problem stringlengths 5 2.09k | answer stringlengths 1 229 |
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Let \( h(x) \) be an even function, and let \( k(x) \) be an odd function. Is \( k(k(h(k(h(x)))))) \) even, odd, or neither? Enter 'odd', 'even', or 'neither'. | odd |
A strictly increasing sequence of positive integers \( a_1 \), \( a_2 \), \( a_3 \), \(\dots\) has the property that for every positive integer \( k \), the subsequence \( a_{2k-1} \), \( a_{2k} \), \( a_{2k+1} \) is geometric and the subsequence \( a_{2k} \), \( a_{2k+1} \), \( a_{2k+2} \) is arithmetic. Find \( a_{23... | 25165824 |
Calculate the area of a trapezoid with bases of 7 cm and 4 cm, and a height of 6 cm. What is the number of square centimeters in the area? | 33 |
Let \(\lambda\) be a constant, \(0 \le \lambda \le 4,\) and let \(f : [0,1] o [0,1]\) be defined by \(f(x) = \lambda x(1 - x)\). Find the values of \(\lambda,\) \(0 \le \lambda \le 4,\) for which there exists an \(x \in [0,1]\) such that \(f(x)
eq x\) but \(f(f(x)) = x.\) | 2 |
Let \( h(x) \) be the function defined by \( h(x) = x^3 - 39x^2 + 580x - 1745 \), and let \( k(x) = h(x + 7) \). Compute the sum of the roots of \( k(x) \). | 39 |
Let \( h(x) \) be an even function, and let \( k(x) \) be an odd function. What is \( k(k(h(k(h(x)))))) \)? Enter 'odd', 'even', or 'neither'. | odd |
Let \( Q(x) \) be a monic polynomial of degree 3. Suppose that \( Q(x) \) has remainder \( R(x) \) when it is divided by \( (x - 1)(x - 4) \), and remainder \( 2R(x) \) when it is divided by \( (x - 2)(x - 3) \). Given that \( Q(0) = 6 \), find \( Q(5) \). | 131 |
A bakery has enough ingredients to make 20 loaves of bread. If the bakery decides to make smaller loaves, how many loaves should they make to feed 15 people if each person is expected to eat the same amount of bread? | 20 |
In the small village of Fluttershade, the local currency system is quite peculiar. Three Blinks are worth the same as five Pivots, and two Pivots are equivalent to seven Squints. If a farmer has 42 Squints, how many Blinks will he receive when exchanged to the local currency? | 7.2 |
What power of 3 is equal to 27? Express your answer as a common fraction. | 3 |
Suppose that $WXYZ$ is a trapezoid in which $\overline{WX}||\overline{YZ}$. Given $\overline{XZ}\perp\overline{WZ}$, $\overline{XZ}$ bisects angle $\angle WXY$, and $[WXYZ]=56$, then compute $[ riangle XWZ]$. | 28 |
A spinner is divided into 8 equal sections numbered 1 through 8. The spinner is spun twice. The first number is divided by 3 to determine the column, and the second number is divided by 4 to determine the row. The column and row then meet to place a checker on the checkerboard. What is the probability that the checker ... | \dfrac{1}{2} |
Let \( b \) be a positive real number such that all the roots of the polynomial \( x^3 + bx^2 + bx + 1 = 0 \) are real. Find the smallest possible value of \( b \). | 3 |
Let $P$ and $Q$ denote the centroid and circumcenter of triangle $XYZ,$ respectively. Let $R$ be the midpoint of $\overline{PQ}.$ Express $XR^2 + YR^2 + ZR^2$ in terms of the side lengths $x,$ $y,$ $z$ and the circumradius $r$ of triangle $XYZ.$ | 3r^2 |
The graphs of \( x^2 + y^2 + 6x - 24y + 72 = 0 \) and \( x^2 - y^2 + 6x + 16y - 46 = 0 \) intersect at four points. Compute the sum of the distances from these four points to the point \((-3, -2)\). | 36 |
Let \( T(x) \) be a quadratic polynomial with real coefficients satisfying \( x^2 + 3x + 4 \le T(x) \le 3x^2 + 6x + 7 \) for all real numbers \( x \), and suppose \( T(9) = 113 \). Find \( T(12) \). | 221 |
What is the sum of the first 12 positive multiples of 11? | 858 |
Let $b$ be a positive real number such that all the roots of \(x^3 + bx^2 + bx + 1 = 0\) are real. Find the smallest possible value of $b.$ | 3 |
