prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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Let \(a, b, c, d\) be rational numbers with \(a > 0\). If for every integer \(n \ge 0\), the number \(an^{3} + bn^{2} + cn + d\) is also an integer, find the minimal value of \(a\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 7 |
The sum of the distances from one vertex of a square with sides of length two to the midpoints of each of the sides of the square is in the form k + m\sqrt{n}. Please find the value of k + m + n. | 9 |
A circular disc with diameter $D$ is placed on an $8 \times 8$ checkerboard with width $D$ so that the centers coincide. Find the number of checkerboard squares which are completely covered by the disc. | 32 |
Vertex \(E\) of equilateral triangle \(ABE\) is in the interior of square \(ABCD\), and \(F\) is the point of intersection of diagonal \(BD\) and line segment \(AE\). If length \(AB\) is \(\sqrt{1+\sqrt{3}}\), then find the area of \(\triangle ABF\). The original answer is in the form \(\frac{\sqrt{k}}{m}\), please giv... | 5 |
How many $10$-digit positive integers can be written by using four $0$s, five $1$s, and one $2$? | 756 |
An uncrossed belt is fitted without slack around two circular pulleys with radii of $14$ inches and $4$ inches. If the distance between the points of contact of the belt with the pulleys is $24$ inches, find the distance between the centers of the pulleys in inches. | 26 |
What is the minimum value of \( \sqrt{x^2 - 4x + 7 - 2\sqrt{2}} + \sqrt{x^2 - 8x + 27 - 6\sqrt{2}} \) where \( x \) is a real number? The original answer is in the form \( k\sqrt{2} \), please find the value of \( k \). | 2 |
The fraction \(\frac{\sqrt{a^2+x^2}-(x^2-a^2)/\sqrt{a^2+x^2}}{a^2+x^2}\) reduces to a simpler form. The reduced expression is in the form \(\frac{k a^2}{(a^2 + x^2)^{\frac{m}{n}}}\). Please find the value of \(k + m + n + p\). | 7 |
The figures $F_1$, $F_2$, $F_3$, and $F_4$ shown are the first in a sequence of figures. For $n \geq 3$, $F_n$ is constructed from $F_{n-1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $F_{n-1}$ had on each side of its outside square. For example, figure $F_3$ has $1... | 761 |
What is the least real number $C$ that satisfies $\sin x \cos x \leq C(\sin^6x+\cos^6x)$ for every real number $x$? | 2 |
In quadrilateral \(ABCD\), \(m\angle B = m\angle C = 120^\circ\), \(AB = 3\), \(BC = 4\), and \(CD = 5\). Find the area of \(ABCD\) in the form \(\frac{k\sqrt{3}}{m}\), where \(k\) and \(m\) are integers. Please provide the value of \(k + m\). | 51 |
A positive integer $n$ not exceeding $100$ is chosen such that if $n \leq 50$, the probability of choosing $n$ is $p$, and if $n > 50$, the probability of choosing $n$ is $3p$. What is the value of the probability that a perfect square is chosen multiplied by 100? | 8 |
A digital display shows the current date as an $8$-digit integer consisting of a $4$-digit year, followed by a $2$-digit month, followed by a $2$-digit date within the month. For example, Arbor Day this year is displayed as 20230428. Find the number of dates in $2023$ for which each digit appears an even number of time... | 9 |
A majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1$. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils ... | 11 |
Let $a, b, c, d, e, f, g,$ and $h$ be distinct elements in the set: \[ \{-7, -5, -3, -2, 2, 4, 6, 13\}. \] Find the minimum possible value of: \[ (a + b + c + d)^2 + (e + f + g + h)^2 \] | 34 |
What is the area of the polygon whose vertices are the points of intersection of the curves $x^2 + y^2 = 25$ and $(x-4)^2 + 9y^2 = 81$? | 27 |
A cryptographer devises the following method for encoding positive integers. First, the integer is expressed in base $5$. Second, a 1-to-1 correspondence is established between the digits that appear in the expressions in base $5$ and the elements of the set $\{V, W, X, Y, Z\}$. Using this correspondence, the cryptogra... | 108 |
