original stringlengths 41 566 | trap stringlengths 71 840 | annotation stringlengths 26 977 | trap type stringclasses 2
values | source stringclasses 5
values | level stringclasses 5
values | split stringclasses 2
values | answer stringlengths 0 53 | additional Information stringclasses 2
values |
|---|---|---|---|---|---|---|---|---|
Determine the largest value of the quadratic expression \(-5x^2 + 40x - 12\) when \(x\) is a real number. | The largest value of the quadratic expression \(f(x) = 5x^2 + cx - 12 \) when \(x\) is a negative integer is 36. What is the value of \(c\)? | A parabola opening upwards doesn't have a maximum value for negative integer \(n\). | Contradiction | MathTrap | 4.0 | train | 68 | |
Consider a real function \(g(y) = \frac{1}{1+\frac{1}{1+\frac{1}{y}}}.\) Find the sum of the three real values of \(y\) that make this function undefined. | Consider the real function
\[
g(y)=\frac{1}{1+\frac{1}{c+\frac{1}{y}}}.
\]
Suppose exactly two real values of \(y\) are excluded from its domain and their sum is \(\frac12\). Determine \(c\). | The expression can be undefined at \(y=0\), at \(y=-1/c\) when \(c\ne0\), and at \(y=-1/(c+1)\) when \(c\ne-1\). For generic \(c\), these are three distinct excluded values. Exactly two occur only when \(c=0\) or \(c=-1\): their sums are respectively \(-1\) and \(1\), never \(\frac12\). Hence no real \(c\) satisfies th... | Contradiction | MathTrap | 4.0 | train | -\frac 32 | |
Suppose \(m\) is a real number such that \(\frac{1}{2} - \frac{1}{3} = \frac{3}{m}\). Find \(m\). | Suppose \(m\), \(n\), and \(k\) are odd integers such that \(\frac{1}{2} - \frac{1}{k} = \frac{m}{n}\). Find a possible value for \(k\). | \[
\frac{1}{2} - \frac{m}{n} = \frac{n - 2m}{2n}, \quad \text{so } k = \frac{2n}{n - 2m}.
\]
Since \( m \) and \( n \) are odd, \( n - 2m \) must be odd, and since \( 2n \) is even, \( k \) must be even.
This contradicts the assumption that \( k \) is odd.
Therefore, such \( m \), \( n \), and \( k \) do not exist.
| Contradiction | MathTrap | 1.0 | train | 18 | |
Find the real number \(a\) such that \((2008+a)^2=a^2\). | If the equation \((q+a)^{2147483647}=a^{2147483647}\) in terms of \(a\) has a unique solution, find the real number \(q\). | Because \(2147483647\) is odd, the map \(t\mapsto t^{2147483647}\) is injective on \(\mathbb R\). Thus the equation implies \(q+a=a\), so \(q=0\). But when \(q=0\), every real \(a\) is a solution; when \(q\ne0\), there is no solution. Therefore the equation can never have exactly one real solution. | Contradiction | MathTrap | 4.0 | train | -1004 | |
Find the real number \(a\) such that \((2008+a)^2=a^2\). | What is the smallest even integer \(p\) for which the equation \((1+a)^p=a^p\) in terms of \(a\) has no solution? | For any even integer \(p\), this equation can be simplified into \(1 + a = \pm a\), which always has a solution of \(a=-1/2\). Thus, no even \(p\) satisfies the condition that the equation has no solution. | Contradiction | MathTrap | 4.0 | train | -1004 | |
An equilateral triangle \(POQ\) has two points \(P\) and \(Q\) on the curve \(y = -\frac{1}{2}x^2\), where \(O\) is the origin. Determine the side length of triangle \(PQO\). | An equilateral triangle \(POQ\) has two distinct points \(P\) and \(Q\) on the curve \(y = -\frac{1}{c}x^2 (x>0),\) where \(O\) is the origin. Given that the side length of triangle \(PQO\) is 1, find \(c\). | Since triangle \(POQ\) is equilateral, then \(P\) and \(Q\) should be on a circle whose center is \(O\). However, a circle can only interact with \(y = -\frac{1}{c}x^2, (x>0)\) on one point. Thus, such an equilateral does not exist. | Contradiction | MathTrap | 5.0 | train | 4\sqrt{3} | |
Consider a regular hexagon where the diagonals can have two distinct lengths. Determine the ratio of the shorter diagonal to the longer diagonal. Provide your answer as a simplified fraction in radical form. | Consider a regular polygon with \(n\) sides, whose diagonals have exactly three distinct lengths. Given that the ratio of the lengths of the shortest and longest diagonals is \(1/2\), find \(n\). Note that a diagonal is a segment connecting two non-adjacent vertices. | Only a regular octagon and regular nonagon have three distinct diagonal lengths, but neither of them satisfies that the ratio of the shortest length to the longest length is 1/2. | Contradiction | MathTrap | 4.0 | train | \frac{\sqrt{3}}{2} | |
A garden is shaped like an equilateral triangle with an area of \(64\sqrt{3}\) square meters. If each side of this triangular garden is shortened by 4 meters, by how many square meters does the area of the garden decrease? | An equilateral triangle has area \(c\sqrt3\) square meters. Each side is shortened by 16 meters, and the area decreases by \(48\sqrt3\) square meters. Determine \(c\). | Let the original side length be \(s\). Then \(\frac{\sqrt3}{4}s^2=c\sqrt3\). The stated decrease gives
\[
\frac{\sqrt3}{4}\bigl(s^2-(s-16)^2\bigr)=48\sqrt3,
\]
which yields \(s=14\). But shortening a side of length 14 by 16 would give a negative side length. Hence no such triangle exists. | Contradiction | MathTrap | 4.0 | train | 28\sqrt{3} | |
In a Cartesian coordinate, consider a triangle with 3 vertices \((1, 7)\), \((13, 16)\), and \((5, m)\), where \(m\) is an integer. Determine the sum of all possible values of \(m\) that minimizes the area of the triangle. | In a Cartesian coordinate, consider a triangle with 3 vertices \((1, 7)\), \((a, 16)\), and \((5, m)\). With a certain fixed \(a\), if \(m=9\) minimizes the area of the triangle, find the value of such an \(a\). | When \(m=9, a=19\) three points are colinear, the triangle doesn't exist, so the area has an infimum 0, but cannot attain, since the colinear case is not a triangle. | Contradiction | MathTrap | 4.0 | train | 20 | |
Consider a triangle \(XYZ\) with side lengths \(XY = 3\), \(YZ = 4\), and \(XZ = 5\). \(E\) is a point on \(XZ\), and \(YE\) is the angle bisector of angle \(XYZ\). If \(YE\) has a length of \(k \sqrt{2}\), what is the value of \(k\)? | Consider a triangle \(XYZ\) with side lengths \(XY = 3\), \(YZ = 4\), and \(XZ = k\). \(E\) is a point on \(XZ\), and \(YE\) is the angle bisector of angle \(XYZ\). If \(YE\) has a length of 4, what is the value of \(k\)? | The internal angle-bisector length from the vertex between sides 3 and 4 is
\[
\ell^2=3\cdot4\left(1-\frac{k^2}{(3+4)^2}\right).
