id int64 1.61k 3.1k | subfield stringclasses 4
values | context null | instruction_seed stringlengths 49 3.39k | solution stringlengths 48 11.9k | final_answer sequencelengths 1 1 | is_multiple_answer bool 2
classes | unit stringclasses 10
values | answer_type stringclasses 6
values | error stringclasses 1
value | original_solution sequencelengths 2 6 | embedding sequencelengths 3.07k 3.07k |
|---|---|---|---|---|---|---|---|---|---|---|---|
1,610 | Algebra | null | Given a positive integer $n$, determine the largest real number $\mu$ satisfying the following condition: for every $4 n$-point configuration $C$ in an open unit square $U$, there exists an open rectangle in $U$, whose sides are parallel to those of $U$, which contains exactly one point of $C$, and has an area greater ... | The required maximum is $\frac{1}{2 n+2}$. To show that the condition in the statement is not met if $\mu>\frac{1}{2 n+2}$, let $U=(0,1) \times(0,1)$, choose a small enough positive $\epsilon$, and consider the configuration $C$ consisting of the $n$ four-element clusters of points $\left(\frac{i}{n+1} \pm \epsilon\rig... | [
"$\\frac{1}{2 n+2}$"
] | false | null | Expression | null | [
"The required maximum is $\\frac{1}{2 n+2}$. To show that the condition in the statement is not met if $\\mu>\\frac{1}{2 n+2}$, let $U=(0,1) \\times(0,1)$, choose a small enough positive $\\epsilon$, and consider the configuration $C$ consisting of the $n$ four-element clusters of points $\\left(\\frac{i}{n+1} \\pm... | [
0.048168737441301346,
0.007420479319989681,
-0.008511526510119438,
0.002534483326599002,
0.04098544269800186,
-0.009616128169000149,
-0.01825641095638275,
0.07898914068937302,
-0.015762588009238243,
-0.011994747444987297,
-0.03437138721346855,
-0.014990045689046383,
0.03328711912035942,
-0... |
1,612 | Number Theory | null |
Find (in closed form) the difference between the number of positive integers at most $2^{2017}$ with even weight and the number of positive integers at most $2^{2017}$ with odd weight. | For every integer $M \geq 0$, let $A_{M}=\sum_{n=-2^{M}+1}^{0}(-1)^{w(n)}$ and let $B_{M}=$ $\sum_{n=1}^{2^{M}}(-1)^{w(n)}$; thus, $B_{M}$ evaluates the difference of the number of even weight integers in the range 1 through $2^{M}$ and the number of odd weight integers in that range.
Notice that
$$
w(n)= \begin... | [
"$2^{1009}$"
] | false | null | Numerical | null | [
"For every integer $M \\geq 0$, let $A_{M}=\\sum_{n=-2^{M}+1}^{0}(-1)^{w(n)}$ and let $B_{M}=$ $\\sum_{n=1}^{2^{M}}(-1)^{w(n)}$; thus, $B_{M}$ evaluates the difference of the number of even weight integers in the range 1 through $2^{M}$ and the number of odd weight integers in that range.\n\n\n\nNotice that\n\n\n\n... | [
-0.030240066349506378,
0.022731304168701172,
-0.010744193568825722,
-0.009027172811329365,
0.020527366548776627,
-0.0062850648537278175,
0.04764091968536377,
0.053304523229599,
-0.013838674873113632,
0.00775862718001008,
0.0030175999272614717,
-0.0021991319954395294,
-0.0008264763746410608,
... |
1,613 | Algebra | null | "Determine all positive integers $n$ satisfying the following condition: for every monic polynomial (...TRUNCATED) | "There is only one such integer, namely, $n=2$. In this case, if $P$ is a constant polynomial, the r(...TRUNCATED) | [
"2"
] | false | null | Numerical | null | ["There is only one such integer, namely, $n=2$. In this case, if $P$ is a constant polynomial, the (...TRUNCATED) | [0.0005820162477903068,-0.0020209711510688066,-0.014775608666241169,0.004828687757253647,0.016822902(...TRUNCATED) |
1,614 | Combinatorics | null | "Let $n$ be an integer greater than 1 and let $X$ be an $n$-element set. A non-empty collection of s(...TRUNCATED) | "The required maximum is $2 n-2$. To describe a $(2 n-2)$-element collection satisfying the required(...TRUNCATED) | [