Let $ABC$ be a triangle and $AD$ be the bisector of the triangle ($D \in (BC)$). Assume that $AB = 14$ cm, $AC = 35$ cm, and $AD = 12$ cm. What is the area of triangle $ABC$ in cm$^2$? The original answer is in \frac{k}{m} format, please give the value of k + m. | 1181 |
A $4 \times 4$ block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums? | 4 |
In triangle $ABC$ shown in the adjoining figure, $M$ is the midpoint of side $BC$, $AB=12$ and $AC=16$. Points $E$ and $F$ are taken on $AC$ and $AB$, respectively, and lines $EF$ and $AM$ intersect at $G$. If $AE=2AF$, find the value of \(\frac{EG}{GF}\). The original answer is in \(\frac{k}{m}\) format, please give t... | 5 |
Two congruent squares are packed into an isosceles right triangle as shown below. Each of the squares has an area of 10. What is the area of the triangle? ```asy \draw((0,0) -- (3\sqrt{10}, 0) -- (0, 3\sqrt{10}) -- cycle); \draw((0,0) -- (2\sqrt{10}, 0) -- (2\sqrt{10}, \sqrt{10}) -- (0, \sqrt{10})); \draw((\sqrt{10}, \... | 45 |
An engineer said he could finish a highway section in $3$ days with his present supply of a certain type of machine. However, with $3$ more of these machines, the job could be done in $2$ days. If the machines all work at the same rate, how many days would it take to do the job with one machine? | 18 |
There are exactly $N$ distinct rational numbers $k$ such that $|k| < 200$ and \[5x^2 + kx + 12 = 0\] has at least one integer solution for $x$. What is the value of $N$? | 78 |
What is the area of the region defined by the inequality $|3x-18|+|2y+7| \leq 3$? | 3 |
Kyle found the sum of the digits of $2014^{2014}$. Then, Shannon found the sum of the digits of Kyle's result. Finally, James found the sum of the digits of Shannon's result. What number did James find? | 7 |
In how many ways can $7$ red and $7$ white balls be distributed into $7$ boxes such that every box contains exactly $2$ balls? | 393 |
Let $f$ be a function from the non-negative integers to the non-negative integers such that $f(nm) = n f(m) + m f(n)$, with the following conditions: $f(10) = 19$, $f(12) = 52$, and $f(15) = 26$. What is the value of $f(8)$? | 36 |
A plane contains points \(A\) and \(B\) with \(AB = 1\). Let \(S\) be the union of all disks of radius \(1\) in the plane that cover \(\overline{AB}\). The original answer is in the form \(k\pi - \frac{\sqrt{m}}{n}\). Please find the value of \(k + m + n\). | 8 |
How many integers $n$ are there such that $n^3 + 8$ has at most 3 positive divisors? Provide your answer as an integer. | 3 |
Lines $L_1, L_2, \dots, L_{100}$ are distinct. All lines $L_{4n}$, where $n$ is a positive integer, are parallel to each other. All lines $L_{4n-3}$, where $n$ is a positive integer, pass through a given point $A$. Find the maximum number of points of intersection of pairs of lines from the complete set $\{L_1, L_2, \d... | 4351 |
The ratio of the radii of two concentric circles is $1:3$. If $\overline{AC}$ is a diameter of the larger circle, $\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB = 12$, find the radius of the larger circle. | 18 |
Sequence $(a_n)$ is defined as $a_{n+1} - 2a_n + a_{n-1} = 7$ for every $n \geq 2$, where $a_1 = 1$, $a_2 = 5$. Find the value of $a_{17}$. | 905 |
Given points \(P(-1,-2)\) and \(Q(4,2)\) in the \(xy\)-plane; point \(R(1,m)\) is taken so that \(PR+RQ\) is a minimum. The original answer is in the form \(-\frac{a}{b}\), where a and b are integers and the fraction is in its simplest form. Please find the value of \(a + b\). | 7 |
What is the value of 100 times the expression $\sqrt{65} - \sqrt{63}$ rounded to the nearest integer? | 13 |