\]
For a nondegenerate triangle, \(1<k<7\), so \(\ell^2<12\) and therefore \(\ell<\sqrt{12}<4\). Thus the angle bisector cannot have length 4, and no such triangle exists. | Contradiction | MathTrap | 5.0 | train | \frac{12}{7} | |
A circle has a radius of 10 and is centered at the point \((-6, 2)\). It intersects with the \(y\)-axis at two points. Find the sum of the \(y\)-coordinates of the two points. | A circle has a radius of 5 and is centered at the point \((5a, 2)\), where \(a\) is a positive integer. It intersects with the \(y\)-axis at two distinct points, and the sum of the \(y\)-coordinates of the two points is 4. Find \(a\). | The center is \((5a,2)\) and the radius is 5. Two distinct intersections with the \(y\)-axis require the center's distance from that axis to be strictly less than 5, so \(|5a|<5\), or \(|a|<1\). This is impossible for a positive integer \(a\). At \(a=1\) the circle is tangent to the axis and has only one intersection. | Contradiction | MathTrap | 4.0 | train | 4 | |
Compute the area of a trapezoid that is enclosed by the following four lines: \(y = x\), \(y = 10\), \(y = 5\), and \(x=0\). Provide the answer as a decimal rounded to the nearest tenth. | The area of a trapezoid enclosed by the following four lines \(y = x\), \(y = x + p\), \(y = 5\) and \(x=0\) is 15. Find the value of \(p\). | These four lines have 5 intersection points (0,0), (0,p), (0,5), (5-p,5), (5,5), but they can make a trapezoid only if 0<p<5. When 0<p<5, the area of this trapezoid cannot be larger than the triangle formed by \(y = x\), \(y = 5\) and \(x=0\), whose area is 12.5. Thus, no \(p\) can make the trapezoid with area 15. | Contradiction | MathTrap | 3.0 | train | 37.5 | The trap problem provided in MathTrap actually has a solution. We further modify it to make it insolvable. |
Consider two points, \(A = (0,1)\) and \(B = (4,1)\). Then the set of points \(Q\) such that the sum of \(QA\) and \(QB\) is 6 forms an ellipse. The equation can be expressed as
\[\frac{(x - m)^2}{a^2} + \frac{(y - n)^2}{b^2} = 1.\]
Calculate the sum \(m + n + a + b\). | Consider two points, \(A = (0,1)\) and \(B = (4,4)\). Then the set of points \(Q\) such that the sum of \(QA\) and \(QB\) is 6 forms an ellipse. The equation can be expressed as
\[\frac{(x - m)^2}{a^2} + \frac{(y - n)^2}{b^2} = 1.\]
Calculate the sum \(m + n + a + b\). | The given points \(A = (0,1)\) and \(B = (4,4)\) define a line that is not parallel to either axis. Thus, the ellipse formed by the set of points \(Q\) satisfying \(QA + QB = 6\) cannot be expressed in the standard form \(\frac{(x - m)^2}{a^2} + \frac{(y - n)^2}{b^2} = 1\) without including an \(xy\) term, as its axes ... | Contradiction | MathTrap | 4.0 | train | 6 + \sqrt{5} | |
Consider a function \( g(t) \) such that for every pair of real numbers \( t \) and \( u \), \[ tg(u) = ug(t) \]. Given that \( g(15) = 20 \), what is the value of \( g(3) \)? | Consider a function \( g(t) \) such that for every pair of real numbers \( t \) and \( u \), \[ g(u+t) = g(u)g(t) + p\] for some constant \( p \). Given that \( g(2) = 3, g(6) = 31 \), what is the value of \( p \)? | Let \( t = 0 \). We get \( g(u)(1 - g(0)) = p \), which means either \( g(u) \) is constant, or \( g(0) = 1 \), and \( p = 0 \). If \( g(u) \) is constant, then \( g(6) \) should equal \( g(2) \). If \( p = 0 \), then the equation becomes \( g(u + t) = g(u)g(t) \). Then \( g(4) = g(2)g(2) = 9,\quad g(6) = g(4)g(2) = 27... | Contradiction | MathTrap | 2.0 | train | 4 | |
What are the coordinates of the focus of the parabola \(x = -\frac{1}{12} y^2\)? | If the parabola \(x + x^2 = -\frac{p}{12} y^2\) has a focus at \((-1/2, 1)\), what is the value of \(p\)? | The curve is an ellipse when \(p>0\), a hyperbola when \(p<0\). It can never be a parabola | Contradiction | MathTrap | 3.0 | train | (-3,0) | |
In a Cartesian coordinate system, a circle intersects the hyperbola \(xy = 1\) at \(\left( 2, \frac{1}{2} \right),\) \(\left( -5, -\frac{1}{5} \right),\) and \(\left( \frac{1}{3}, 3 \right), and the fourth point is unknown. What are the coordinates of the fourth intersection point? | In a Cartesian coordinate system, the hyperbola \(xy = 1\) intersects \(x^2 + y^2 + ky = p\) at \(\left( 2, \frac{1}{2} \right),\) \(\left( -5, \frac{1}{5} \right),\) and two other points. What is the value of \(k\)? | The point \( \left( -5, \frac{1}{5} \right) \) is not on the curve \( xy = 1 \). | Contradiction | MathTrap | 5.0 | train | \left( -\frac{3}{10}, -\frac{10}{3} \right) | |
Two positive integers have a greatest common divisor of \(9\) and a least common multiple of \(7!\). If one of the two numbers is \(315\), find the other one. | Two positive integers have a greatest common divisor of \(99\) and a least common multiple of \(240\). Find the product of the two numbers. | 240 is not divisible by 99, so there is a contradiction. | Contradiction | MathTrap | 3.0 | train | 144 | |
Suppose the first day of a particular month falls on a Monday. Determine the day of the week that corresponds to the twenty-third day of that month. | Suppose the first day of a particular month falls on a Monday, and the (30 + p)th day of this month is Saturday (\( p\) is the integer). Determine the value of \( p\). | If the 1st is Monday, then the \(n\)th day is \(\text{Monday} + (n - 1) \bmod 7\). So the \((30 + p)\)th day being Saturday means \[29 + p \equiv 5 \pmod{7} \Rightarrow p \equiv 4 \pmod{7}.\] The smallest non-negative solution is \(p = 4\), making it the 34th day. This is invalid, since months have at most 31 days. | Contradiction | MathTrap | 2.0 | train | \text{Tuesday} | |
Determine the final nonzero digit that appears after the decimal in the decimal representation of \(\frac{141}{400}\). | If the final nonzero digit after the decimal in the decimal representation of \(\frac{19}{a}\) is 9, identify the largest possible positive integer value for \(a\). | \(a\) can be 10 to the power of any positive integer, so there doesn't exist a maximum value. | Contradiction | MathTrap | 2.0 | train | 5 | |
Determine the final nonzero digit that appears after the decimal in the decimal representation of \(\frac{141}{400}\). | If the final nonzero digit after the decimal in the decimal representation of \(\frac{a}{402}\) is 9, identify the smallest possible positive integer value for \(a\). | If \(a\) is not divisible by \(201\) (\(67 \times 3\)), the decimal representation of \(\frac{a}{402}\) is infinite and repeating, so there is no final nonzero digit. If \(a\) is divisible by \(201\), the final nonzero digit must be \(5\), meaning no such \(a\) satisfies the condition of the final digit being \(9\). | Contradiction | MathTrap | 2.0 | train | 5 | |
Determine the greatest three-digit integer such that the multiplication of its digits equals \(12\). | Determine the greatest integer such that the multiplication of its digits equals \(12\). | Since you can insert "1" into a number increasing its value but not changing its digits product value, there is no biggest number for the product of whose digits is \(12\). | Contradiction | MathTrap | 2.0 | train | 621 | |
Determine the greatest three-digit integer such that the multiplication of its digits equals \(12\). | Determine the greatest four-digit integer such that the multiplication of its digits equals \(11\). | 11 is a prime, but a digit must be from \(0\) to \(9\) and can't be 11, so there doesn't exist a number in which the product of the digits is \(11\) | Contradiction | MathTrap | 2.0 | train | 621 | |
A certain integer, when divided by \(7\), results in a quotient of \(28\) and a remainder of \(6\). What is this integer? | 225, when divided by \(7\), results in a quotient of \(31\) and a remainder of \(p\). Find \(p\). | The remainder can't be larger than or equal to divisor, but in this problem the remainder is 8 > the divisor 7, so there is a contradiction. | Contradiction | MathTrap | 2.0 | train | 202 | |
A certain integer, when divided by \(7\), results in a quotient of \(28\) and leaves a remainder of \(6\). What is this integer? | 225, when divided by \(7\), results in a quotient \(p\) and a remainder of \(4\). Find \(p\). | The remainder is 1, not 4. | Contradiction | MathTrap | 2.0 | train | 202 | |
In a regular octagon \(UVWXYZPQ\), the sides \(\overline{UQ}\) and \(\overline{WX}\) are extended until they intersect at a point labeled \(M\). Find the angle at point \(M\) in degrees. | \(A\) is one of the vertices of a regular hexagon \(UVWXYZ\). The sides of the hexagon \(\overline{UV}\) and \(\overline{YA}\) are extended until they intersect at a point labeled \(M\). If angle \(M\) is \(\pi/2\), which vertex is \(A\)? | In a regular hexagon, no two sides (or their extensions) intersect at a right angle. Since angle \(M\) is given as \(\frac{\pi}{2}\), the point \(M\) cannot exist, making the question invalid. | Contradiction | MathTrap | 5.0 | train | 45^\circ | |
Consider a convex pentagon whose interior angles are given by \(a+1\), \(2a\), \(3a\), \(4a\), and \(5a-1\) degrees. Find the greatest angle in degrees. | Consider a convex pentagon four of whose interior angles are given by \(\pi/2\), \(\pi/2\), \(3\pi/4\) and \(\pi/6\) degrees. Find the greatest angle in degrees. | As calculated, the largest angle in the pentagon measures is \(13\pi/12\), which is contradictory to the statement that the pentagon is convex. | Contradiction | MathTrap | 4.0 | train | 179 | |
Consider a sequence of five positive integers where the median is 3, and the average is 11. Find the greatest value that the maximum in this sequence can take. | Consider a sequence of five integers where the median is \(p\), and the average is 4. Find \(p\) if the greatest value that the maximum in this sequence can take is 12. | Let the sorted integers be \(a\le b\le p\le d\le e\), with total sum 20. Since negative integers are allowed, take \(a=b=-K\) and \(p=d=0\). Then \(e=20+2K\), which becomes arbitrarily large as \(K\to\infty\). Thus the largest entry has no finite maximum, contrary to the premise that its maximum is 12. | Contradiction | MathTrap | 5.0 | train | 47 | |
Determine the period of the function \( f(x) = \tan \left( \frac{x}{2} \right) \). | If the period of \(f(x) = \cos \frac{x^2}{bx}\) is \(\pi\), determine the value of \(b\). | For \(b\ne0\), the expression simplifies to \(f(x)=\cos(x/b)\) only for \(x\ne0\); its domain is \(\mathbb R\setminus\{0\}\). A period \(T\ne0\) must preserve the domain under translation, but \(-T\) is in the domain while \(-T+T=0\) is not. Hence the function has no period. For \(b=0\), the expression is undefined. Th... | Contradiction | MathTrap | 1.0 | train | 2 \pi | |