Let \( q(x) = (x^2 - 1)p(x) - x \) be a polynomial of degree 7, where \( p(x) \) is a polynomial of degree 5 satisfying \( p(n) = rac{n}{n^2 - 1} \) for \( n = 2, 3, 5, 6, 7 \). Given that \( q(5) = 0 \), find \( q(8) \). | 0 |
Find all values of \( b \) that satisfy the equation \( b = \sqrt{15 - 4b} + 6 \). | No real solutions |
The distances from a point Q to five of the vertices of a regular dodecahedron are 4, 7, 10, 12, and 15. Find the distance from Q to the sixth vertex. | 12 |
Let \(P(x)\) be a monic polynomial of degree 3. Suppose that \(P(x)\) has remainder \(R(x) = ax + b\) when it is divided by \((x - 1)(x - 4)\), and remainder \(2R(x) = 2ax + 2b\) when it is divided by \((x - 2)(x - 3)\). Given that \(P(0) = 5\), find \(P(5)\). | 100 |
In the town of Mathville, the currency is based on unique units. One Coin is equal to 5 Pennies, and 2 Pennies are equal to 9 Quarters. How many Coins are equivalent to 18 Quarters? | 0.8 |
Find the area of the quadrilateral formed by the four points of tangency between the circle \(x^2 + y^2 = 3\) and the parabola \(y^2 = 12x\). | 12 |
Two numbers, \(a\) and \(b\) are selected at random from the interval \((0, 4)\). What is the probability that a triangle with sides of length 1, \(a\), and \(b\) exists? | \frac{9}{16} |
Find the least common multiple (LCM) of 36 and 48. | 144 |
A palindrome is a number that reads the same forwards and backwards. Find a set of three consecutive positive integers whose sum is a three-digit palindrome. If the sum is less than 320, determine the greatest possible value for the largest of the three integers in the set. | 95 |
In the diagram, \(AB\) is parallel to \(CD\). What is the measure of \(\angle BCD\) in degrees? | 90 |
In the currency system of Mathland, the conversion rates are as follows: 1 Gem is equal to 5 Jubos, and 9 Jubos are equal to 14 Kops. How many Gems are required to purchase 49 Kops? | 6.3 |
For what real values of $x$ is $-5 < x^4 + 4x^2 < 26$ satisfied? Express your answer in interval notation. | (-\sqrt{-2 + \sqrt{30}}, \sqrt{-2 + \sqrt{30}}) |
In 2000, a bottle of water could be purchased in France for 60 euros. The same water would have cost \$3.00 in the U.S. At the equivalent exchange rate between the euros and the dollar, how many dollars would be equivalent to 5000 euros? | 250 |
There are two boxes, A and B. Box A contains two red balls and three blue balls. Box B contains one red ball and two blue balls. A ball is drawn from Box A and placed into Box B. Then, a ball is drawn from Box B. What is the probability that the two balls drawn (one from Box A and one from Box B) are of the same color? | \frac{13}{20} |
Let \( A \) be the union of the set of all points inside an equilateral triangle with side length 3 units and the set of all points less than 0.5 units away from a point on the perimeter of the triangle. What, in units, is the perimeter of \( A \)? | 12 |
The product of integers 180 and $k$ is a perfect cube. What is the smallest possible positive value of $k$? | 150 |
If \(\sqrt[3]{3} = a + \cfrac{1}{b + \cfrac{1}{c + \cfrac{1}{d + \dotsb}}}\), where \(a,\) \(b,\) \(c,\) \(d\) are positive integers, compute \(b\). | 2 |
Let \( g(x) = 2x^8 + 3x^7 - 4x^6 + 5x^5 - 6x^4 + 7x^3 - 8x^2 + 9x - 10 \). Find the remainder when \( g(x) \) is divided by \( x^2 - 1 \). | 21x - 23 |
A truck is carrying a load that weighs 12,000 pounds. The truck can hold a maximum weight of 15,000 pounds. While driving, the truck picks up an additional load that weighs 30% of the initial load. How long can the truck continue to drive before the truck reaches its maximum weight limit? | 0 |
Find the greatest integer less than $(\sqrt{10} + \sqrt{6})^6.$ | 7000 |
Calculate the total surface area of a right circular cylinder with a radius of 5 cm and a height of 10 cm. Use the formula for the surface area of a cylinder: \( ext{Surface Area} = 2\pi r(r + h) \). | 150\pi cm^2 |
What real values of $x$ are not in the domain of the function $g(x) = rac{1}{|x^2 - 5x + 6| + |x^2 - 4x + 4|}?$ | 2 |
