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
$$\sqrt{a^2+Cac+c^2}+\sqrt{b^2+Cbd+d^2}<\sqrt{a^2+Cad+d^2... | [
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1.4798... | [
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1.3052002191... | 2025-08-13 |
2,759,385 | "Find all triplets of positive integers $(x, y, z)$ such that $x^2 + y^2 + x + y + z = xyz + 1$.\n\n(...TRUNCATED) | "[hide=\"Solution (using Vieta's Jumping Root Method)\"]\n$\\wedge$ means 'and'.\n$\\mathbb{N*}$ mea(...TRUNCATED) | [1.5163580179214478,-0.8464680910110474,2.840644121170044,-0.5195057988166809,-4.691343307495117,2.3(...TRUNCATED) | [2.2642714977264404,-1.6876134872436523,2.9104230403900146,1.4544355869293213,-3.2917864322662354,-3(...TRUNCATED) | 2025-08-13 |
2,759,387 | "Find all positive integers $n$ such that the set $S=\\{1,2,3, \\dots 2n\\}$ can be divided into $2$(...TRUNCATED) | "We claim the answer is all $n \\not\\equiv 5 \\pmod 6$. Let $\\sum_{i \\in S_1} i=A$ and $\\sum_{i (...TRUNCATED) | [0.19847874343395233,0.33447733521461487,3.6585869789123535,0.5112766623497009,-2.2067501544952393,0(...TRUNCATED) | [1.8218730688095093,-0.9144459366798401,3.8882343769073486,0.20985819399356842,-1.307905673980713,-3(...TRUNCATED) | 2025-08-13 |
2,759,392 | "We say that a positive integer $n$ is [i]memorable[/i] if it has a binary representation with stric(...TRUNCATED) | $n^2=2^k \cdot a_k + ... + 2^1 \cdot a_1 + 2^0 a_0$
Next number$$\boxed {(2^{k+2} + 1)n}$$ | [3.113323926925659,0.22853334248065948,-0.8531686067581177,-0.33961135149002075,1.084518551826477,-1(...TRUNCATED) | [-0.08653967827558517,-4.138226509094238,0.4340161979198456,5.275760173797607,0.1228422299027443,3.5(...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) | [1.515586256980896,-0.7212414741516113,0.2294681966304779,2.379037380218506,7.056363105773926,-3.701(...TRUNCATED) | [1.1114997863769531,-2.816742420196533,2.611332654953003,1.5169991254806519,1.1030797958374023,-0.28(...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) | [0.7487195134162903,-2.083986520767212,5.544702053070068,0.18988420069217682,-1.4947723150253296,-4.(...TRUNCATED) | [1.2954787015914917,-3.276646614074707,4.920186519622803,0.6686939597129822,0.19753865897655487,-1.8(...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) | [0.9271271228790283,0.3525989353656769,7.038249492645264,0.9833080768585205,0.4143315553665161,-2.78(...TRUNCATED) | [-0.04525904357433319,2.9520444869995117,7.694033145904541,1.4301867485046387,-2.054389238357544,-1.(...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) | [-1.9107069969177246,-0.7070751786231995,5.653172492980957,2.2436838150024414,0.8559641242027283,-0.(...TRUNCATED) | [0.5639886260032654,0.9873212575912476,5.394408226013184,2.8923983573913574,-4.825655460357666,-2.62(...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) | [3.206362009048462,2.08489990234375,1.23214852809906,-1.3026267290115356,0.6776032447814941,-0.49237(...TRUNCATED) | [2.492079496383667,-0.06130904331803322,-0.6913279294967651,-2.3991751670837402,0.22098447382450104,(...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) | [0.09749334305524826,2.634800672531128,6.977302074432373,2.7685608863830566,-2.30715012550354,-5.300(...TRUNCATED) | [0.9937731027603149,2.56622052192688,6.993828296661377,1.7658838033676147,-2.7875819206237793,-5.082(...TRUNCATED) | 2025-08-13 |
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