conversations listlengths 2 2 |
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[
{
"from": "human",
"value": "Let $a$ be the smallest positive integer congruent to $3$ (mod $7$) that can be expressed as the sum of two squares. Let $b$ be the smallest positive integer congruent to $4$ (mod $7$) that can be expressed as the difference of two cubes. What is $b-a$?"
},
{
"from": "gp... |
[
{
"from": "human",
"value": "There is a string of length n+1 of characters 'A' and 'B'. The first character and last character of the string are equal to 'A'.\n\nYou are given m indices p_1, p_2, …, p_m (0-indexation) denoting the other indices of characters 'A' in the string.\n\nLet's denote the minimum di... |
[
{
"from": "human",
"value": "A quadrilateral $ABCD$ is drawn such that $AB=9$, $BC=8$, $CD=7$, and $DA=6$. The diagonals $AC$ and $BD$ intersect at a point $O$. Determine the length of $AO$ if $AO$ is perpendicular to $BD$."
},
{
"from": "gpt",
"value": "<think> Okay, let's try to tackle this ge... |
[{"from":"human","value":"In the coordinate plane, consider points $D = (0, 0)$, $E = (8, 0)$, and $(...TRUNCATED) |
[{"from":"human","value":"Consider the polynomial $x^3+px^2+q=0$ with three distinct real roots $r$,(...TRUNCATED) |
[{"from":"human","value":"Let $P$ be the center of a circle inscribed in a square $ABCD$ with side l(...TRUNCATED) |
[{"from":"human","value":"A bag contains an infinite number of red, blue, green, and white marbles. (...TRUNCATED) |
[{"from":"human","value":"In the equation\n\\[2y^3-6xy^2+6x^2y-x^3-2=0,\\]\nthe roots for $y$ are di(...TRUNCATED) |
[{"from":"human","value":"A square pyramid with a square base $ABCD$ and vertex $E$ has eight edges (...TRUNCATED) |
[{"from":"human","value":"Two circles of radius 5 and 3 respectively are centred at (-3,0) and (4,0)(...TRUNCATED) |
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