Find all solutions to \[\sin \left( an^{-1} (x) + an^{-1} (x^2)
ight) = rac{1}{3}.\] | within \boxed{} |
Let \(ABC\) be a triangle with centroid \(G\) and orthocenter \(H\). Let \(F\) be the midpoint of \(\overline{GH}\). Express \(AF^2 + BF^2 + CF^2\) in terms of the side lengths \(a\), \(b\), \(c\) and the circumradius \(R\) of triangle \(ABC\). | \frac{2}{9}(a^2 + b^2 + c^2) + R^2 |
Find all values of \( y \) that satisfy the equation \( y = \sqrt{13 - y} + 5 \). | 4 |
Let \(g(x) = \left\lfloor \left( rac{3}{4}
ight)^x
ight
floor\) be a function defined for all \(x\) in \([0, \infty)\). How many distinct values exist in the range of \(g(x)\)? | 2 |
The medians AD, BE, and CF of triangle ABC intersect at the centroid G. The line through G that is parallel to BC intersects AB and AC at M and N, respectively. If the area of triangle ABC is 144, then find the area of triangle AMN. | 16 |
Let \(\lambda\) be a constant, \(0 \le \lambda \le 4,\) and let \(f : [0,1] o [0,1]\) be defined by \(f(x) = \lambda x(1 - x)\). Find the values of \(\lambda,\) \(0 \le \lambda \le 4,\) for which there exists an \(x \in [0,1]\) such that \(f(x)
eq x\) but \(f(f(x)) = x\). | 2 |
Let \( E \) be the ellipse given by the equation \( 2x^2 + y^2 = 1 \). Find the value of the constant \( m \) if there exists a circle that passes through the foci of \( E \) and is tangent to \( E \) at two points on the \( x \)-axis. | \frac{\sqrt{2}}{4} |
Assume \(0 < r < 4\). Below are five equations for \(x\). Which equation has the largest solution \(x\)?
\( extbf{(A)}\ 4(1 + r)^x = 12\qquad extbf{(B)}\ 4(1 + r/15)^x = 12\qquad extbf{(C)}\ 4(1 + 2r)^x = 12\) \( extbf{(D)}\ 4(1 + \sqrt{r})^x = 12\qquad extbf{(E)}\ 4(1 + 1/r)^x = 12\) | E |
The parabola with equation \(y = 2x^2 - 4x + 3\) and vertex \((1, 1)\) is reflected about the line \(y = 1\). This results in the parabola with equation \(y = dx^2 + ex + f\). Express \(2 + (-4) + 3 + d + e + f\) in terms of \(k\). | 2 |
In the diagram, line segment MT is parallel to line segment NP. What is the measure of angle MNP in degrees? | answer |
A biased six-sided die has faces numbered 1, 2, 3, 4, 5, and 6. The probability of landing on the face 3 is less than 1/6, while the probability of landing on the face 4 is greater than 1/6. The probabilities of landing on the other faces (1, 2, 5, and 6) are each 1/6. When two such dice are rolled, the probability of ... | \frac{1}{6} |
Let \(a\) and \(b\) be positive real numbers such that \(a^3b = 6\). Find the minimum value of \(a^6 + b^6\). | 12 |
Consider a polynomial \( p(x) = 3x^3 + bx^2 + cx + d \) with integer coefficients, where the roots are distinct integers. Given that the leading coefficient is \(3\) and the constant term is \(24\), find the least possible value of \( |b| \). | 3 |
Let \( g(x) = 2x^6 + 3x^5 - 4x^4 + 5x^3 - 6x^2 + 7x - 8 \). Without using long division (which would be horrible!), find the remainder when \( g(x) \) is divided by \( x^2 - 1 \). | 15x - 16 |
Let \( w \) be a complex number such that \( |w| = 1 \). Find the maximum value of \( |2 + w| + |2 - w + w^2| \). | 5 |
Let \( g(x) = \left\lfloor x^{ rac{3}{2}}
ight
floor \) be a function defined for all \( x \geq 0 \). How many distinct values exist in the range of \( g(x) \)? | \infty |
If \(0 < r < 2\), which of the following equations for \(x\) has the largest solution? | A |
The average of Alice's, Bob's, and Charlie's ages is 8. Two years ago, Charlie was the same age as Alice is now. In two years, Bob's age will be \( rac{2}{3}\) of Alice's age at that time. How many years old is Charlie now? | 10.5 |
If a 12-inch ruler is divided into 6 equal parts, what fraction of the ruler is 2 inches? Express your answer as a common fraction. | \frac{1}{6} |
Let \(a, b,\) and \(c\) be positive real numbers such that \(abc = 8.\) Find the minimum value of \(a^2 + 2b^2 + 2c^2.\) | 20 |
