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
$$\sqrt{a^2+Cac+c^2}+\sqrt{b^2+Cbd+d^2}<\sqrt{a^2+Cad+d^2... | [
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-0.5973... | [
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2.4122843742370605,
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1.3221749067306519,
-1.9... | 2025-08-13 |
2,759,385 | "Find all triplets of positive integers $(x, y, z)$ such that $x^2 + y^2 + x + y + z = xyz + 1$.\n\n(...TRUNCATED) | "[hide=\"Solution (using Vieta's Jumping Root Method)\"]\n$\\wedge$ means 'and'.\n$\\mathbb{N*}$ mea(...TRUNCATED) | [2.247650146484375,-2.404554843902588,2.9078915119171143,-0.368393212556839,-0.19213421642780304,3.0(...TRUNCATED) | [3.5803115367889404,-1.786057949066162,3.064851760864258,-0.7349568009376526,-0.020212676376104355,0(...TRUNCATED) | 2025-08-13 |
2,759,387 | "Find all positive integers $n$ such that the set $S=\\{1,2,3, \\dots 2n\\}$ can be divided into $2$(...TRUNCATED) | "We claim the answer is all $n \\not\\equiv 5 \\pmod 6$. Let $\\sum_{i \\in S_1} i=A$ and $\\sum_{i (...TRUNCATED) | [0.8473052978515625,-0.6242504715919495,3.025387763977051,-1.6584482192993164,-0.36547398567199707,0(...TRUNCATED) | [1.4945396184921265,-1.503811240196228,2.8866512775421143,-0.8992555737495422,-0.4290386736392975,-0(...TRUNCATED) | 2025-08-13 |
2,759,392 | "We say that a positive integer $n$ is [i]memorable[/i] if it has a binary representation with stric(...TRUNCATED) | $n^2=2^k \cdot a_k + ... + 2^1 \cdot a_1 + 2^0 a_0$
Next number$$\boxed {(2^{k+2} + 1)n}$$ | [1.1597073078155518,-0.3185015320777893,0.4900343418121338,0.1601572036743164,2.1969027519226074,-0.(...TRUNCATED) | [1.1040416955947876,-0.3887858986854553,3.074641704559326,2.9836463928222656,3.067077159881592,2.036(...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) | [1.5185445547103882,-1.5460915565490723,-0.5258828997612,0.44297856092453003,2.7116048336029053,-2.0(...TRUNCATED) | [1.6797196865081787,-1.657285451889038,2.239624500274658,-0.21988515555858612,2.132533550262451,0.14(...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) | [-0.43257954716682434,-1.502730131149292,1.673073410987854,-1.4519996643066406,-0.979162335395813,-1(...TRUNCATED) | [2.366600751876831,-0.834926187992096,3.2803986072540283,1.142236590385437,1.461019515991211,-0.5229(...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) | [2.452089309692383,0.7593040466308594,4.593801975250244,0.062079135328531265,2.0502712726593018,-0.6(...TRUNCATED) | [1.3848474025726318,1.2545137405395508,3.9389567375183105,0.6719993948936462,1.1515780687332153,0.85(...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) | [-0.7207275032997131,-2.0369503498077393,2.6196796894073486,0.3759940564632416,0.6756381392478943,-0(...TRUNCATED) | [2.7822563648223877,0.19562554359436035,3.413719892501831,0.09681824594736099,-1.3589121103286743,-0(...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) | [-0.3696599304676056,-0.7566507458686829,-0.20475082099437714,-1.49534273147583,1.5492558479309082,-(...TRUNCATED) | [0.08449532836675644,-1.2525818347930908,-1.9136296510696411,-2.072153329849243,1.0837246179580688,-(...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) | [-0.3884572386741638,0.3098502457141876,3.8736438751220703,1.4167803525924683,-1.476656436920166,-0.(...TRUNCATED) | [1.2754790782928467,1.738067388534546,3.859797239303589,1.5618445873260498,0.9746736884117126,-1.708(...TRUNCATED) | 2025-08-13 |
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