Determine the period of the function \( f(a) = \tan \frac{a}{2} \). | Determine the period of the function \( g(a) = \sin \lvert a \rvert \). | The period doesn't exist. Since the period of \(\sin a\) is \(2\pi\), the period for \(g(a)\) can only be multiples of \(2\pi\). Consider \(a = \pi/2\), \(\sin \lvert \pi/2 \rvert =1\), but for any potential period that is multiples of \(2\pi\) and positive \(k\), let \(b = \left(a - 2k\pi\right) = \left(\pi/2 - 2k\pi\... | Contradiction | MathTrap | 1.0 | train | 2 \pi | |
Consider \(y\) such that \(\tan (y) = \frac{m}{n}\) and \(\tan(2y) = \frac{n}{m + n}.\) Given that the smallest positive value of \(y\) equals \(\tan^{-1} p.\) Find the value of \(p.\) | Consider an angle \(y\) such that \(\tan (y) = \frac{m}{n}\) and \(\tan(2y) = \frac{n}{m}.\) What is the largest possible value of \(y\)? | The conditions determine \(\tan y\) but not a largest positive representative of the angle. From the double-angle identity one obtains \(\tan y=\pm1/\sqrt3\), so every solution is accompanied by \(y+k\pi\) for arbitrarily large integers \(k\). Therefore the set of positive solutions is unbounded and has no largest elem... | Contradiction | MathTrap | 4.0 | train | \frac{1}{3} | |
Consider two functions, \( f(x) \) and \( g(x) \), whose graphs intersect at the points \((2,2)\), \((4,6)\), \((6,12)\), and \((8,12)\). Determine the sum of the coordinates of the point where the graphs of \( f(2x) \) and \( 2g(x) \) intersect. | Consider two functions, \( f(x) \) and \( g(x) \), whose graphs intersect at the points \((2,2)\), \((4,6)\), \((6,12)\), and \((8,4)\). Determine the sum of the coordinates of the point where the graphs of \( f(2x) \) and \( 2g(x) \) intersect. | The four shared points specify only isolated values of \(f\) and \(g\). They do not determine either function away from those inputs, so the graphs of \(f(2x)\) and \(2g(x)\) can be completed in continuously many ways, producing different intersections or none at all. No intersection point is forced by the stated data. | Missing conditions | MathTrap | 5.0 | train | 16 | |
Consider a cube \(PQRSTUVX\) with a total volume of 1 cubic unit. Determine the volume of the pyramid formed by the vertices \(PQR\) and \(X\). | Consider a cube \(PQRSTUVX\) with a total volume of 1 cubic unit. Determine the volume of the pyramid formed by the vertices \(PQVX\). | \(PQVX\) is a rectangle, not a pyramid. | Contradiction | MathTrap | 4.0 | train | \frac16 | |
Consider a polynomial \( p \) with a degree of 5, and assume that \( p(x) = (x^2 - 7x + 10) \times q(x) \), where \( q(x) \) is another polynomial of degree \( c \). What is the value of \( c \)? | Consider a polynomial \( p \) with a degree of 5, and assume that \( p(x) = (x^2 - 7x + 10) \times q(x) (x+1) \), where \(q(x)\) is another polynomial of degree \(c\). In addition, assume that: 1. \(p(x)\) has no repeated roots; and 2. \(q(2) = 0\). What is the value of the integer \(c\)? | Since \(x^2 - 7x + 10 = (x-2)(x-5)\), it contradicts “no repeated roots”. No choice of \(c\) can reconcile these. | Contradiction | MATH | 3.0 | train | 3 | |
Consider the quadratic equation \(2y^2 + 3y - 5 = 0\). If \(p\) and \(q\) be the roots of this equation, determine the value of \((p-1)(q-1)\). | Consider the quadratic equation \(2y^2 + 3y - 5 = 0\). If \(p\) and \(q\) are both integers roots of this equation, determine the value of \((p-1)(q-1)\). | The roots \(p\) and \(q\) of the quadratic equation \(2y^2 + 3y - 5 = 0\) are not integers. The two roots are \(1\) and \(-\frac{5}{2}\). | Contradiction | MATH | 3.0 | train | 0 | |
Given the vectors \(\mathbf{a} = \begin{pmatrix} -10 \\ 6 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 15 \\ -9 \end{pmatrix}\), determine the projection of \(\mathbf{a}\) onto \(\mathbf{b}\). | Given the vectors \(\mathbf{a} = \begin{pmatrix} -10 \\ 6 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 15 \\ -9 \end{pmatrix}\), find the projection of \(\mathbf{a}\) onto \(\mathbf{b}\) such that the resulting projection is perpendicular to \(\mathbf{b}\). | The projection of a vector \(\mathbf{a}\) onto another vector \(\mathbf{b}\) is always parallel to \(\mathbf{b}\). Therefore, it is impossible for the projection to be orthogonal to \(\mathbf{b}\) unless \(\mathbf{a}\) is the zero vector, which it is not in this case. | Contradiction | MATH | 3.0 | train | \begin{pmatrix} -10 \\ 6 \end{pmatrix} | |
Consider two nonzero vectors \(\mathbf{x}\) and \(\mathbf{y}\) such that the sum \(\mathbf{x} + \mathbf{y}\) is perpendicular to \(\mathbf{y}\), and the vector \(\mathbf{x} + 2\mathbf{y}\) is perpendicular to \(\mathbf{x}\). What is the ratio \(\frac{\|\mathbf{x}\|}{\|\mathbf{y}\|}\)? | Consider two nonzero vectors \(\mathbf{x}\) and \(\mathbf{y}\) such that the sum \(\mathbf{x} + \mathbf{y}\) is perpendicular to \(\mathbf{y}\). What is the ratio \(\frac{\|\mathbf{x}\|}{\|\mathbf{y}\|}\)? | Since \(\mathbf{x} + \mathbf{y}\) is perpendicular to \(\mathbf{y}\), one can consider \|\mathbf{x}+ \mathbf{y}\| and \|\mathbf{y}\| as the length of two legs of a right triangle. Then the ratio of the hypotenuse and one leg is definitely underdetermined. | Missing conditions | MATH | 3.0 | train | \sqrt{2} | |
Determine a quadratic equation with real coefficients where the highest degree term is \(x^2\) and one of its solutions is \(5-4i\). | Determine a quadratic equation with real coefficients where the leading term is \(x^2\) that has \(5-4i\) as one solution and \(4+5i\) as the other solution. | The complex roots of a quadratic with real coefficients must be conjugates of each other. Thus, the two complex roots cannot be \(5-4i\) and \(4+5i\). | Contradiction | MATH | 3.0 | train | x^2-10x+41=0 | |
The line \( y = -\frac{3}{4}x + 9 \) intersects the horizontal axis at point \( A \) and the vertical axis at point \( B \). A point \( M(u, v) \) lies on the segment \( AB \). If the area of triangle \( \triangle AOB \) is three times the area of triangle \( \triangle MOA \), what is the value of \( u + v \)? | The line \( y = -\frac{3}{4}x + 9 \) intersects the horizontal axis at point \( A \) and the vertical axis at point \( B \). A point \( M(u, v) \) lies on the segment \( AB \). If the area of triangle \( \triangle AOB \) is three times the area of triangle \( \triangle MOB \) and \(M\) is the midpoint of \(AB\), what i... | The condition that the area of triangle \( \triangle AOB \) is three times the area of triangle \( \triangle MOB \) indicate that \(M\) is the trisection point of \(AB\), instead of midpoint. | Contradiction | MATH | 4.0 | train | 11 | |
Find the real solution for \( a \) in the equation
\[ \tan^{-1} \frac{1}{a} + \tan^{-1} \frac{1}{a^3} = \frac{\pi}{4} \] | Find the real solution for \( a \) in the equation
\[ \tan^{-1} \frac{1}{a} + \tan^{-1} (2a - 1) = \frac{\pi}{4}. \] | Applying the tangent-addition formula to a putative solution gives
\[
\frac{\frac1a+(2a-1)}{1-\frac1a(2a-1)}=1,
\]
which simplifies to \(a=0\). But \(a=0\) is outside the domain because \(1/a\) is undefined. The exceptional case where the denominator in the addition formula vanishes is \(a=1\), for which the left side ... | Contradiction | MATH | 4.0 | train | \frac{1 + \sqrt{5}}{2} | |
What is the largest value of \(\cos \theta + 2 \sin \theta\) over all real value of \(\theta\)'s? | What is the largest value of \(\theta \cos \theta + 2 \sin \theta\) over all positive integer \(\theta\)'s? | There are infinitely many positive integers \(n\) for which \(\cos n>1/2\) (the residues of integers modulo \(2\pi\) are dense). Along such a subsequence,
\[
n\cos n+2\sin n>\frac n2-2,
\]
which tends to infinity. Thus the expression is unbounded above and has no largest value. | Contradiction | MATH | 3.0 | train | \sqrt{5} | |
Consider three polynomials \(p\), \(q\), and \(r\), where \(r(x) = p(x) \times q(x)\). Given that the constant term of \(p(x)\) is \(-4\) and the constant term of \(r(x)\) is 3, determine the value of \(q(0)\). | Consider three polynomials \(p\), \(q\), and \(r\) where \(r(x) = p(x) \times q(x)\). If the constant term of \(p(x)\) is \(-4\) and the constant term of \(r(x)\) is 3, determine the value of \(q(1)\). | One only can determine that \(q(0)= - \frac{3}{4}\). There is no enough information to determine \(q(1)\). | Missing conditions | MATH | 4.0 | train | -\frac{3}{4} | |
Suppose \(a\), \(b\), and \(c\) are positive real numbers such that:
\[
\ln a - \ln b = m,
\]
\[
\ln b - \ln c = 15,
\]
\[
\ln c - \ln a = -7,
\]
where \(m\) is a real number. Determine the value of \(m\). | Suppose \(a\), \(b\), and \(c\) are positive real numbers such that:
\[
\ln a + \ln b = m,
\]
\[
\ln b - \ln c = 15,
\]
\[
\ln c - \ln a = -7,
\]
where \(m\) is a real number. Determine the value of \(m\). | Let \(x=\ln a\), \(y=\ln b\), and \(z=\ln c\). The last two equations give \(y=z+15\) and \(x=z+7\). Hence
\[
m=x+y=2z+22.
\]
Because \(z\) is arbitrary, \(m\) can take every real value. The conditions do not determine a unique answer. | Missing conditions | MATH | 4.0 | train | -8 | |
In a Cartesian coordinate, the points \(P(-1, -2)\) and \(Q(3, 2)\) represent the ends of a diameter of a circle. Calculate the circle's area in square units, and provide your answer using \(\pi\). | In a coordinate grid, the points \((-1, -2)\), \((-2, 1)\), \((2, 1)\), and \((2, -2)\) lie on a circle. Calculate the circle's area in square units, and provide your answer using \(\pi\). | The first three points determine the circle \(x^2+y^2=5\). The fourth point \((2,-2)\) satisfies \(x^2+y^2=8\), so it is not on that circle. Therefore no circle passes through all four stated points. | Contradiction | MATH | 4.0 | train | 8\pi | |
Suppose that we have a collection of \(n\) vectors, each of length 1, denoted by \(\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n\). They satisfy \[ \mathbf{a}_1 + \mathbf{a}_2 + \ldots + \mathbf{a}_n = \begin{pmatrix} 6 \\ -5 \end{pmatrix}.\] What is the minimum value of \(n\)? | Suppose that we have a collection of \(n\) vectors, each of length 1, denoted by \(\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n\). They satisfy \[ \mathbf{a}_1 + \mathbf{a}_2 + \ldots + \mathbf{a}_n = \begin{pmatrix} 6 \\ -5 \end{pmatrix}.\] What is the maximum value of \(n\)? | Once one representation by unit vectors exists, any number of canceling pairs \(\mathbf u,-\mathbf u\) may be appended without changing the sum. For example, the vector \((6,-5)\), whose norm is less than 8, can be expressed as a sum of finitely many unit vectors; adding canceling pairs then makes \(n\) arbitrarily lar... | Contradiction | MATH | 4.0 | train | 8 | |
What is the range of the function
\[f(a) = \frac{2a + 7}{a - 3}?\]. Express the answer in the format of interval. | The maximum value of the function \[f(a) = \frac{2a + 7}{a - c}.\] over all positive integers within its domain is 46, where \(c\) is a positive integer. What is the value of \(c\)? | For positive integers \(a\ne c\),
\[
f(a)=2+\frac{2c+7}{a-c}.
