Datasets:

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ksuinbragimova

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aops_dataset_lgai

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"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)

"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)

"We say that a positive integer $n$ is [i]memorable[/i] if it has a binary representation with stric(...TRUNCATED)

"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)

"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)

"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)

"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)

"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)

"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)