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 2025-08-13 00:00:00 2025-08-13 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
$$\sqrt{a^2+Cac+c^2}+\sqrt{b^2+Cbd+d^2}<\sqrt{a^2+Cad+d^2... | [
1.5864040851593018,
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0.574711799... | [
2.725477695465088,
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3.06655216217041,
0.7593258023262024,
1.9007607698440552,
2.6914403438568115,
0.42406603693... | 2025-08-13 |
2,759,385 | "Find all triplets of positive integers $(x, y, z)$ such that $x^2 + y^2 + x + y + z = xyz + 1$.\n\n(...TRUNCATED) | "[hide=\"Solution (using Vieta's Jumping Root Method)\"]\n$\\wedge$ means 'and'.\n$\\mathbb{N*}$ mea(...TRUNCATED) | [4.764293193817139,-4.475564956665039,3.3570284843444824,-0.9703601598739624,-2.3251242637634277,1.7(...TRUNCATED) | [7.970137119293213,-4.192419052124023,4.296323299407959,-1.237061858177185,-2.0492563247680664,-2.94(...TRUNCATED) | 2025-08-13 |
2,759,387 | "Find all positive integers $n$ such that the set $S=\\{1,2,3, \\dots 2n\\}$ can be divided into $2$(...TRUNCATED) | "We claim the answer is all $n \\not\\equiv 5 \\pmod 6$. Let $\\sum_{i \\in S_1} i=A$ and $\\sum_{i (...TRUNCATED) | [2.111485481262207,-0.49331340193748474,4.633695602416992,-3.5197033882141113,-1.8725008964538574,-1(...TRUNCATED) | [2.9286842346191406,-2.3048110008239746,6.155924320220947,-1.4521645307540894,-1.3363571166992188,-5(...TRUNCATED) | 2025-08-13 |
2,759,392 | "We say that a positive integer $n$ is [i]memorable[/i] if it has a binary representation with stric(...TRUNCATED) | $n^2=2^k \cdot a_k + ... + 2^1 \cdot a_1 + 2^0 a_0$
Next number$$\boxed {(2^{k+2} + 1)n}$$ | [2.3853397369384766,0.34974735975265503,-1.0572028160095215,-1.1914173364639282,3.8546717166900635,-(...TRUNCATED) | [2.046057939529419,-2.289414882659912,4.638077259063721,5.5990376472473145,4.893801212310791,0.04615(...TRUNCATED) | 2025-08-13 |
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) | [3.7872605323791504,-5.148733615875244,-2.1199958324432373,0.5472174286842346,3.7204790115356445,-7.(...TRUNCATED) | [3.691632032394409,-4.398102760314941,3.1211116313934326,0.7098188400268555,3.0458157062530518,-2.55(...TRUNCATED) | 2025-08-13 |
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) | [0.9708786010742188,-2.83416748046875,1.3498958349227905,-2.8463776111602783,-4.023007869720459,-9.4(...TRUNCATED) | [5.26491641998291,-0.8277864456176758,5.766354560852051,2.516669273376465,0.11046724021434784,-5.467(...TRUNCATED) | 2025-08-13 |
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) | [5.069219589233398,1.4736937284469604,6.843637466430664,0.27406585216522217,4.268244743347168,-4.340(...TRUNCATED) | [3.066310167312622,2.9179937839508057,6.50152063369751,1.766244649887085,1.4445717334747314,-1.28826(...TRUNCATED) | 2025-08-13 |
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) | [0.19948633015155792,-5.15372896194458,5.0688395500183105,1.3277244567871094,-1.0633233785629272,-3.(...TRUNCATED) | [5.980508804321289,-2.749767303466797,6.402554988861084,0.29425710439682007,-4.244509696960449,-5.30(...TRUNCATED) | 2025-08-13 |
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) | [1.3041223287582397,-1.2887600660324097,-1.4358494281768799,-3.1086313724517822,1.0523375272750854,-(...TRUNCATED) | [0.5712248086929321,-2.4529929161071777,-4.623786926269531,-4.6245880126953125,0.7670878767967224,-8(...TRUNCATED) | 2025-08-13 |
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) | [-0.39755910634994507,0.42208632826805115,6.556204319000244,1.8377008438110352,-3.9873085021972656,-(...TRUNCATED) | [2.8459553718566895,3.111429452896118,7.078856468200684,3.5853254795074463,0.16762378811836243,-8.32(...TRUNCATED) | 2025-08-13 |
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