\]
Values with \(a<c\) are below 2, while for \(a>c\) the maximum occurs at \(a=c+1\), giving \(2c+9\). Setting this equal to 46 yields \(c=37/2\), not an integer. Therefore no positive integer \(c\) satisfies the premise. | Contradiction | MATH | 3.0 | train | (-\infty,2) \cup (2,\infty) | |
A geometric sequence starts with \(3\), \(\dfrac{9}{2}\), \(\dfrac{27}{4}\), \(\dfrac{81}{8}\), \(\ldots\). What is the value of the eighth term? Provide your answer as a fraction in simplest form. | A geometric sequence starts with \(3\), \(\dfrac{9}{2}\), \(\dfrac{27}{4}\), \(\dfrac{81}{7}\), \(\ldots\). What is the value of the eighth term? Provide your answer as a fraction in simplest form. | The given sequence is not geometric. | Contradiction | MATH | 3.0 | train | \frac{6561}{128} | |
Consider real numbers \(x\), \(y\), and \(z\) and a matrix
\[\begin{pmatrix} x & y & z \\ y & z & x \\ z & x & y \end{pmatrix}\]. If the matrix is singular, provide all possible values of
\[\frac{x}{y + z} + \frac{y}{x + z} + \frac{z}{x + y}.\], and separate them by commas. | Consider real numbers \(x\), \(y\), and \(z\) and a matrix
\[\begin{pmatrix} x & y & z \\ x & y & z \\ z & x & y \end{pmatrix}\]. If the matrix is singular, compute
\[\frac{x}{y + z} + \frac{y}{x + z} + \frac{z}{x + y}.\] | The matrix is always singular since row 2 is same as row 1, so it cannot provide any information to compute the desired formula. | Missing conditions | MATH | 4.0 | train | -3, \frac{3}{2} | |
Consider seven variables \( a_1, a_2, \ldots, a_7 \) that are real numbers, satisfying the following equations:
\[
\begin{aligned}
a_1 + 4a_2 + 9a_3 + 16a_4 + 25a_5 + 36a_6 + 49a_7 &= 1, \\
4a_1 + 9a_2 + 16a_3 + 25a_4 + 36a_5 + 49a_6 + 64a_7 &= 12, \\
9a_1 + 16a_2 + 25a_3 + 36a_4 + 49a_5 + 64a_6 + 81a_7 &= 12... | Consider seven variables \( a_1, a_2, \ldots, a_7 \) that are real numbers, satisfying the following equations:
\[
\begin{aligned}
a_1 + 4a_2 + 9a_3 + 16a_4 + 25a_5 + 36a_6 + 49a_7 &= 1, \\
4a_1 + 9a_2 + 16a_3 + 25a_4 + 36a_5 + 49a_6 + 64a_7 &= 12, \\
9a_1 + 16a_2 + 25a_3 + 36a_4 + 49a_5 + 64a_6 + 81a_7 &= 12... | Let matrix \( A \) be
\[A = \begin{bmatrix}
1 & 4 & 9 & 16 & 25 & 36 & 49 \\
4 & 9 & 16 & 25 & 36 & 49 & 64 \\
9 & 16 & 25 & 36 & 49 & 64 & 81
\end{bmatrix}\]
and let \( \mathbf{a} = [a_1, a_2, \ldots, a_7]^T \). The vector \( \mathbf{v} = [16, 25, 36, 49, 64, 81, 121] \) does not lie in the row space of \(... | Missing conditions | MATH | 4.0 | train | 334 | |
An integer \(B\) is referred to as delightful if there is a finite sequence of consecutive integers, including \(B\), whose sum equals 2014. Find the smallest delightful integer. | An integer \(B\) is referred to as delightful if there is a finite sequence of consecutive odd integers, including \(B\), whose sum equals 2022. Find the smallest delightful integer. | A sum of an odd number of odd integers is odd, so it cannot equal 2022. A sum of an even number of consecutive odd integers is divisible by 4, whereas \(2022\equiv2\pmod4\). Hence no finite sequence of consecutive odd integers sums to 2022, so no delightful integer exists. | Contradiction | MATH | 4.0 | train | -2013 | |
\(a\) and \(b\) are two distinct real numbers such that the determinant of the matrix
\[
\begin{pmatrix} 1 & 4 & 9 \\ 3 & a & b \\ 3 & b & a \end{pmatrix}\] is \(0\).
What is the value of \(a + b\)? | \(a\) and \(b\) are two distinct real numbers such that the determinant of the matrix
\[\begin{pmatrix} 1 & 4 & 9 \\ 3 & a & b \\ 3 & b & a \end{pmatrix}\] is \(1\).
What is the value of \(a + b\)? | Expanding the determinant gives
\[
(a-b)(a+b-39)=1.
\]
Let \(t=a-b\ne0\). Then \(a+b=39+1/t\), which varies continuously with the nonzero real parameter \(t\). Thus the determinant condition does not determine \(a+b\). | Missing conditions | MATH | 3.0 | train | 39 | |
A rectangular prism has integer side lengths. The height is 3, and the surface area is 52. Determine the volume of it. | A rectangular prism has integer side lengths. The height is 3, and the surface area is 26. Determine the volume of it. | Denote height, length, width as h,l,w, then \(lw+3l+3w = 13\) or \((l+3)(w+3)=22\). The only two factorizations are \(2\times 11\) and \(1 \times 22\), but both \((l+3)\) and \((w+3)\) are larger than 3, leading to contradiction. | Contradiction | MATH | 4.0 | train | 24 | |
What is the smallest possible value of \( a(a + 1)(a + 2)(a + 3) \) for any real number \( a \)? | What is the minimum of \( a(a + 1)(a + 2) \cdots (a + n) \) when \( a > 0 \)? \( n \) is a positive integer. | When \( a>0 \), the given function increases monotonically, so the infimum is the limit as \( a \to 0^+ \), which is 0, but this value is not attained since \( a > 0 \). | Contradiction | MATH | 3.0 | train | -1 | |
Two positive integers \( a \) and \( b \), multiply to give 56, with \( a < b \). If \(\frac{7}{a} + \frac{14}{b} = 4\), find \( a \). | Positive integers \(a<b\) satisfy \(ab=56\) and
\[
\frac7a+\frac{14}{b}=3.
\]
Determine \(a\). | Using \(ab=56\), the second equation becomes \(b+2a=24\). The factor pairs with \(a<b\) are \((1,56),(2,28),(4,14),(7,8)\), and none satisfies \(b+2a=24\). Hence no such positive integers exist. | Contradiction | MATH | 4.0 | train | 2 | |
Consider real numbers \(x,\) \(y,\) \(z,\) \(u,\) \(v,\) and \(w\) such that the determinant of the matrix
\[ \begin{pmatrix} u & y & z \\ x & v & z \\ x & y & w \end{pmatrix}\] is 0. Given that \(x \neq u,\) \(y \neq v,\) and \(z \neq w,\) determine the value of \(\frac{u}{u - x} + \frac{v}{v - y} + \frac{w}{w - z}.... | Consider real numbers \(x,\) \(y,\) \(z,\) \(u,\) \(v,\) and \(w\) such that the determinant of the matrix
\[ \begin{pmatrix} u & y & z \\ x & v & z \\ x & y & w \end{pmatrix}\] is 1. Given that \(x \neq u,\) \(y \neq v,\) and \(z \neq w,\) determine the value of \(\frac{u}{u - x} + \frac{v}{v - y} + \frac{w}{w - z}.... | Fix \(x=0\), \(y=0\), and \(z=1\). The determinant condition becomes \(uvw=1\), while the requested expression is
\[
1+1+\frac{w}{w-1}=3+\frac1{w-1}.
\]
For any real \(w\ne0,1\), choose nonzero \(u,v\) with \(uv=1/w\). All stated inequalities hold, but the requested value varies continuously with \(w\). Therefore it is... | Missing conditions | MATH | 4.0 | train | 2 | |
A quadrilateral has four vertices with the following coordinates: \(P(0,1)\), \(Q(1,3)\), \(R(5,2)\), and \(S(4,0)\). Determine the area of the quadrilateral \(PQRS\). | A nondegenerate quadrilateral has consecutive vertices \(P(0,1)\), \(Q(1,3)\), \(R(5,2)\), and \(S(2,5)\). Determine its area. | The points \(P,Q,S\) are collinear because they all lie on \(y=2x+1\), with \(Q\) between \(P\) and \(S\). Therefore the four points cannot be the consecutive vertices of a nondegenerate quadrilateral, so the requested area is undefined. | Contradiction | MATH | 4.0 | train | 9 | |
Consider two positive real numbers, \(a\) and \(b\), such that \(a + b = 35\). Determine the ordered pair \((a,b)\) that results in the maximum value of \(a^5 b^2\). | Consider two integers \(a\) and \(b\), such that \(a + b = 35\). Determine the ordered pair \((a,b)\) that results in the maximum value of \(a^5 b^2\). | There's no maximum for \( a^5b^2 \). One can always increase \(a\) and decrease \(b\) (make \(b\) negative) to make \( a^5b^2 \) infinitely large. | Contradiction | MATH | 4.0 | train | (25,10) | |
The roots of the quadratic equation of \(y\), \(y^2 + ky = -14\), are integers. Given that \(k\) is a positive integer, find the maximum possible value of \(k\). | The roots of the quadratic equation of \(y\), \(y^2 + 4ky = -14\), are integers. Given that \(k\) is a positive integer, find the maximum possible value of \(k\). | If the roots are integers \(r,s\), then \(rs=14\) and \(r+s=-4k\). The possible integer sums from factor pairs of 14 are \(\pm15\) and \(\pm9\), none divisible by 4. Hence no positive integer \(k\) can make both roots integers. | Contradiction | MATH | 4.0 | train | 15 | |
Consider \( n \) real numbers \( a_1, a_2, \ldots, a_n \), and \( |a_i| < 1 \) for \( i = 1, 2, \dots, n \).