Consider the sequence defined by the recurrence relation \( g(n + 1) = (-1)^{n + 1} n - 2g(n) \) for \( n \geq 1 \), and \( g(1) = g(1985) \). Compute the sum \( g(1) + g(2) + g(3) + \dots + g(1984) \). | 0 |
Find the greatest integer less than \((\sqrt{3} + \sqrt{2})^6\). (Do not use a calculator!) | 970 |
If \(\arccos x + \arccos 2x + \arccos 3x = \pi\), then \(x\) satisfies a cubic polynomial of the form \(ax^3 + bx^2 + cx + d = 0\), where \(a, b, c, d\) are integers, and \(a
eq 0\). Find the smallest possible value of \(|a| + |b| + |c| + |d|\). | 27 |
The lines \(l\) and \(m\) are parallel, and the triangle \(ABC\) with vertices at points \(A(0, 0)\), \(B(4, 0)\), and \(C(2, 6)\) is inscribed in a circle. The radius of the circle is \(6\) units. Find the length of segment \(CD\) given that \(D\) is the midpoint of line segment \(AB\). | 6 |
A local bakery baked 240 cupcakes for a school event. Each cake tray can hold 16 cupcakes. If the bakery wants to ensure that each tray is completely full, how many additional cupcakes should they bake to achieve this? | 0 |
The graphs of \(x^2 + y^2 - 4x - 16y + 64 = 0\) and \(x^2 - y^2 - 4x + 8y - 8 = 0\) intersect at four points. Compute the sum of the distances from these four points to the point \((2,4)\). | 8 |
Let \( w \) be a complex number such that \( |w| = 1 \). Find the maximum value of \( |w + 1| + |w^2 - w + 1| \). | 3 |
Find the shortest distance from the point (-3, 4) to the line 3x + 4y - 12 = 0. Express your answer in simplest radical form. | 1 |
What fraction of 3 feet is 4 inches? Express your answer as a common fraction. | \frac{1}{9} |
A store owner has 50 liters of lemonade and needs to sell it in containers, each holding 2/5 of a liter. How many full containers can the store owner fill? | 125 |
Let \( w \) be a complex number such that \( |w| = 1 \). Find the minimum value of \( |1 + w| + |1 - w + w^3| \). | 1 |
Simplify the expression \(\sin(30^\circ) \sin(60^\circ) \sin(90^\circ) \sin(120^\circ) \sin(150^\circ)\). | \frac{3}{16} |
A point \( P \) is at a distance of 3, 7, 8, 9, and 10 from five of the vertices of a regular tetrahedron. Find the distance from \( P \) to the sixth vertex. | 3 |
Simplify \(\cos \left( rac{\pi}{10}
ight) \cos \left( rac{\pi}{5}
ight) \cos \left( rac{3\pi}{10}
ight)\). | \frac{1}{8} |
In the diagram below, we have \(\overline{XY} \parallel \overline{WZ}\), \(\angle V = 55^\circ\), and \(\angle W = 60^\circ\). Find the measure of \(\angle XYZ\) in degrees. | 60 |
The asymptotes of a hyperbola are $y = 3x + 1$ and $y = 16 - 3x$. Also, the hyperbola passes through the point $(3,5)$. Find the distance between the foci of the hyperbola. | \frac{20}{3} |
In triangle \(ABC\), the area is given by \( (a + b - c)(a - b + c) \), where \(a\), \(b\), and \(c\) are the sides of triangle \(ABC\). Compute \( an A\). | \frac{8}{15} |
A car travels 60 miles per hour. Another car travels 80 miles per hour. Initially, both cars start at the same point but travel in opposite directions. After how many minutes will they be 120 miles apart from each other? | 51.43 |
What is the average of the two smallest positive integer solutions to the congruence $$12v \equiv 48 \pmod{100}~?$$ | 16.5 |
What is the number of square inches in the area of this trapezoid? | 2 |
What is the shortest distance from the point $(8, 0)$ to the line $y = 3x-3$? Express your answer in simplest radical form. | \frac{21\sqrt{10}}{10} |
A fair six-sided die is rolled twice. The first roll determines a number from 1 to 6, and the second roll determines another number from 1 to 6. Each number is then divided by 3, and the remainder is used to determine a position on a 2x2 grid where a square is colored green if the remainder is 0 and red otherwise. What... | \frac{1}{9} |
The average of Dani's, Eve's, and Fred's ages is 7. Three years ago, Fred was the same age as Dani is now. In three years, Eve's age will be 3/4 of Dani's age at that time. How many years old is Fred now? | 10 |