If the distinct non-zero numbers $x(y - z)$, $y(z - x)$, $z(x - y)$ form a geometric progression with common ratio $r$, find the equation that $r$ satisfies. The equation is in the form $ax^2 + bx + c = 0$. Please give the value of $a + b + c$. | 3 |
How many pairs of real numbers $(x, y)$ are there such that \( x^{4} - 2^{-y^{2}} x^{2} - \lfloor x^{2} \rfloor + 1 = 0 \), where \( \lfloor a \rfloor \) denotes the greatest integer not exceeding \( a \)? | 2 |
How many ordered triples $(a, b, c)$ of non-zero real numbers have the property that each number is the product of the other two? | 4 |
A laser is placed at the point \((3,5)\). The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the \(y\)-axis, then hit and bounce off the \(x\)-axis, and finally hit the point \((7,5)\). Find the total distance the beam will travel along this path. The original answer is in the form \(... | 12 |
In $\triangle ABC$ with a right angle at $C$, the altitude $CH$ and the median $CM$ trisect the right angle. If the area of $\triangle CHM$ is $K$, find the area of $\triangle ABC$ in terms of $K$. The answer is in the form of $aK$, please provide the value of $a$. | 4 |
A fair standard six-sided die is tossed three times. Given that the sum of the first two tosses equals the third, find the probability that at least one $2$ is tossed. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 23 |
How many positive integers $n$ are there such that $n!(2n+1)$ and $221$ are relatively prime? Provide your answer as an integer. | 10 |
A house worth $9000 is sold by Mr. A to Mr. B at a 10\% loss. Mr. B sells the house back to Mr. A at a 10\% gain. How much money does Mr. A lose in total from these two transactions? | 810 |
How many polynomial functions $f$ of degree $\ge 1$ satisfy the following conditions? \[ f(x^2) = [f(x)]^2 = f(f(x)) \] Provide your answer as an integer. | 1 |
A circle with a radius of $10$ inches has its center at the vertex $C$ of an equilateral triangle $\triangle ABC$ and passes through the other two vertices. The side $AC$ extended through $C$ intersects the circle at $D$. Determine the measure of angle $\angle ADB$ in degrees. If the answer is $x^\circ$, what is the va... | 30 |
After rationalizing the numerator of \(\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}}\), the denominator can be expressed in the form \( k + \sqrt{m} \). Please find the value of \( k + m \). | 9 |
Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, *special* if $a$ and $b$ are positive integers whose sum is 15. Find how many distinct integers can be written as the sum of two, not necessarily different, special fractions. | 11 |
Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run $100$ meters. They next meet after Sally has run $150$ meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters? | 350 |
For how many ordered pairs of positive integers \((m,n)\), does \(m \cdot n\) divide \(2008 \cdot 2009 \cdot 2010\)? Express your answer as a product of powers of prime numbers in the form \(2^k \cdot 3^m \cdot 5^n\). Find the value of \(k + m + n\). | 9 |
In the mobile below, the two cross beams and the seven supporting strings are all massless. The hanging objects are $M_1 = 400 \, \text{g}$, $M_2 = 200 \, \text{g}$, and $M_4 = 500 \, \text{g}$. What is the value of $M_3$ for the system to be in static equilibrium? If the answer is $x \, \text{g}$, what is the value of... | 700 |
There are two positive numbers that may be inserted between $3$ and $9$ such that the first three numbers form a geometric progression, while the last three numbers form an arithmetic progression. If the sum of those two positive numbers is $x$, find the value of $100x$. | 1125 |
A solid cube has a side length of $3$ inches. A $2$-inch by $2$-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid? | 7 |