Given that \[|a_1| + |a_2| + \dots + |a_n| = 19 + |a_1 + a_2 + \dots + a_n|,\] determine the smallest possible value of \( n \). | Consider \( n \) real numbers \( a_1, a_2, \ldots, a_n \), and \( |a_i| < 1 \) for \( i = 1, 2, \dots, n \).
Given that \[|a_1| + |a_2| + \dots + |a_n| = 19 + |a_1 + a_2 + \dots + a_n|,\] determine the maximum possible value of \( n \). | For every even integer \(n>19\), take half the terms equal to \(19/n\) and half equal to \(-19/n\). Then every \(|a_i|<1\), the sum of absolute values is 19, and the ordinary sum is 0, so the equation holds. Since such even \(n\) are arbitrarily large, no maximum exists. | Contradiction | MATH | 4.0 | train | 20 | |
Consider the points \( M = (8,0,0) \), \( N = (0,-4,0) \), \( P = (0,0,6) \), and \( Q = (0,0,0) \) in a three-dimensional space. Determine the coordinate of a point \( X \) such that the distances from \( X \) to each of the points \( M \), \( N \), \( P \), and \( Q \) are identical. | Consider the points \( M = (8,0,0) \), \( N = (0,-4,0) \), \( P = (0,0,6) \), and \(X\) with unknown coordinates in a three-dimensional space. Given that the distances from \( X \) to each of the points \( M \), \( N \), \( P \) are identical, and that the coordinates of \(X\) are all integers, find the coordinates of ... | Equating squared distances gives \(2x+y=6\) and \(4x-3z=7\). All integer solutions are
\[
X_k=(3k+1,\,4-6k,\,4k-1),\qquad k\in\mathbb Z.
\]
Their squared distances from the origin are \(61k^2-50k+18\), which are unbounded as \(|k|\to\infty\). Hence there is no farthest such point. | Contradiction | MATH | 3.0 | train | (4,-2,3) | |
Two functions \(f(x)\) and \(g(x)\) satisfy \(f(g(x)) = x^2\) and \(g(f(x)) = x^3\) for all \(x \ge 1\). Given that \(g(16) = 16\), what is the value of \(\bigl[g(4)\bigr]^3\)? | Two functions \(h(x)\) and \(k(x)\) satisfy \(h(k(x)) = x^2\) and \(k(h(x)) = x^3\) for all \(x \ge 1\). Given that \(k(16) = 16\), what is the value of \(\bigl[k(8)\bigr]^3\)? | Applying \(k()\) to the first condition, and substituting \(x = k(x)\) into the second condition, we can obtain the functional relation \[\boxed{k(x^2) = \bigl(k(x)\bigr)^3}.\] From \(k(16) = 16 = \bigl(k(4)\bigr)^3\) we get \[k(4) = \sqrt[3]{16}.\] Similarly, from \(\bigl(k(4)\bigr)^3 = k(16)\) we get \[k(2) = 16^{1/9... | Contradiction | MATH | 3.0 | train | 16 | |
Consider a quadrilateral in a Cartesian coordinate with vertices at the points \((1, 3)\), \((1, 1)\), \((2, 1)\) and \((2006, 2007)\). Find the area of this quadrilateral in square units. | Consider a quadrilateral in a Cartesian coordinate with vertices at the points \((1, 3)\), \((1, 1)\), \((2, 1)\) and \((1000,-1995)\). Find the area of this quadrilateral in square units. | \((1000,-1995)\) lies on the line formed by \((1,3),(2,1)\), so the four points cannot form a quadrilateral. | Contradiction | MATH | 4.0 | train | 3008 | |
Consider a parabola \(y = 9 - x^2 \). A point \(M\) is on this parabola, and the parabola's tangent line at \(M\) intersects the x-axis and y-axis, forming a triangular region with these axes whose area is 25. What is the \(x\)-coordinate of \(M\) if x is a positive rational number? | Consider a parabola \(y = 9 - x^2 \). A point \(M\) is on this parabola, and the parabola's tangent line at \(M (a, 9 - a^2) \), where \(a\) is a positive rational number, intersects the x-axis and y-axis, forming a triangular region with these axes whose area is 26. What is the value of \(a\)? | Slope at \(x=a\): \(m=-2a\).
Tangent line at \(M(a,9-a^2)\): \( y-(9-a^2)=-2a(x-a)\quad\Rightarrow\quad y=-2ax+(a^2+9). \) Intercepts: \( x\text{-int}: \; \frac{a^2+9}{2a},\qquad y\text{-int}: \; a^2+9. \) Triangle area: \( \frac12\left(\frac{a^2+9}{2a}\right)(a^2+9)=26
\;\Longrightarrow\;
\frac{(a^2+9)^2}{4a}=26... | Contradiction | MATH | 4.0 | train | 1 | |
Consider the matrix \(\mathbf{B} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}.\) There are positive real numbers \(u\) and \(v\) such that
\[(u \mathbf{I} + v \mathbf{B})^2 = \mathbf{B}.\] Determine the ordered pair \((u, v).\) | Consider the matrix \(\mathbf{B} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}.\) There exist integers \(u\) and \(v\) such that \[(u \mathbf{I} + v \mathbf{B})^2 = \mathbf{B}.\] Determine the ordered pair \((u, v)\). | Solving for the equation would yield \[u^2 - v^2 = 1,\quad 2uv = 1,\quad uv = 1/2.\] So \(u, v\) cannot be integers. | Contradiction | MATH | 4.0 | train | \left( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right) | |
\(m, n\) are integers where \(m < n\). Given that
\[\sqrt{1 + \sqrt{21 + 12 \sqrt{3}}} = \sqrt{m} + \sqrt{n}\], what are the values of \(m, n\)? Express your answer in the form of ordered pair \((m, n)\). | \(m, n\) are integers where \(m < n\). Given that
\[\sqrt{1 + \sqrt{21 + 12 \sqrt{3}}} = \sqrt{m} - \sqrt{n}\], what are the values of \(m, n\)? Express your answer in the form of ordered pair \((m, n)\). | \[
\sqrt{1 + \sqrt{21 + 12\sqrt{3}}}
= \sqrt{m} - \sqrt{n},\quad m,n\in\mathbb{Z}^+.
\]
Squaring both sides gives
\[
\sqrt{21 + 12\sqrt{3}}
= (m + n - 1) - 2\sqrt{mn}.
\]
Squaring again yields
\[
21 + 12\sqrt{3}
= (m + n - 1)^2 + 4mn - 4(m + n - 1)\sqrt{mn}.
\]
Comparing the irrational parts... | Contradiction | MATH | 4.0 | train | (1,3) | |
Consider four positive integers: \(X\), \(Y\), \(X-Y\), and \(X+Y\). Each of these numbers is a prime number. Which of the following must be true about the sum of these four prime numbers:
- \(\bullet\) A. Is an even number.
- \(\bullet\) B. Is divisible by 3.
- \(\bullet\) C. Is divisible by 5.