"$2n-2$"
] | false | null | Expression | null | ["The required maximum is $2 n-2$. To describe a $(2 n-2)$-element collection satisfying the require(...TRUNCATED) | [0.004021350294351578,0.02398652769625187,-0.012074163183569908,0.008770790882408619,0.0570067688822(...TRUNCATED) |
1,618 | Number Theory | null | "Determine all prime numbers $p$ and all positive integers $x$ and $y$ satisfying $x^{3}+y^{3}=$ $p((...TRUNCATED) | "Up to a swap of the first two entries, the only solutions are $(x, y, p)=(1,8,19)$, $(x, y, p)=(2,7(...TRUNCATED) | [
"$(1,8,19), (2,7,13), (4,5,7)$"
] | true | null | Tuple | null | ["Up to a swap of the first two entries, the only solutions are $(x, y, p)=(1,8,19)$, $(x, y, p)=(2,(...TRUNCATED) | [-0.007929147221148014,0.04637183994054794,-0.02168758399784565,0.016284827142953873,-0.004426651168(...TRUNCATED) |
1,620 | Algebra | null | "Let $n \\geqslant 2$ be an integer, and let $f$ be a $4 n$-variable polynomial with real coefficien(...TRUNCATED) | "The smallest possible degree is $2 n$. In what follows, we will frequently write $A_{i}=$ $\\left(x(...TRUNCATED) | [
"$2n$"
] | false | null | Expression | null | ["The smallest possible degree is $2 n$. In what follows, we will frequently write $A_{i}=$ $\\left((...TRUNCATED) | [-0.006142050959169865,-0.0018429165938869119,-0.013766873627901077,0.021578548476099968,0.038431517(...TRUNCATED) |
1,631 | Algebra | null | "For a positive integer $a$, define a sequence of integers $x_{1}, x_{2}, \\ldots$ by letting $x_{1}(...TRUNCATED) | "The largest such is $k=2$. Notice first that if $y_{i}$ is prime, then $x_{i}$ is prime as well. Ac(...TRUNCATED) | [
"2"
] | false | null | Numerical | null | ["The largest such is $k=2$. Notice first that if $y_{i}$ is prime, then $x_{i}$ is prime as well. A(...TRUNCATED) | [0.003750513307750225,0.028965672478079796,-0.007666734047234058,-0.03210306912660599,0.011577431112(...TRUNCATED) |
1,641 | Combinatorics | null | "Let $n$ be a positive integer and fix $2 n$ distinct points on a circumference. Split these points (...TRUNCATED) | "The required number is $\\left(\\begin{array}{c}2 n \\\\ n\\end{array}\\right)$. To prove this, tra(...TRUNCATED) | [
"$\\binom{2n}{n}$"
] | false | null | Expression | null | ["The required number is $\\left(\\begin{array}{c}2 n \\\\ n\\end{array}\\right)$. To prove this, tr(...TRUNCATED) | [-0.003075103973969817,0.007351972628384829,-0.01785295456647873,0.007621887605637312,0.042389519512(...TRUNCATED) |
1,644 | Combinatorics | null | "Given positive integers $m$ and $n \\geq m$, determine the largest number of dominoes $(1 \\times 2(...TRUNCATED) | "The required maximum is $m n-\\lfloor m / 2\\rfloor$ and is achieved by the brick-like vertically s(...TRUNCATED) | [
"$m n-\\lfloor m / 2\\rfloor$"
] | false | null | Expression | null | ["The required maximum is $m n-\\lfloor m / 2\\rfloor$ and is achieved by the brick-like vertically (...TRUNCATED) | [0.0007347069913521409,-0.015164145268499851,-0.017298970371484756,0.0168306864798069,0.042751595377(...TRUNCATED) |
1,645 | Algebra | null | "A cubic sequence is a sequence of integers given by $a_{n}=n^{3}+b n^{2}+c n+d$, where $b, c$ and $(...TRUNCATED) | "The only possible value of $a_{2015} \\cdot a_{2016}$ is 0 . For simplicity, by performing a transl(...TRUNCATED) | [
"0"
] | false | null | Numerical | null | ["The only possible value of $a_{2015} \\cdot a_{2016}$ is 0 . For simplicity, by performing a trans(...TRUNCATED) | [-0.017904028296470642,-0.005215521436184645,-0.015464928932487965,0.024767238646745682,0.0207972154(...TRUNCATED) |
End of preview. Expand in Data Studio
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