Let $f(x) = x^2 + 3x + 2$ and let $S$ be the set of integers $\{0, 1, 2, \dots, 25\}$. Find the number of members $s$ of $S$ such that $f(s)$ has remainder zero when divided by 6. | 17 |
A point in the plane, both of whose rectangular coordinates are integers with absolute values less than or equal to four, is chosen at random, with all such points having an equal probability of being chosen. Find the probability that the distance from the point to the origin is at most two units. Express your answer i... | 94 |
How many positive integers $n$ are there such that $n$ is a multiple of $5$, and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n$? Provide your answer as an integer. | 48 |
Right triangle $ABC$ has side lengths $BC=6$, $AC=8$, and $AB=10$. A circle centered at $O$ is tangent to line $BC$ at $B$ and passes through $A$. A circle centered at $P$ is tangent to line $AC$ at $A$ and passes through $B$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 47 |
A rock is dropped off a cliff of height \(h\). As it falls, a camera takes several photographs at random intervals. In each picture, the distance the rock has fallen is measured. Let the average (expected value) of all these distances be \(kh\). If the number of photographs taken is huge, find \(k\). The original answe... | 4 |
Find the number of all positive integers $n$ such that $n + s(n) = 2016$, where $s(n)$ is the sum of all digits of $n$. | 2 |
The graph of $y = x^6 - 10x^5 + 29x^4 - 4x^3 + ax^2$ lies above the line $y = bx + c$ except at three values of $x$, where the graph and the line intersect. What is the largest of those values? | 4 |
Let \[ S_1 = \{ (x,y)\ | \ \log_{10} (1 + x^2 + y^2) \le 1 + \log_{10}(x + y)\} \] and \[ S_2 = \{ (x,y)\ | \ \log_{10} (2 + x^2 + y^2) \le 2 + \log_{10}(x + y)\}. \] Find the ratio of the area of $S_2$ to the area of $S_1$. | 102 |
Small lights are hung on a string 6 inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on, continuing this pattern of 2 red lights followed by 3 green lights. How many feet separate the third red light and the 21st red light? If the answer is x feet, what is the value of 10x?... | 225 |
In the given diagram, semi-circles are constructed on diameters $\overline{AB}$, $\overline{AC}$, and $\overline{CB}$, such that they are mutually tangent. If $\overline{CD} \bot \overline{AB}$, find the ratio of the shaded area to the area of a circle with $\overline{CD}$ as its radius. The original answer is in the f... | 5 |
Let \(a\) and \(b\) be positive reals such that: \[ a = 1 + \frac{a}{b} \] \[ b = 3 + \frac{4 + a}{b - 2} \] The original answer is in the form \(\sqrt[3]{k}\). Please find the value of \(k\). | 2 |
For how many integers $n$, are there four distinct real numbers satisfying the equation $|x^2 - 4x - 7| = n$? | 10 |
If $x, y > 0$, $\log_y x + \log_x y = \frac{10}{3}$ and $xy = 144$, then find the value of $\frac{x+y}{2}$. The original answer is in the form of k\sqrt{m}, please give the value of k + m. | 16 |
$n$ integers are arranged along a circle such that each number is equal to the absolute value of the difference of the two numbers following it in the clockwise direction. If the sum of all numbers is $278$, find how many different values $n$ can take. | 2 |
The sum of four two-digit numbers is $221$. None of the eight digits is $0$ and no two of them are the same. Which digit is **not** included among the eight digits? | 4 |
Points \(A\), \(B\), and \(C\) are on a circle with radius \(r\). They are positioned such that \(AB = AC\), \(AB > r\), and the length of the minor arc \(BC\) is \(r\). If angles are measured in radians, find the value of \(\frac{AB}{BC}\). The original answer is in the form \(\frac{k}{m}\csc\left(\frac{n}{p}\right)\)... | 8 |