- \(\bullet\) D... | Three successive odd integers that are all prime numbers, each greater than 200. What are the smallest possible values for these numbers? | There are no three consecutive odd numbers larger than 3 that can all be primes. If the first one cannot be divided by 3, then the remainder is either 1 or 2, indicating that either the second one is divisible by 3 or the third one. | Contradiction | MATH | 3.0 | train | E | |
Consider a triangle \(XYZ\) with side lengths \(XY = \sqrt{30}\), \(XZ = \sqrt{6}\), and \(YZ = \sqrt{15}\). There is a point \(P\) such that \(\overline{XP}\) bisects \(\overline{YZ}\), and \(\angle XPY\) is a right angle (\(P\) is not on \(YZ\); \(XP\) intersects \(YZ\) at some other point \(M\), and \(M\) is the mid... | Consider a triangle \(XYZ\) with side lengths \(XY = \sqrt{30}\), \(XZ = \sqrt{6}\), and \(YZ = \sqrt{15}\). \(P\) is the midpoint of \(\overline{YZ}\), and \(\angle XPY\) is a right angle. What is the ratio of the area of triangle \(XPY\) to the area of triangle \(XYZ\)? | Since \(P\) is the midpoint of \(\overline{YZ}\), and \(\angle XPY\) is a right angle, triangle \(XPY\) is congruent to \(XPZ\), so \(\overline{XY} = \overline{XZ}\), which contradicts the given lengths. | Contradiction | MATH | 4.0 | train | \frac{27}{38} | |
Consider four consecutive positive integers whose multiplication ends with a digit of 4. Given that this multiplication exceeds 1000, determine the sum of the smallest set of these four numbers. | Consider four consecutive primes whose multiplication ends with a digit of 4. Given that this multiplication exceeds 1000, determine the sum of the smallest set of these four numbers. | All primes, except for 2, are odd. You can eliminate the case of 2 since the product of 2, 3, 5, 7 does not end with 4. The product of four odd number should be odd, so it cannot end with 6. | Contradiction | MATH | 4.0 | train | 30 | |
Consider four distinct numbers \( w \), \( x \), \( y \), and \( z \) such that when you add them in pairs, you get the results 10, 18, 19, 20, 21, and 29. Find these four numbers in ascending order. (Use a comma followed by a space to separate each number) | Consider four distinct numbers \( w \), \( x \), \( y \), and \( z \) such that when you add them in pairs, you get the results 10, 18, 19, 21, 21, and 29. Find these four numbers in ascending order. (Use a comma followed by a space to separate each number) | Label the four numbers so that \(w < x < y < z\). The smallest pair‐sum is \(w + x\) and the largest is \(y + z\). The remaining four sums each combine one element from \(\{w, x\}\) with one from \(\{y, z\}\). Among those four, \(x + z\) is the largest, and the others—\(w + z\), \(x + y\), and \(w + y\)—are all strictl... | Contradiction | MATH | 4.0 | train | 4, 6, 14, 15 | |
Consider base-10 numbers 36, 64, and 81. Each of these numbers can be expressed in other bases such that they appear in the same digits \(\triangle\Box\Box\), where \(\triangle\) and \(\Box\) are two different digits ranging from 0 to 9. What are the three digits \(\triangle\Box\Box\)? | Consider base-10 numbers 36, 64, and 81. Each of these numbers can be expressed in other bases such that they appear in the same digits \(\triangle\Box\Box\), where \(\triangle\) and \(\Box\) are two different digits ranging from 1 to 9. What are the three digits \(\triangle\Box\Box\)? | Note that only the bases 4, 5, and 6 use three digits to represent the number 36. (This follows from \(b^2\leq 36<b^3\), which expresses the condition that 36 has 3 digits in base \(b\)). The representations are \(100_6\), \(121_5\) and \(210_4\), only one of which fits the form \(\triangle\Box\Box\), but it involves 0... | Contradiction | MATH | 4.0 | train | 100 | |
Consider three vectors \(\mathbf{p},\) \(\mathbf{q},\) and \(\mathbf{r}\) whose magnitudes are \(\frac{1}{\sqrt{2}},\) \(\frac{1}{\sqrt{3}},\) and \( \frac{1}{\sqrt{6}}\), respectively. We also have \(\|\mathbf{p} + \mathbf{q} + \mathbf{r}\| = 1,\) and \[\mathbf{p} = m (\mathbf{q} \times \mathbf{r})\] for some scalar \... | Consider three 3-dimensional vectors \(\mathbf{p},\) \(\mathbf{q},\) and \(\mathbf{r}\) whose magnitudes are \(\frac{1}{\sqrt{2}},\) \(\frac{1}{\sqrt{3}},\) and \( \frac{1}{\sqrt{6}}\), respectively. We also have \(\|\mathbf{p} + \mathbf{q} + \mathbf{r}\| = 1,\) and \[\mathbf{p} \cdot \mathbf{q} = 0\]. What is the angl... | One can imagine the vectors as rigid rods: rods \(\mathbf{p}\) and \(\mathbf{q}\) are joined at their common endpoint, forming a right angle there. A third rod, \(\mathbf{r}\), is connected at the joint with \(\mathbf{q}\). The endpoints of \(\mathbf{p}\) (its tail) and \(\mathbf{r}\) (its head) are fixed in space, so ... | Missing conditions | MATH | 4.0 | train | 90^\circ | |
In triangle \(XYZ\), sides \(XY\) and \(XZ\) are both 10 units long, and side \(YZ\) measures 12 units. Point \(P\) is located strictly between \(X\) and \(Y\) on line segment \(\overline{XY}\), and point \(Q\) is located strictly between \(X\) and \(Z\) on line segment \(\overline{XZ}\) such that \(XP = PQ = QZ\). Det... | In triangle \(XYZ\), sides \(XY\) and \(XZ\) are both 10 units long, and side \(YZ\) measures 12 units. Point \(P\) is located strictly between \(X\) and \(Y\) on line segment \(\overline{XY}\), and point \(Q\) is located strictly between \(X\) and \(Z\) on line segment \(\overline{XZ}\) such that \(XP = QZ\). Determin... | the single condition \(XP=QZ\) leaves one free parameter, so \(PQ\) can take infinitely many values. | Missing conditions | MATH | 4.0 | train | \frac{250}{39} | |
Let \(M, N, P, Q\) be four points on a circle.
\(\overline{MQ}\) is a diameter with length \(36\) inches.
\(\overline{MN} \parallel \overline{PQ}\).
\(\angle NMQ = 50^\circ\).
Determine the length of the minor arc \(\widehat{MN}\) in inches, in terms of \(\pi\).
| Let \(M, N, P, Q\) be four points on a circle.
\(\overline{MQ}\) is a diameter with length \(36\) inches.
\(\overline{MN} \parallel \overline{PQ}\).
\(\angle NMQ = 100^\circ\).
Determine the length of the minor arc \(\widehat{MN}\) in inches, in terms of \(\pi\).
| The angle \(\angle NMQ\) cannot be \(100^{\circ}\). Since \overline{MQ} is a diameter, \(\angle MNQ=90^{\circ}\), both \(\angle NMQ\) and \(\angle NQM\) are acute angles. | Contradiction | MATH | 4.0 | train | 8\pi | |
A circular garden has a diameter of 16 meters. By how many meters should the radius be shortened to reduce the garden's area by \(48\pi\) square meters? | If a circular garden has a diameter of 16 meters, by how many meters should the radius be shortened to reduce the garden's area by \(100\pi\) square meters? | Let the original radius be \(R = 8\). Suppose you decrease the radius by \(x\), so the new radius is \(R - x = 8 - x\). The decrease in area is
\[
\Delta A
= \pi\bigl(R^2 - (R - x)^2\bigr)
= \pi\bigl(8^2 - (8 - x)^2\bigr)
= \pi\bigl(64 - (64 - 16x + x^2)\bigr)
= \pi\,(16x - x^2).
\]
Setting this ... | Contradiction | MATH | 4.0 | train | 4 | |
Compute the length of the segment joining the two vertices of the hyperbola defined by \( 9a^2 + 54a - b^2 + 10b + 55 = 0.\) | Compute the length of the segment joining the two vertices of the hyperbola defined by \(9a^2 + 54a + b^2 + 10b + 55 = 0.\) | Upon completing the square, the modified equation \(9a^2 + 54a + b^2 + 10b + 55 = 0\) represents an ellipse, not a hyperbola. | Contradiction | MATH | 3.0 | train | \frac{2}{3} | |
A square sheet of cardboard has an isosceles right triangle cut out from each of its corners, resulting in a rectangular shape. Given that the diagonal of the rectangle measures 12 units, determine the total area of all four triangles that were removed, expressed in square units. | A square sheet of cardboard has an isosceles right triangle cut out from each of its corners, resulting in a rectangular shape. If the side length of the original square is 12 units, determine the total area of all four triangles that were removed, expressed in square units. | The areas cut off to form a rectangle must consist of two groups of congruent triangles.
Let the side length of the original square be \(y\), and let the leg length of one of the cut-off triangles be \(x\).
Then there will be two triangles with leg length \(x\) and two triangles with leg length \(y - x\).
The total ... | Missing conditions | MATH | 4.0 | train | 72 | |
Determine the count of elements in the sequence \(\{3,13,23,33, \ldots\}\) that can be expressed as the subtraction of one prime number from another. | Suppose \(a\) is the sum of elements in the arithmetic sequence \(\{5,15,25,35, \ldots\}\) that can be expressed as the subtraction of one prime number from another. Then compute \(\log{(-a^2+2a+2025)}\) | Every term of the sequence is of the form \(10k + 5\), which is always odd.
Suppose \(10k + 5 = p - q\), where \(p\) and \(q\) are primes and \(p > q\).
If both \(p\) and \(q\) are odd, then \(p - q\) is even — which contradicts the fact that \(10k + 5\) is odd.
Therefore, \(q\) must be 2. Then:
\\[
10k + 5... | Contradiction | MATH | 4.0 | train | 1 | |
Alex, Ben, Chloe, and Dave decide to measure their weights in pairs. Alex and Ben together weigh 260 pounds, Ben and Chloe together weigh 245 pounds, and Chloe and Dave together weigh 270 pounds. What is the combined weight of Alex and Dave? | Alex, Ben, Chloe, and Dave decide to measure their weights in pairs. Alex and Ben together weigh 260 pounds, Ben and Chloe together weigh 245 pounds, and Chloe and Dave together weigh 270 pounds. How much does Alex weigh? | 3 equations cannot solve for 4 unknown numbers | Contradiction | MATH | 3.0 | train | 285 | |
Determine a matrix \(\mathbf{N} = \begin{pmatrix} x & y \\ 0 & z \end{pmatrix}\) such that when cubed, it equals
\[
\mathbf{N}^3 = \begin{pmatrix} 8 & -57 \\ 0 & 27 \end{pmatrix}.
\] | Determine a matrix \(\mathbf{N} = \begin{pmatrix} x & y \\ 0 & z \end{pmatrix}\) such that its cube results in
\[
\mathbf{N}^3 = \begin{pmatrix} 8 & -57 \\ 1 & 27 \end{pmatrix}.
\] | Because any matrix of the form \(\mathbf{N}=\begin{pmatrix}x&y\\0&z\end{pmatrix}\) is upper triangular, every power of it is also upper triangular. In particular, the \((2,1)\) entry of \(\mathbf{N}^3\) must be \(0\).
Compute it explicitly to see this:
\(\mathbf{N}^2=\begin{pmatrix}x^2 & y(x+z)\\ 0 & z^2\end{... | Contradiction | MATH | 3.0 | train | \begin{pmatrix} 2 & -3 \\ 0 & 3 \end{pmatrix} | |
Consider three distinct positive integers \( p \), \( q \), and \( r \) such that \( p < q < r \). They are selected in a way that the following system of equations holds:
\[
2u + v = 2003 \quad \text{and} \quad v = |u-p| + |u-q| + |u-r|
\]
This system has exactly one solution. Note that \(u,v\) are real number... | Consider three distinct positive integers \( p \), \( q \), and \( r \) such that \( p < q < r \). They are selected in a way that the following system of equations holds:
\[
4u + v = 2003 \quad v = |u-p| + |u-q| + |u-r|
\]
This system has exactly two solutions. Note that \(u,v\) are real numbers. Determine th... | Set \(f(u)=|u-p|+|u-q|+|u-r|\).
