id int64 2 3.28M | problem stringlengths 30 5.17k | solution stringlengths 8 20.3k | problem_vector listlengths 1.02k 1.02k | solution_vector listlengths 1.02k 1.02k | last_modified stringdate 2025-08-11 00:00:00 2025-08-11 00:00:00 |
|---|---|---|---|---|---|
57,271 | Let $k$ be a positive integer and $M_k$ the set of all the integers that are between $2 \cdot k^2 + k$ and $2 \cdot k^2 + 3 \cdot k,$ both included. Is it possible to partition $M_k$ into 2 subsets $A$ and $B$ such that
\[ \sum_{x \in A} x^2 = \sum_{x \in B} x^2. \] | The answer is positive. Take \[ A = \{2k^2 + k,\ 2k^2 + k + 1,\ \dots,\ 2k^2 + 2k\} \] and \[ B = \{2k^2 + 2k + 1,\ 2k^2 + 2k + 2,\ \dots,\ 2k^2 + 3k\}. \]
Then \[ \sum_{x \in A}x^2 = \sum_{x \in B}x^2 = \frac{k}{6}\left(24k^4 + 60k^3 + 50k^2 + 15k + 1\right). \] | [
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57,272 | An $ n \times n, n \geq 2$ chessboard is numbered by the numbers $ 1, 2, \ldots, n^2$ (and every number occurs). Prove that there exist two neighbouring (with common edge) squares such that their numbers differ by at least $ n.$ | Here's an alternate way to use the idea in love_sc1's solution, where it suffices to partition the board into two consecutive intervals of numbers with at least $n$ borders. Let $S_k$ be the set of squares containing a number at most $k$. We consider the maximal $k$ such that $S_k$ contains neither a full row or a full... | [
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57,273 | Let $ n$ be an even positive integer. Let $ A_1, A_2, \ldots, A_{n \plus{} 1}$ be sets having $ n$ elements each such that any two of them have exactly one element in common while every element of their union belongs to at least two of the given sets. For which $ n$ can one assign to every element of the union one of t... | First of all, the incidence structure is determined uniquely by the given conditions: every element of $A=\bigcup A_i$ belongs to [i]precisely[/i] two $A_i$'s, and $|A|=\frac{n(n+1)}2$. We can see this as follows:
Suppose we can find $x\in A_1\cap A_2\cap A_3$. Then $A_1\cup A_2\cup A_3$ has, except for $x$, at leas... | [
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57,274 | In a given tedrahedron $ ABCD$ let $ K$ and $ L$ be the centres of edges $ AB$ and $ CD$ respectively. Prove that every plane that contains the line $ KL$ divides the tedrahedron into two parts of equal volume. | Let $p$ be the plane through $AB$ parallel to $CD$. Any plane $p'$ parallel to $p$ which intersects the segment $KL$ will intersect the tetrahedron in a parallelogram with center on the line $KL$. Any plane $q$ through $KL$ will pass through the center of the parallelogram and hence divide it into two equal parts. So s... | [
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57,276 | "Let $ a$ be the greatest positive root of the equation $ x^3 \\minus{} 3 \\cdot x^2 \\plus{} 1 \\eq(...TRUNCATED) | "Here's a sketch of an answer. Let $a > b > c$ be the three real roots of $f(x) = x^3 - 3x^2 + 1$ a(...TRUNCATED) | [-0.005721735768020153,-0.023692406713962555,-0.002080986276268959,0.026376424357295036,0.0082322834(...TRUNCATED) | [-0.009241919033229351,-0.021124469116330147,-0.0035529611632227898,0.02683972381055355,-0.007079737(...TRUNCATED) | 2025-08-11 |
57,277 | "If $a_0$ is a positive real number, consider the sequence $\\{a_n\\}$ defined by:\r\n\r\n\\[ a_{n+1(...TRUNCATED) | "I claim $a=2$. We need to show that conditions (i) and (ii) are satisfied. \n\n[b]Condition (i).[/b(...TRUNCATED) | [0.022382721304893494,-0.00932216551154852,0.0017708478262647986,0.0176790002733469,-0.0168904587626(...TRUNCATED) | [0.005641154479235411,0.014216535724699497,-0.0007001379271969199,0.043039772659540176,-0.0421609766(...TRUNCATED) | 2025-08-11 |
57,278 | "Let $ u_1, u_2, \\ldots, u_m$ be $ m$ vectors in the plane, each of length $ \\leq 1,$ with zero su(...TRUNCATED) | "I think post #5 is missing some detail.\n\n[hide =Solution]\n\nWLOG suppose that $u_1 + u_2 + \\cdo(...TRUNCATED) | [0.051676906645298004,-0.0011020437814295292,-0.006477417424321175,0.03320226073265076,0.01035994756(...TRUNCATED) | [0.02195059135556221,-0.011205174960196018,-0.0034953956492245197,0.025424722582101822,0.01784763485(...TRUNCATED) | 2025-08-11 |
57,279 | "Show that there do not exist more than $27$ half-lines (or rays) emanating from the origin in the $(...TRUNCATED) | "The area of the whole unit sphere is less than $26.28$ times the area of a spherical cap of (spheri(...TRUNCATED) | [-0.017196130007505417,-0.01568634994328022,-0.007116302847862244,0.058195408433675766,0.05397160723(...TRUNCATED) | [0.011304437182843685,-0.044200289994478226,-0.008504729717969894,0.06981875002384186,0.048961095511(...TRUNCATED) | 2025-08-11 |
57,284 | "Let $T$ be a triangle with inscribed circle $C.$ A square with sides of length $a$ is circumscribed(...TRUNCATED) | "This isn't true when the square shares a side with $T$, verify it with an equilateral triangle. Alt(...TRUNCATED) | [0.026185594499111176,0.02002521976828575,-0.001863322569988668,0.030465321615338326,0.0338117182254(...TRUNCATED) | [0.0413215272128582,0.008137323893606663,0.0027474169619381428,0.057471245527267456,0.01918337866663(...TRUNCATED) | 2025-08-11 |
57,285 | "Let $1 \\leq k \\leq n.$ Consider all finite sequences of positive integers with sum $n.$ Find $T(n(...TRUNCATED) | Bump bump ! | [-0.0008027150179259479,-0.04909360781311989,-0.004574501421302557,-0.012738458812236786,-0.03427037(...TRUNCATED) | [-0.004749286454170942,-0.0357050821185112,-0.014455660246312618,-0.000979508156888187,0.05963527411(...TRUNCATED) | 2025-08-11 |
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