The graphs of \(x^2 + y^2 + 4x - 16y + 48 = 0\) and \(x^2 - y^2 + 4x + 8y - 22 = 0\) intersect at four points. Compute the sum of the distances from these four points to the point \((-2,4)\). | 8 |
Triangle XYZ has sides x, y, and z. The area of triangle XYZ is given by \(x^2 - (y - z)^2\), where x, y, and z are the sides of the triangle. Compute \( an X.\) | 1 |
Find the minimum value of the expression $\sqrt{x^2 + 900} + \sqrt{y^2 + 1600} + \sqrt{x^2 + y^2 - 60x - 80y + 850}$ for $0 \le x \le 30$ and $0 \le y \le 40$. | 70\sqrt{2} |
Let $XYZW$ be a regular octahedron with side length 3. The plane parallel to edges $XY$ and $ZW$ and lying halfway between them cuts $XYZW$ into two pieces. Find the surface area of one of these pieces. | \frac{9\sqrt{3}}{2} |
In the kingdom of Mathland, the currency system is peculiar. One Gold Coin is equal to 5 Silver Coins, and 2 Silver Coins are equal to 3 Copper Coins. In Gold Coins, what is the value of 48 Copper Coins? | 6.4 |
A fair six-sided die is rolled twice, so that the numbers obtained are randomly determined (with each number on the die being equally likely). The first number is divided by 4, determining one of the remainders 1, 2, or 3, marking the columns of a 3x3 grid. The second number is divided by 3, determining one of the rema... | \frac{1}{9} |
In the diagram below, we have \(\overline{AB}\parallel\overline{CD}\), \(\angle A= 50^\circ\), and \(\angle C =30^\circ\). Find the measure of \(\angle BCD\) in degrees. | 50 |
In the diagram, two circles, each with center $E$, have radii of 1 and 3. The total area of the shaded region is \(\frac{7}{12}\) of the area of the larger circle. How many degrees are in the measure of \(\angle AEC\)? | 120 |
Six points \(A\), \(B\), \(C\), \(D\), \(E\), and \(O\) lie on a flat field. \(A\) is directly north of \(O\), \(B\) is directly west of \(O\), \(C\) is directly south of \(O\), \(D\) is directly east of \(O\), \(E\) is directly northeast of \(O\), and \(F\) is directly southeast of \(O\). The distance between \(C\) an... | 130 |
Six points \(X\), \(Y\), \(Z\), \(W\), \(V\), and \(O\) lie on a flat field. \(X\) is directly north of \(O\), \(Y\) is directly west of \(O\), \(Z\) is directly south of \(O\), and \(W\) is directly east of \(O\). The distance between \(Z\) and \(W\) is 120 m. A hot-air balloon is positioned in the air at \(U\) direct... | 85 |
Let \( g(x) = \left\lfloor \left( - rac{3}{4}
ight)^x
ight
floor \) be a function that is defined for all values of \( x \) in \( [0, \infty) \) such that \( g(x) \) is a real number. How many distinct values exist in the range of \( g(x) \)? | 2 |
Determine the sum of all complex solutions of the equation
$ rac{2}{x^2 - 2} + rac{3}{x^2 - 3} + rac{4}{x^2 - 4} = 2010x - 4.$ | 0 |
Three runners, \(C\), \(D\), and \(E\), start at point \(O\) on a linear track and run in the same direction. Runner \(D\) runs twice as fast as runner \(C\), and runner \(E\) runs four times as fast as runner \(D\). An observer stands at point \(Q\) such that \(\overline{OQ}\) is perpendicular to the track. Find the m... | 90 |
Mia wants to divide $8$ by $ rac{4}{5}$, but she cannot remember how to do that. By what number should she multiply $8$ to get the answer? | \frac{5}{4} |
In the diagram, two circles, each with center $E$, have radii of 3 and 6. The total area of the shaded region is \( rac{7}{24}\) of the area of the larger circle. How many degrees are in the measure of \(\angle AEC\)? | 60 |
In triangle \( ABC \), \( AB = 18 \), \( AC = 10 \), and \( BC = 14 \). Let \( D \) be the foot of the altitude from \( C \) to \( AB \). Find the area of triangle \( ACD \). | \frac{133 \sqrt{11}}{18} |
Let \( g(x) = x^9 + 5x^8 - 8x^7 + 7x^6 - x^5 - 12x^4 + 4x^3 - 8x^2 + 12x - 5 \). Find the remainder when \( g(x) \) is divided by \( x^2 - 1 \). | 7x - 10 |
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