From \(4u+v=2003\) and \(v=f(u)\) we get
\(f(u)+4u=2003.\)
Because \(p<q<r\), \(f(u)\) is piecewise linear with slopes \(-3,-1,1,3\) on the intervals \((-\infty,p], [p,q], [q,r], [r,\infty)\).
Hence \(f(u)+4u\) has slopes
\((-3)+4=1,\quad (-1)+4=3,\quad 1+4=5,\quad 3+4... | Contradiction | MATH | 4.0 | train | 1002 | |
Determine the smallest positive integer that is a multiple of 72 and possesses exactly 16 divisors. | Determine the smallest positive integer that is a multiple of 72 and possesses exactly 14 divisors. | The prime factorization of 72 is \(2^3\times 3^2\), which has \((3+1)(2+1)=12\) factors.
If we multiply by 2, we get \(2^4\times 3^2\), which has \((4+1)(2+1)=15\) factors. Multiplying by any higher power of 2 gives a product with more than 16 positive factors.
Multiplying by 3 gives \(2^3\times 3^3\), which has \(... | Contradiction | MATH | 3.0 | train | 216 | |
Consider the quartic equation given by
\[
py^4 + qy^3 + ry^2 + sy + t = 0,
\]
where all coefficients are integers. The equation has roots at \(-2\), \(5\), \(9\), and \(-1/3\). If \(t\) is a positive integer, determine the smallest value that \(t\) can take. | Consider the quartic equation given by
\[
py^4 + qy^3 + ry^2 + sy + t = 0,
\]
where all coefficients are integers. The equation has roots at \(-2\), \(5\), \(9\), and \(-1/3\). If \(t\) is a positive integer, determine the largest value that \(t\) can take. | By taking
\[
p = 3k \quad (k \in \mathbb{Z}^+),
\]
the quartic can be written as
\[
P(y) = k\,(y + 2)(y - 5)(y - 9)(3y + 1),
\]
and then
\[
t = 30p = 30 \cdot (3k) = 90k.
\]
Since \(k\) can be any positive integer, \(t = 90k\) can be made arbitrarily large. Hence there is no largest possib... | Contradiction | MATH | 4.0 | train | 90 | |
Determine all values of \( y \) that satisfy the equation:
\[
\sqrt{y + 3 - 4 \sqrt{y - 1}} + \sqrt{y + 8 - 6 \sqrt{y - 1}} = 1.
\] | Determine all values of \( y \) that satisfy the equation:
\[
\sqrt{y + 3 - 4 \sqrt{y - 1}} + \sqrt{y + 8 - 6 \sqrt{y - 2}} = 1.
\] | \[
\begin{aligned}
&y + 3 - 4\sqrt{y - 1} \ge 0,\quad y + 8 - 6\sqrt{y - 2} \ge 0, \\
&y - 1 \ge 0,\quad y - 2 \ge 0 \;\Longrightarrow\; y \ge 2. \\
&y + 3 - 4\sqrt{y - 1} = (\sqrt{y - 1} - 2)^2, \\
&y + 8 - 6\sqrt{y - 2} = (\sqrt{y - 2} - 3)^2 + 1. \\
&\sqrt{(\sqrt{y - 1} - 2)^2} + \sqrt{(\sqrt{y - 2} - 3)... | Contradiction | MATH | 4.0 | train | [5,10] | |
Consider a scenario where a positive integer \( y \) satisfies the equation
\[
1^{y+2} + 2^{y+1} + 3^{y-1} + 4^y = 1170.
\]
Determine the value of \( y \). | Consider a scenario where a positive integer \( y \) satisfies the equation
\[
1^{y+2} + 2^{y+4} + 3^{y+3} + 4^{y-1} = 1170.
\]
Find the value of \( y \). | Let
\[
S(y) = 1^{y+2} + 2^{y+4} + 3^{y+3} + 4^{y-1}
\]
We have \(1^{y+2} = 1\), so the equation becomes
\[
2^{y+4} + 3^{y+3} + 4^{y-1} = 1169.
\]
Checking small positive \(y\):
- \(y = 1\): \(2^5 + 3^4 + 4^0 = 32 + 81 + 1 = 114\)
- \(y = 2\): \(2^6 + 3^5 + 4^1 = 64 + 243 + 4 = 311\)
-... | Contradiction | MATH | 4.0 | train | 5 | |
Consider a triangle \(XYZ\) where \(\angle X = 45^\circ\). A point \(Q\) is located on the segment \(\overline{YZ}\) such that \(QY = 3\) and \(QZ = 5\). Let \(M\) be the center of the circle that passes through points \(X\), \(Y\), and \(Z\). Find the distance from \(M\) to \(Q\). | Consider a triangle \(XYZ\) where \(\angle Z = 45^\circ\). A point \(Q\) is located on the segment \(\overline{YZ}\) such that \(QY = 3\) and \(QZ = 5\). Let \(M\) be the center of the circle that passes through points \(X\), \(Y\), and \(Z\). Find the distance from \(M\) to \(Q\). | With only \(\angle Z = 45^\circ\) and the division \(YQ = 3\), \(QZ = 5\) on \(YZ\), the position of \(X\) (and hence of the circumcenter \(M\)) is not fully fixed, so \(MQ\) would still depend on the length \(XZ\),
\[
MQ = \sqrt{1 + \left(\tfrac{XZ}{\sqrt{2}} - 4\right)^2}.
\] | Missing conditions | MATH | 4.0 | train | \sqrt{17} | |
Determine the radius of the circle described by the equation
\[
a^2 - 4a + b^2 - 6b - 36 = 0.
\] | Determine the radius of the circle described by the equation
\[
a^2 - 4a + b^2 - 6b + ab - 36 = 0.
\] | There is an \(ab\) term in the equation, so it is an ellipse, not a circle, and hence does not have radius. | Contradiction | MATH | 4.0 | train | 7 | |
Consider the polynomial equation \(y^3 + 8y^2 - 4y + a = 0\). This equation has three solutions, and one of these solutions is equal to the sum of the other two solutions. Determine the value of the constant \(a\). | Consider the polynomial equation \(y^3 + 8y^2 - 4y + a = 0\). This equation has three solutions, where one solution is equal to the sum of the other two solutions, and another solution is equal to the product of the other two solutions. Determine the value of the constant \(a\). | Let the three roots be \(r_1, r_2, r_3\). By Vieta’s formulas for \(y^3 + 8y^2 - 4y + a = 0,\) we have
\[
r_1 + r_2 + r_3 = -8, \quad
r_1r_2 + r_2r_3 + r_3r_1 = -4, \quad
r_1r_2r_3 = -a.
\]
WLOG, assume \(r_1 = r_2 + r_3\) and \(r_2 = r_1r_3\). Combine these with the first Vieta equation.
One can obt... | Contradiction | MATH | 4.0 | train | -80 | |
Determine the count of the initial 200 positive integers that leave a remainder of 1 when divided by 9. | Among the initial \(n\) positive integers, there are exactly 100 numbers that leave the same remainder as \(n+1\) when divided by 9. Identify the 10th smallest possible value of n. | There are only nine values of \(n\) that satisfy the given conditions (900-908), so there is no tenth smallest \(n\). All the \(k\) that satisfy \(k\equiv n+1\pmod9\) can be represented as \(k=n+1-9m\). Since there are 100 such \(k\) among \(1,2,\dots,n\), \(m\) can be \(1,2,\dots,100\), and \(m=101\) would give \(k\le... | Contradiction | MATH | 4.0 | train | 23 | |
Consider an infinite sequence of positive integers \( n \) that satisfy the equation
\[\cos^2 (n^2 + 36)^\circ = 1.\]
Identify the two smallest values of \( n \) that solve this equation, and list them separated by commas. | Determine the second smallest positive integer \( n \) such that \( \cos^2 (n^2 + c)^\circ = 1 \) for a fixed constant \(c\), given that the smallest solution (the smallest such \(n\) ) is 12 and the 100th smallest solution is 114514. | For any solution \(k\), we must have \(k^2 + c = 180m,\quad m\in\mathbb Z,\) since \(\cos^2\theta=1\) iff \(\theta\equiv0\pmod{180^\circ}\). Since \(k=12\) and \(k=114514\) satisfiy the equation above, there exist integers \(m_1,m_2\) with
\(
12^2 + c = 180 m_1,\qquad
114514^2 + c = 180 m_2.
\)
Subtracting gi... | Contradiction | MATH | 3.0 | train | 46009 | |
Consider the number 49, which can be expressed as a sum of smaller perfect squares in several ways. One such expression involves six terms: \(25 + 9 + 9 + 4 + 1 + 1\). If each term is a perfect square ranging from 1 to 48, determine the minimum number of perfect square terms less than 49 that sum up to 49. | Consider the number 49, which can be expressed as sums and differences of smaller perfect squares in various configurations. One such expression involves seven terms: \(25 + 25 - 16 + 9 + 4 + 1 + 1\). If each term is a perfect square ranging from 1 to 48, determine the maximum number of such terms (with both additions ... | Since \(1^2 = 1\) is allowed, one can keep adding and subtracting 1 indefinitely while maintaining the total sum at 49. This leads to arbitrarily large numbers of terms. | Contradiction | MATH | 4.0 | train | 3 | |
Consider the polynomial \( q(y) = y^3 + dy^2 + ey + f \), where \( d \), \( e \), and \( f \) are complex numbers. Assume that the following conditions hold true:
\[
q(2009 + 9002\pi i) = q(2009) = q(9002) = 0
\]
Determine how many zeros of the polynomial \( y^{12} + dy^8 + ey^4 + f \) are not real numbers. | Consider the polynomial \( q(y) = y^3 + dy^2 + ey + f \), where \( d \), \( e \), and \( f \) are complex numbers. It is given that:
\[
q(2009 + 9002\pi i) = q(2009) = 0
\]
If the polynomial \( y^{12} + dy^8 + ey^4 + f \) has 6 distinct non-real zeros, identify 4 possible sets of values for \( d \), \( e \), an... | Let the new polynomial be \(Q(y)\), and three roots of \(q(y)\) as
\(r_1 = 2009 + 9002\pi i\),
\(r_2 = 2009\),
and an unknown root \(r_3\).