Suppose \(S\) cans of soda can be purchased from a vending machine for \(Q\) quarters. Find the expression that describes the number of cans of soda that can be purchased for \(D\) dollars, where \(1\) dollar is worth \(4\) quarters. The original answer is in \(\frac{kDS}{mQ}\) format, please give the value of k + m + ... | 5 |
Let $ABCD$ be a trapezoid with $AB$ and $CD$ parallel, $\angle D = 2 \angle B$, $AD = 5$, and $CD = 2$. Find the length of $AB$. | 7 |
Pentagon $ABCDE$ is inscribed in a circle such that $ACDE$ is a square with an area of $12$. What is the largest possible area of pentagon $ABCDE$? The original answer is in the form \(k + m\sqrt{n}\), please give the value of \(k + m + n\). | 14 |
Each circle in an infinite sequence with decreasing radii is tangent externally to the one following it and to both sides of a given right angle. Find the ratio of the area of the first circle to the sum of areas of all other circles in the sequence in the form $(k+m\sqrt{2}):1$. What is the value of $k+m$? | 28 |
How many integers $n$ with $0 \leq n < 840$ are there such that $840$ divides $n^8-n^4+n-1$? Provide your answer as an integer. | 2 |
How many non-negative integer triples $(m,n,k)$ are there such that $m^3 - n^3 = 9^k + 123$? | 1 |
Consider sequences of positive real numbers of the form $x, 2000, y, \ldots,$ in which every term after the first is 1 less than the product of its two immediate neighbors. Find the number of different values of $x$ for which the term 2001 appears somewhere in the sequence. | 4 |
In the five-sided star shown, the letters $A,B,C,D,$ and $E$ are replaced by the numbers $3,5,6,7,$ and $9$, although not necessarily in this order. The sums of the numbers at the ends of the line segments $\overline{AB},\overline{BC},\overline{CD},\overline{DE},$ and $\overline{EA}$ form an arithmetic sequence, althou... | 12 |
The polynomial \((x+y)^9\) is expanded in decreasing powers of \(x\). The second and third terms have equal values when evaluated at \(x=p\) and \(y=q\), where \(p\) and \(q\) are positive numbers whose sum is one. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\). | 9 |
How many nonnegative integers can be written in the form $a_7\cdot3^7 + a_6\cdot3^6 + a_5\cdot3^5 + a_4\cdot3^4 + a_3\cdot3^3 + a_2\cdot3^2 + a_1\cdot3^1 + a_0\cdot3^0$, where $a_i \in \{-1,0,1\}$ for $0 \le i \le 7$? | 3281 |
The number $n$ is called a composite number if it can be written in the form $n = a \times b$, where $a, b$ are positive integers greater than $1$. Write the number $2013$ as a sum of $m$ composite numbers. What is the largest possible value of $m$? Please provide the value of m. | 502 |
Triangle $ABC$ has side-lengths $AB=12$, $BC=24$, and $AC=18$. The line through the incenter of $\triangle ABC$ parallel to $\overline{BC}$ intersects $\overline{AB}$ at $M$ and $\overline{AC}$ at $N$. Find the perimeter of $\triangle AMN$. | 30 |
Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region. If the area is $x\pi$, what is the value of $x$? | 2 |
What is the average length in letters of a word in this question? Provide your answer as an integer. | 4 |
A grasshopper jumps either $364$ or $715$ units on the real number line. If it starts from the point $0$, find the smallest distance that the grasshopper can be away from the point $2010$. | 5 |
The first 9 positive odd integers are placed in a magic square such that the sum of the numbers in each row, column, and diagonal is equal. Find the value of $A + E$. \[ \begin{array}{|c|c|c|} \hline A & 1 & B \\ \hline 5 & C & 13 \\ \hline D & E & 3 \\ \hline \end{array} \] | 32 |
In the following equation, each of the letters represents uniquely a different digit in base ten: \[ (YE) \cdot (ME) = TTT \] Find the sum $E+M+T+Y$. | 21 |
