387771606,
... | 2025-08-25 |
2,310,426 | Find the number of complex solutions to
\[
\frac{z^3 - 1}{z^2 + z - 2} = 0.
\] | "[quote=GameBot]Find the number of complex solutions to\n\\[\\frac{z^3 - 1}{z^2 + z - 2} = 0.\\][/qu(...TRUNCATED) | [2.0834505558013916,-3.3445932865142822,1.9409775733947754,-1.191853642463684,-0.9529173374176025,-2(...TRUNCATED) | [1.644652247428894,-1.660526990890503,3.654900550842285,-2.7071850299835205,-2.7177391052246094,-0.7(...TRUNCATED) | 2025-08-25 |
2,310,435 | "Let \\(ABC\\) be a triangle with circumcircle \\((O)\\). Let \\(P\\) be an arbitrary interior point(...TRUNCATED) | "I have labelled $X,Y,Z$ as $K_a,K_b,K_c$ in my solution.\n[quote=tutubixu9198]Let $ABC$ be a triang(...TRUNCATED) | [-2.5846006870269775,0.5811729431152344,2.724564790725708,-0.5052332282066345,-0.18091700971126556,0(...TRUNCATED) | [-1.2537554502487183,0.010456682182848454,2.883819103240967,-0.5252880454063416,0.2914458215236664,0(...TRUNCATED) | 2025-08-25 |
2,310,441 | "Prove that\n\\[\n\\operatorname{Area}(\\triangle ABC)=(s-a)r_a,\n\\]\nwhere \\(s=\\dfrac{a+b+c}{2}\(...TRUNCATED) | "[quote=franzliszt]You have $[ABC]=rs$ where $r$ is the inradius. Consider the homothety sending the(...TRUNCATED) | [0.7101249694824219,1.0419400930404663,2.2005584239959717,-0.2358989715576172,1.1500260829925537,1.2(...TRUNCATED) | [2.1102700233459473,1.3601102828979492,2.2644081115722656,1.670838475227356,1.8061946630477905,4.670(...TRUNCATED) | 2025-08-25 |
231,045 | "A rectangular cow pasture is enclosed on three sides by a fence and the fourth side is part of the (...TRUNCATED) | "The maximum amount of fence the farmer can have is 1200/5=240 feet. Since a square maximizes area, (...TRUNCATED) | [0.6065350770950317,-1.4337310791015625,-0.3184584379196167,0.09022809565067291,3.272578716278076,1.(...TRUNCATED) | [2.2133007049560547,-0.6139740943908691,1.525972843170166,-0.0005062657874077559,2.0105016231536865,(...TRUNCATED) | 2025-08-25 |
23,105 | "Problem: Find all positive integer $n$ and prime number $p$ such that : \r\nAny $a_1,a_2,...,a_n\\i(...TRUNCATED) | "I think that $n<3$ and $p=4k+3$.\r\n\r\nLet's first show that for $n\\ge 3$, all residues $\\pmod p(...TRUNCATED) | [0.13819745182991028,-2.2105321884155273,3.237818717956543,-1.740485668182373,1.626124620437622,-1.6(...TRUNCATED) | [0.9701294302940369,-1.1182256937026978,1.9331145286560059,-0.2944217920303345,0.13968773186206818,-(...TRUNCATED) | 2025-08-25 |
231,050 | "A rectangular piece of paper \\(ADEF\\) is folded so that corner \\(D\\) meets the opposite edge \\(...TRUNCATED) | "mathwizarddude, i believe your wrong.\r\n\r\n[u]Here is my solution[/u]\r\n\r\nHere is my diagram\r(...TRUNCATED) | [0.4302624762058258,-1.4925003051757812,0.5256382822990417,0.001766242552548647,1.6222848892211914,0(...TRUNCATED) | [0.10762961208820343,-1.0328435897827148,-0.10803539305925369,-0.294585257768631,0.9480099081993103,(...TRUNCATED) | 2025-08-25 |
2,310,536 | "For each positive integer \\(n\\), the mean of the first \\(n\\) terms of a sequence is \\(n\\). Wh(...TRUNCATED) | "[hide=Better Solution]We notice that: $$a_1+a_2+\\cdots+a_n=n^2.$$ Therefore: $$a_1+a_2+\\cdots+a_{(...TRUNCATED) | [0.6082778573036194,-0.6106126308441162,2.9449315071105957,0.7773684859275818,0.4191873073577881,0.8(...TRUNCATED) | [1.6345138549804688,0.8930032253265381,4.8069539070129395,2.5494296550750732,-0.6012194156646729,-0.(...TRUNCATED) | 2025-08-25 |
231,054 | "Two regular polygons with the same number of sides have side lengths 48 m and 55 m, respectively. A(...TRUNCATED) | "[hide=\"most likely incorrect\"]The area of the regular $ n$-gon is directly proportional to the sq(...TRUNCATED) | [1.071570873260498,0.116093710064888,1.7603089809417725,-1.4118642807006836,1.7666045427322388,3.086(...TRUNCATED) | [-1.1959125995635986,0.33045363426208496,3.972243309020996,0.2752837538719177,0.7204051613807678,4.5(...TRUNCATED) | 2025-08-25 |
2,310,555 | "Let \\(f:\\mathbb{R}\\to\\mathbb{R}\\) be a differentiable function such that\n\\[\n\\lim_{x\\to 2}(...TRUNCATED) | "If $y=f(x)$ intersects the line $6x-y=4$, then $f(2)=8$ (i). Upon inspection, we have:\n\n$\\lim_{(...TRUNCATED) | [0.4626089930534363,-1.1113663911819458,1.946232795715332,-0.02194814570248127,1.8518993854522705,-2(...TRUNCATED) | [2.315070152282715,0.06585457921028137,1.0778002738952637,1.943103313446045,1.5350759029388428,-1.79(...TRUNCATED) | 2025-08-25 |
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