id int64 2 3.28M | problem stringlengths 27 6.88k | solution stringlengths 2 38.6k | problem_vector listlengths 4.1k 4.1k | solution_vector listlengths 4.1k 4.1k | last_modified stringdate 2025-08-14 00:00:00 2025-08-14 00:00:00 |
|---|---|---|---|---|---|
2,759,383 | Given an integer $n\geq2$, let $x_1<x_2<\cdots<x_n$ and $y_1<y_2<\cdots<y_n$ be positive reals. Prove that for every value $C\in (-2,2)$ (by taking $y_{n+1}=y_1$) it holds that
$\hspace{122px}\sum_{i=1}^{n}\sqrt{x_i^2+Cx_iy_i+y_i^2}<\sum_{i=1}^{n}\sqrt{x_i^2+Cx_iy_{i+1}+y_{i+1}^2}$.
[i]Proposed by Mirko Petrusevski[/... | We can use a similar argument as in the proof of rearrangement inequality. Letting $f(x,y)=\sqrt{x^2+Cxy+y^2}$, it suffices to show the case $n=2$, which corresponds to a single transposition in the general case.
Al we have to show that if $a<b$ and $c<d$, then
$$\sqrt{a^2+Cac+c^2}+\sqrt{b^2+Cbd+d^2}<\sqrt{a^2+Cad+d^2... | [
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-1.2919... | [
2.584754467010498,
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0.2750685214996338,
-1.1016607284545898,
0.6838734745979309,
-0.4... | 2025-08-14 |
2,759,385 | "Find all triplets of positive integers $(x, y, z)$ such that $x^2 + y^2 + x + y + z = xyz + 1$.\n\n(...TRUNCATED) | "[hide=\"Solution (using Vieta's Jumping Root Method)\"]\n$\\wedge$ means 'and'.\n$\\mathbb{N*}$ mea(...TRUNCATED) | [3.8406872749328613,2.6366140842437744,1.1908632516860962,2.6150217056274414,-2.004608631134033,-0.7(...TRUNCATED) | [3.421250343322754,3.5935146808624268,1.4051666259765625,1.9070409536361694,-1.1022286415100098,-3.4(...TRUNCATED) | 2025-08-14 |
2,759,387 | "Find all positive integers $n$ such that the set $S=\\{1,2,3, \\dots 2n\\}$ can be divided into $2$(...TRUNCATED) | "We claim the answer is all $n \\not\\equiv 5 \\pmod 6$. Let $\\sum_{i \\in S_1} i=A$ and $\\sum_{i (...TRUNCATED) | [2.8764142990112305,5.541140556335449,0.6559031009674072,2.227391004562378,-1.2833075523376465,-0.45(...TRUNCATED) | [1.6946182250976562,5.884676456451416,1.2068837881088257,1.3908380270004272,-1.1831142902374268,-2.6(...TRUNCATED) | 2025-08-14 |
2,759,392 | "We say that a positive integer $n$ is [i]memorable[/i] if it has a binary representation with stric(...TRUNCATED) | $n^2=2^k \cdot a_k + ... + 2^1 \cdot a_1 + 2^0 a_0$
Next number$$\boxed {(2^{k+2} + 1)n}$$ | [3.096621036529541,5.191338539123535,0.7049055099487305,0.2388538122177124,-1.288427710533142,-1.773(...TRUNCATED) | [2.9405951499938965,3.800574541091919,1.7823249101638794,2.7608306407928467,0.6639795303344727,-0.39(...TRUNCATED) | 2025-08-14 |
2,777,211 | "Quadratic polynomials $P(x)$ and $Q(x)$ have leading coefficients of $2$ and $-2$, respectively. Th(...TRUNCATED) | "Mine.\n\n[hide=\"Solution\"]The polynomial $R(x) := P(x) + Q(x)$ is linear, and we may compute $R(1(...TRUNCATED) | [2.173774242401123,2.1772871017456055,-0.4245903491973877,2.934870958328247,1.2993478775024414,-2.36(...TRUNCATED) | [2.359318733215332,2.7608158588409424,0.6290521621704102,3.003706693649292,-0.23152370750904083,-1.6(...TRUNCATED) | 2025-08-14 |
2,777,219 | "Let $w = \\frac{\\sqrt{3}+i}{2}$ and $z=\\frac{-1+i\\sqrt{3}}{2}$, where $i=\\sqrt{-1}$. Find the n(...TRUNCATED) | "[hide=Sketch]Rewrite in exponential form using $e^{i\\theta}=\\cos{\\theta}+i\\sin{\\theta}$. Take (...TRUNCATED) | [3.13726544380188,3.2649459838867188,1.712881326675415,2.817673921585083,-0.5546501278877258,-3.0627(...TRUNCATED) | [3.4776642322540283,3.4496490955352783,0.9854793548583984,2.2770326137542725,-0.41138237714767456,-1(...TRUNCATED) | 2025-08-14 |
2,777,232 | "Let $ABCD$ be a parallelogram with $\\angle BAD < 90^{\\circ}$. A circle tangent to sides $\\overli(...TRUNCATED) | "Mine.\n\n[hide=\"Solution\"]\nLet $X$, $Y$, and $Z$ denote the tangency points of $\\omega$ with $A(...TRUNCATED) | [1.8989360332489014,3.3031234741210938,0.831558108329773,1.368220567703247,0.40913209319114685,-2.72(...TRUNCATED) | [0.6189562678337097,1.891210913658142,0.8687095642089844,1.5216370820999146,-1.3057236671447754,-2.0(...TRUNCATED) | 2025-08-14 |
2,782,948 | "Twenty distinct points are marked on a circle and labeled $1$ through $20$ in clockwise order. A l(...TRUNCATED) | "Of the three labels $a<b<c$ at the vertices of such a triangle, let $p_1=b-a$ be the distance betwe(...TRUNCATED) | [3.1571245193481445,4.880073547363281,1.363282322883606,2.240176200866699,1.2754417657852173,0.88129(...TRUNCATED) | [3.54367733001709,4.186898708343506,2.0293972492218018,2.112478733062744,-0.0005308897816576064,-2.4(...TRUNCATED) | 2025-08-14 |
2,793,705 | "Ten birds land on a $10$-meter-long wire, each at a random point chosen uniformly along the wire. ((...TRUNCATED) | "The probability of placing the birds on a $10$-meter wire such that they are all more than a meter (...TRUNCATED) | [4.652280807495117,3.279249429702759,-3.889354705810547,-0.05047110468149185,1.9416744709014893,0.05(...TRUNCATED) | [4.103112697601318,2.940695285797119,-3.478959321975708,0.07919123768806458,0.8211680054664612,-0.69(...TRUNCATED) | 2025-08-14 |
2,352,936 | "For a given value $t$, we consider number sequences $a_1, a_2, a_3,...$ such that $a_{n+1} =\\frac{(...TRUNCATED) | "[b](a)[/b] $a_{n+1} =\\frac{a_n +2}{a_n + 1}$\n$\\frac43 \\le \\frac{a_1 +2}{a_1 + 1} \\le \\frac32(...TRUNCATED) | [2.3670594692230225,5.603346347808838,2.4277169704437256,2.1943423748016357,-1.7302054166793823,-2.4(...TRUNCATED) | [2.0393564701080322,7.920342922210693,2.359890937805176,2.1742947101593018,0.29074588418006897,-4.73(...TRUNCATED) | 2025-08-14 |
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