Right triangle \(ACD\) with a right angle at \(C\) is constructed outwards on the hypotenuse \(\overline{AC}\) of isosceles right triangle \(ABC\) with leg length \(1\), as shown, so that the two triangles have equal perimeters. Find the value of \(\sin(2\angle BAD)\). The original answer is in \(\frac{k}{m}\) format, ... | 16 |
The figure below is used to fold into a pyramid and consists of four equilateral triangles erected around a square with an area of 9. What is the length of the dashed path shown? | 24 |
A square with an area of $40$ is inscribed in a semicircle. What is the area of a square that could be inscribed in the entire circle with the same radius? Provide your answer as a number. | 100 |
How many ordered triples $(x, y, z)$ of positive integers satisfy the following conditions? - $\text{lcm}(x, y) = 72$ - $\text{lcm}(x, z) = 600$ - $\text{lcm}(y, z) = 900$ Find the number of such ordered triples. | 15 |
Line \(\ell_1\) has equation \(3x-2y=1\) and goes through \(A=(-1,-2)\). Line \(\ell_2\) has equation \(y=1\) and meets line \(\ell_1\) at point \(B\). Line \(\ell_3\) has positive slope, goes through point \(A\), and meets \(\ell_2\) at point \(C\). The area of \(\triangle ABC\) is \(3\). The slope of \(\ell_3\) is in... | 7 |
A point \((x, y)\) is to be chosen in the coordinate plane so that it is equally distant from the x-axis, the y-axis, and the line \(x+y = 2\). The original answer is in the form \(k - \sqrt{m}\). Please find the value of \(k + m\). | 4 |
With $400$ members voting, the House of Representatives defeated a bill. A re-vote, with the same members voting, resulted in passage of the bill by twice the margin by which it was originally defeated. The number voting for the bill on the re-vote was $\frac{12}{11}$ of the number voting against it originally. How man... | 60 |
Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\frac{1}{2}$ is $\frac{a-b\pi}{c}$, where $a$, $b$, and $c$ are positive integers and $\gcd(a,b,c)=1$. Find the value of $a+b+c$. | 59 |
What is the sum of the roots of \(z^{12} = 64\) that have a positive real part? Express your answer in the form \(k\sqrt{m} + n\sqrt{p}\), where k, m, n, and p are integers. Please provide the value of k + m + n + p. | 11 |
$A$, $B$, and $C$ are three piles of rocks. The mean weight of the rocks in $A$ is 40 pounds, the mean weight of the rocks in $B$ is 50 pounds. The mean weight of the rocks in the combined piles $A$ and $B$ is 43 pounds, and the mean weight of the rocks in the combined piles $A$ and $C$ is 44 pounds. Find the greatest ... | 59 |
Let \(ABCDA'B'C'D'\) be a rectangular prism with \(|AB|=2|BC|\). \(E\) is a point on the edge \([BB']\) satisfying \(|EB'|=6|EB|\). Let \(F\) and \(F'\) be the feet of the perpendiculars from \(E\) at \(\triangle AEC\) and \(\triangle A'EC'\), respectively. If \(m(\widehat{FEF'})=60^{\circ}\), find the value of \(|BC|/... | 20 |
If $yz:zx:xy = 1:2:3$, find the ratio $\frac{x}{yz}:\frac{y}{zx}$. The original answer is in the format m:n, please give the value of m + n. | 5 |
A right rectangular prism has edge lengths \(\log_2 x\), \(\log_3 x\), and \(\log_4 x\). The surface area and volume of the prism are numerically equal. What is the value of \( x \)? | 576 |
What is the probability that a random arrangement of the letters in the word 'ARROW' will have both R's next to each other? Express your answer in the form of a simplified fraction \(\frac{k}{m}\). Please give the value of k + m. | 7 |
In $\triangle ABC$, $\angle A = 100^\circ$, $\angle B = 50^\circ$, $\angle C = 30^\circ$, $\overline{AH}$ is an altitude, and $\overline{BM}$ is a median. If the answer is $x^\circ$, what is the value of $x$ for $\angle MHC$? | 30 |
Find the number of triples \((a,b,c)\) of positive integers which satisfy the simultaneous equations: \[ ab + bc = 44, \] \[ ac + bc = 23. \] | 2 |
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