Since \(Q(y) = q(y^4)\), \(Q(y)\) has four non-real roots coming from \(y^4 = r_1\), and two purely imaginary roots \(\pm i \cdot 2009^{1/4}\), coming from \(y^4 = r_... | Contradiction | MATH | 4.0 | train | 8 | |
Assuming
\[\sin \left( \alpha + \frac{\pi}{4} \right) = \frac{1}{3}\]
calculate \(\sin(2\alpha)\). | Assuming \(\alpha\) is a multiple of \(\pi/4\), if
\(
\sin \left( \alpha + \frac{\pi}{4} \right) = \frac{1}{3},
\)
calculate \(\sin(2\alpha)\). | Since \(\alpha\) is an integer multiple of \(\frac{\pi}{4}\), \(\alpha + \frac{\pi}{4}\) is also a multiple of \(\frac{\pi}{4}\). But \(\sin(k\frac{\pi}{4})\) for any integer \(k\) only takes values within \(0, \pm\frac{1}{\sqrt{2}}, \pm1\), never \(\frac{1}{3}\). So the equation has no solution, and \(\sin 2\alpha\) i... | Contradiction | MATH | 3.0 | train | -\frac{7}{9} | |
Consider a square with the points \((1, -2)\) and \((-4, 10)\) that are adjacent to each other. Determine the perimeter around the square. | Consider a rectangle with the points \((1, 7)\) and \((4, 10)\) that are diagonally opposite corners. If one of the rectangle's edges measures \(\frac{\sqrt{2}}{k}\), where \(k\) is a positive integer, determine all potential points where the rectangle crosses the \(x\)-axis. | All four vertices must lie on the circle having the segment from \((1,7)\) to \((4,10)\) as a diameter. Its center is \((2.5,8.5)\) and its radius is \(\frac{3\sqrt2}{2}\), so its lowest \(y\) coordinate is \(8.5-\frac{3\sqrt2}{2}\approx6.38>0\). Therefore the circle never meets the \(x\) axis, and no such rectangle ca... | Contradiction | MATH | 3.0 | train | 52 | |
Determine the value of \(a+b+c\) given the equations
\[
6a + 5b - 2c = -4,
\quad
4a + 5b + 12c = 54.
\] | Determine the value of \(a + b + e^c\) given the equations
\[
6a + 5b - 2e^c = -4,
\quad
4a + 4b + 13e^c = 54.
\] | Let \(t = e^c,\; S = a + b + t.\) Then the given equations and the target sum can be written as dot-products in \(\mathbb{R}^3\):
\[
(6,\,5,\,-2)\cdot(a, b, t) = -4, \quad
(4,\,4,\,13)\cdot(a, b, t) = 54, \quad
(1,\,1,\,1)\cdot(a, b, t) = S.
\]
The first two row-vectors span a 2-dimensional subspace of ... | Contradiction | MATH | 4.0 | train | 5 | |
Consider three positive real numbers \(x,\) \(y,\) and \(z\). Determine the smallest possible value of the expression
\[\frac{x + y}{z} + \frac{x + z}{y} + \frac{y + z}{x}.\] | Consider three nonzero real numbers \(x,\) \(y,\) and \(z\). Determine the smallest possible value of the expression
\[\frac{x + y}{z} + \frac{x + z}{y} + \frac{y + z}{x}.\] | The expression is unbounded below. For instance, set \( x = y = -1,\quad z = \varepsilon > 0 \).
Then
\[
\frac{x + y}{z} = \frac{-2}{\varepsilon} \longrightarrow -\infty \quad (\varepsilon \to 0^+),
\]
while the other two terms remain finite. Hence the total sum can be made arbitrarily negative. | Contradiction | MATH | 3.0 | train | 6 | |
Determine a pair of values \((x, y)\) that satisfy the following equations:
\[
\begin{aligned}
5x &= -7 - 2y, \\
3x &= 4y - 25.
\end{aligned}
\] | Determine a pair of values \((x, y)\) that satisfy the following equations:
\[
\begin{aligned}
5x &= -7 - 2y^2, \\
3x^2 &= 4y - 25.
\end{aligned}
\] | Solve the first equation for
\[
x = \frac{-7 - 2y^2}{5}.
\]
Substituting into
\[
3x^2 = 4y - 25
\]
gives
\[
12y^4 + 84y^2 - 100y + 772 = 0.
\]
Call
\[
L(y) = 12y^4 + 84y^2 + 772, \qquad R(y) = 100y.
\]
Notice that \(L(y)\) can be considered as a quadratic in \(w\),
\[
L(w... | Contradiction | MATH | 3.0 | train | (-3,4) | |
Consider three complex numbers \( x, y, z \) that satisfy the following conditions:
\[
xy + 4y = -16, \\
yz + 4z = -16, \\
zx + 4x = -16.
\]
Determine all possible values for the product \( xyz \), listing them separated by commas. | Consider three positive real numbers \( x, y, z \) that satisfy the following conditions: \( xy + 4y = -16, yz + 4z = -16, zx + 4x = -16, x = y = z. \) Determine all possible values for the product \( xyz \), separated by commas. | Substituting \(x=y\) into \(xy+4y=-16\) yields \(x^2+4x+16=0\). The discriminant is \(\Delta = 4^2 - 4\cdot1\cdot16 = 16 - 64 = -48 < 0\). Thus, there is no real solution. | Contradiction | MATH | 4.0 | train | 64 | |
Consider a triangle \(XYZ\) where the side lengths satisfy the equation:
\[ x^4 + y^4 + z^4 = 2z^2 (x^2 + y^2).\]
Determine the possible measures of \(\angle Z\) in degrees, and list them separated by commas. | Consider a triangle \(XYZ\) where the side lengths satisfy the equation:
\[ x^4 + y^4 + z^4 = 2x^2 (z^2 + y^2).\]
Determine the measure of \(\angle Z\) in degrees, and list it separated by commas. | The given conditions are not enough to compute the value of angle \(Z\). Substituting the Law of Cosines \(z^2 = x^2 + y^2 - 2xy\cos Z\) into the given identity \(x^4 + y^4 + z^4 = 2x^2(y^2 + z^2)\) to obtain
\[
x^4 + y^4 + \bigl(x^2 + y^2 - 2xy\cos Z\bigr)^2
= 2x^2\bigl[y^2 + \bigl(x^2 + y^2 - 2xy\cos Z\bigr)\bigr]... | Missing conditions | MATH | 4.0 | train | 45^\circ, 135^\circ | |
Consider a road described by the equation \(y = 4x - 7\). Another road, represented by the equation \(y = ax + b\), intersects the first road at a right angle at the point \((2,1)\). Determine the \(y\)-value of the location on this second road where the \(x\)-value is 6. | Consider a road described by the equation \(y = 4x - 8\). Another road, represented by the equation \(y = ax + b\), intersects the first road at a right angle at the point \((2,1)\). Determine the \(y\)-value of the location on this second road where the \(x\)-value is 6. | Line \(l\) has the equation \(y = 4x - 8\), and line \(m\) has the equation \(y = ax + b\). The question claims they intersect at \((2,1)\), but this point does not lie on \(l\) because substituting \(x = 2\) into \(y = 4x - 8\) gives \(y = 0 \neq 1\). | Contradiction | MATH | 4.0 | train | 0 | |
Consider a quadratic equation given by \(\frac12y^2 + 99y + k = 0\). If the solutions to this equation are \(y = -99 + \sqrt{8001}\) and \(y = -99 - \sqrt{8001}\), determine the value of \(k\). | Consider a quadratic equation given by \(y^2 + 99y + k = 0\). If the solutions to this equation are \(y = -99 + \sqrt{8001}\) and \(y = -99 - \sqrt{8001}\), determine the value of \(k\). | The sum of the two roots must equal \(-b/a=-99\) using Vieta's formula, but the sum of two given roots is -198, leading to contradiction. | Contradiction | MATH | 4.0 | train | 900 | |
Determine the maximum value of \( y \) that satisfies the equation \(\lfloor y \rfloor = 5 + 100 (y - \lfloor y \rfloor)\), where the fractional part of \( y \) is defined as \( y - \lfloor y \rfloor \). | Assuming there are exactly three integer values of \( y \) that satisfy the equation \(\lfloor y \rfloor = 5 + b (y - \lfloor y \rfloor)\), where the fractional part of \( y \) is \( y - \lfloor y \rfloor \), identify all possible integer values for \( b \) | There is only one integer solution 5. For any integer \(y\),\(y - \lfloor y \rfloor)\) is always 0, so the equation becomes \(\lfloor y \rfloor = 5\). So 5 is the only integer solution. Therefore, it is impossible to have three integer solutions. | Contradiction | MATH | 4.0 | train | 104.99 |
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IDM-batch2
Second batch of IDM math trap items, in the same schema as GYASBGFUHAADSGADF/IDM-batch1.
Dataset summary
- Rows: 152
- File:
upload_batch2.jsonl - Trap types: Contradiction (130), Missing conditions (22)
- Splits (field
split): train (139), test (13)
Fields
| Field | Description |
|---|---|
original |
Original math problem |
trap |
Trap / modified problem |
annotation |
Explanation of the trap |
trap type |
Trap category |
source |
Problem source |
level |
Difficulty level |
split |
train or test |
answer |
Answer to the original problem |
additional Information |
Optional notes |
Load
from datasets import load_dataset
ds = load_dataset("GYASBGFUHAADSGADF/IDM-batch2")
print(ds)
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