id int64 8 3.28M | problem stringlengths 27 6.88k | solution stringlengths 2 18.5k | problem_vector listlengths 2.56k 2.56k | solution_vector listlengths 2.56k 2.56k | last_modified stringdate 2025-08-11 00:00:00 2025-08-11 00:00:00 |
|---|---|---|---|---|---|
2,938,164 | $A$ and $B$ are any two subsets of $\{1, 2,...,n - 1\}$ such that $|A| +|B|> n - 1$. Prove that one can find $a$ in $A$ and $b$ in $B$ such that $a + b = n$. | solve by pigeonhole theorem | [
-0.00026232824893668294,
-0.02358950488269329,
0.03291097283363342,
0.00207824376411736,
-0.0012796608498319983,
0.0058540585450828075,
0.022114703431725502,
-0.0013532480224967003,
0.026578526943922043,
-0.008225547149777412,
-0.005503180902451277,
-0.03919283300638199,
-0.00080091523705050... | [
-0.000055420870921807364,
-0.012941347435116768,
0.006195012480020523,
0.0018772119656205177,
-0.00021870696218684316,
-0.01659204065799713,
0.07339799404144287,
-0.02332882396876812,
0.029605599120259285,
-0.000789682671893388,
-0.018469160422682762,
-0.044614680111408234,
-0.00106661149766... | 2025-08-11 |
2,938,166 | "$n$ and $d$ are positive integers such that $d$ divides $2n^2$. Prove that $n^2 + d$ cannot be a sq(...TRUNCATED) | Solved [url=https://artofproblemsolving.com/community/c6h3217220p29434468]here[/url] | [-0.0002518120454624295,0.003980560693889856,-0.00779159227386117,-0.0074425749480724335,-0.00115161(...TRUNCATED) | [-0.0003775336663238704,0.009270886890590191,0.025100916624069214,0.002040078165009618,-0.0014948676(...TRUNCATED) | 2025-08-11 |
2,938,175 | "Prove that if the two angles on the base of a trapezoid are different, then the diagonal starting f(...TRUNCATED) | "Let $AD = x, AA' = DD' = h$ such that $AA', DD' \\perp BC$. If $\\angle{BCD} = \\alpha, \\angle{CB(...TRUNCATED) | [-0.0003102815826423466,-0.006979442201554775,0.06440596282482147,-0.050994422286748886,-0.000710641(...TRUNCATED) | [0.00013255559315439314,-0.05150195211172104,0.058058962225914,-0.028362635523080826,0.0009820505511(...TRUNCATED) | 2025-08-11 |
2,938,197 | "Show that if $m$ and $n$ are integers such that $m^2 + mn + n^2$ is divisible by $9$, then they mus(...TRUNCATED) | See [url]https://artofproblemsolving.com/community/c6t29718f6h3195541_divisibility_property[/url] | [-0.00010285666212439537,-0.009017234668135643,0.015358276665210724,-0.014029296115040779,-0.0002136(...TRUNCATED) | [-0.0003170346317347139,-0.007638048380613327,0.008222043514251709,-0.015985233709216118,-0.00135052(...TRUNCATED) | 2025-08-11 |
2,939,024 | What integers $a, b, c$ satisfy $a^2 + b^2 + c^2 + 3 < ab + 3b + 2c$ ? | "First we deal with the five trivial cases $a=-2,-1,0,1,2$.\nIf $a=-2$ the given inequality rearrang(...TRUNCATED) | [-0.000011111567800980993,-0.004334710072726011,0.024420922622084618,-0.009970844723284245,-0.000178(...TRUNCATED) | [-0.00011053955677198246,0.01072610542178154,0.020996658131480217,0.0035634201485663652,-0.000594090(...TRUNCATED) | 2025-08-11 |
2,939,043 | Show that if $2 + 2\sqrt{28n^2 + 1}$ is an integer, then it is a square (for $n$ an integer). | solved [url=https://artofproblemsolving.com/community/c6h114176p648653]here[/url] | [-0.0001686343312030658,0.004293898120522499,0.01033051498234272,0.012641001492738724,-0.00056530424(...TRUNCATED) | [-0.0003416523686610162,-0.02177795022726059,0.033287614583969116,0.027159616351127625,-0.0014778386(...TRUNCATED) | 2025-08-11 |
2,941,134 | "A straight line cuts the side $AB$ of the triangle $ABC$ at $C_1$, the side $AC$ at $B_1$ and the l(...TRUNCATED) | posted for the image links | [0.0001099009532481432,-0.023852979764342308,0.035420022904872894,0.022881925106048584,0.00087813421(...TRUNCATED) | [-0.00003919271330232732,-0.014801980927586555,0.019121192395687103,0.01361827738583088,-0.000151065(...TRUNCATED) | 2025-08-11 |
2,941,117 | "For what positive integers $n, k$ (with $k < n$) are the binomial coefficients $${n \\choose k- 1}(...TRUNCATED) | "Let us begin by expanding each of the above combinatorial terms according to:\n\n$\\frac{n!}{(k - 1(...TRUNCATED) | [-0.00005293775029713288,-0.018918851390480995,0.03696373105049133,0.01840592734515667,5.81827725909(...TRUNCATED) | [0.000039606267819181085,0.001005828962661326,0.01636846363544464,0.015635494142770767,0.00035873672(...TRUNCATED) | 2025-08-11 |
2,941,142 | "A triangle has inradius $r$ and circumradius $R$. Its longest altitude has length $H$. Show that if(...TRUNCATED) | solved [url=https://artofproblemsolving.com/community/c6h294413p1593465]here[/url] | [-0.00013179991219658405,-0.008523539640009403,0.06965568661689758,0.044649574905633926,-0.000744265(...TRUNCATED) | [-0.00020531556219793856,-0.008829857222735882,0.030983300879597664,0.012966508977115154,-0.00061552(...TRUNCATED) | 2025-08-11 |
2,941,144 | "$f$ is a real-valued function defined on the reals such that $f(x) \\le x$ and $f(x + y) \\le f(x) (...TRUNCATED) | "solved [url=https://artofproblemsolving.com/community/c6h1109982p5052117]here[/url] and [url=https:(...TRUNCATED) | [-0.0002978646953124553,0.008942551910877228,0.02468571625649929,0.020823346450924873,-0.00186711072(...TRUNCATED) | [-0.00036636783624999225,-0.0018784776329994202,0.03559751436114311,0.021788276731967926,-0.00184733(...TRUNCATED) | 2025-08-11 |
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