problem stringlengths 19 983 | answer stringlengths 1 8.19k |
|---|---|
A point $(x, y)$ is randomly selected such that $0 \le x \le 6$ and $0 \le y \le 3$. What is the probability that $x + y \le 3$? | \frac{1}{4} |
Let \(\mu\) be a constant, \(0 \le \mu \le 2\), and let \(g : [0,1] o [0,1]\) be defined by \(g(x) = \mu x(1 - x)\). Find the values of \(\mu\), \(0 \le \mu \le 2\), for which there exists an \(x \in [0,1]\) such that \(g(x)
eq x\) but \(g(g(x)) = x\). | 1 |
Let \( S \) be the set of points \((a, b)\) with \( 0 \le a, b \le 1 \) such that the equation \( x^4 + ax^3 - bx^2 + ax + 1 = 0 \) has at least one real root. Determine the area of the graph of \( S \). | 1 |
Let \(\mathbf{u}\) and \(\mathbf{v}\) be vectors such that the angle between \(\mathbf{u}\) and \(\mathbf{v}\) is \(30^\circ,\) and the angle between \(\mathbf{v}\) and \(\mathbf{u} - \mathbf{v}\) is \(60^\circ.\) Find the angle between \(\mathbf{u}\) and \(\mathbf{u} - \mathbf{v}.\) | 120^\circ |
Seven books are placed on three shelves: one shelf for novels, one for non-fiction, and one for reference. How many ways can the books be arranged so that all three shelves have at least one book? | 1806 |
Let \( z \) be a complex number such that \( |z| = 1 \). Find the maximum value of \( |1 + z| + |1 - \overline{z} + z^2| \). | 3 |
When rolling a certain unfair six-sided die with faces numbered 1, 2, 3, 4, 5, and 6, the probability of obtaining face \( F \) is \( rac{1}{4} \), the probability of obtaining the face opposite face \( F \) is \( rac{1}{4} \), the probability of obtaining each of the other faces is \( rac{1}{6} \), and the sum of t... | 5 |
A car travels along a straight road from point \( O \) to point \( P \) at a constant speed \( v \). At the same time, a cyclist starts from point \( O \) and travels in the same direction at a constant speed \( 2v \). An observer stands at point \( Q \) such that \( \overline{OQ} \) is perpendicular to the road. Find ... | 90 |
In triangle \(XYZ\), \(XY = 13\), \(XZ = 14\), and \(YZ = 15\). Let \(W\) be the foot of the altitude from \(X\) to \(YZ\). Find the area of triangle \(XWY\). | 42 |
A strictly increasing sequence of positive integers $b_1, b_2, b_3, \dots$ has the property that for every positive integer $k$, the subsequence $b_{2k-1}, b_{2k}, b_{2k+1}$ is geometric with a common ratio of 1.5, and the subsequence $b_{2k}, b_{2k+1}, b_{2k+2}$ is arithmetic with a common difference of 10. Suppose th... | 202 |
Three points \( A \), \( B \), and \( C \) lie on a coordinate plane. Point \( A \) is at \( (0, 0) \), point \( B \) is at \( (4, 0) \), and point \( C \) is at \( (0, 3) \). A line segment \( AB \) is drawn. The length of \( AB \) is given as 5 units. Point \( D \) lies on the line segment \( AB \) such that the leng... | (3, 0) |
Let \(a, b, c, d\) be distinct complex numbers on the unit circle such that \(|a| = |b| = |c| = |d| = 1\) and \(a + b + c + d = 0\). Find the maximum value of \(|(a + b)(a + c)(b + c)|\). | 8 |
The product of integers 270 and \( k \) is a perfect cube. What is the smallest possible positive value of \( k \)? | 100 |
In regular octagon LMNOPQRS, extending the sides of the octagon, as shown, forms a star. What is the measure of angle \( B \) in the figure? | 45^\circ |
The coordinates of three vertices of a parallelogram are (3, 4), (5, 9), and (7, 5). The fourth vertex, $(x, y)$, satisfies $x > 7$. What is the value of $x + y$? | 19 |
Let \(G\) and \(H\) denote the centroid and orthocenter of triangle \(ABC\), respectively. 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 circumradius \(R\) of triangle \(ABC\). | \frac{2}{3} (a^2 + b^2 + c^2) - 3R^2 |
If \(\omega^{101} = 1\) and \(\omega
eq 1\), then evaluate \( rac{1}{1 + \omega} + rac{1}{1 + \omega^2} + \dots + rac{1}{1 + \omega^{100}}\). | -\frac{1}{2} |
Three points, \(P\), \(Q\), and \(R\), are chosen randomly and independently on the circumference of a circle. What is the probability that segments \(PQ\) and \(QR\) intersect? | \frac{1}{3} |
A farmer has a rectangular field that measures 60 meters in length and 40 meters in width. He decides to plant wheat in half of the field and corn in the other half. The wheat requires 1 meter^2 of space to grow 10 kg, while the corn requires 1 meter^2 of space to grow 20 kg. Calculate the total amount of wheat and cor... | 36000 \, kg |
Rectangle \(EFGH\) has center \(I\) and \(EF/EH=k\). A point is randomly chosen from the interior of rectangle \(EFGH\). What is the probability that it is closer to \(I\) than to any of the four vertices? Consider the new problem and the model's reasoning to ensure that the same type of calculation error is likely to ... | \frac{1}{4} |
The medians BD and CE of triangle ABC intersect at the centroid G. The line through G that is parallel to AC intersects AB and BC at M and N, respectively. If the area of triangle ABC is 144, find the area of triangle CGN. | 24 |
Let \( E_1 \) and \( E_2 \) be the foci of the hyperbola \( 3x^2 - y^2 = 9 \). Suppose there is a circle that passes through \( E_1 \) and \( E_2 \) and is tangent to the hyperbola at two points on the \( y \)-axis. Compute the radius of the circle. | 3 |
A group of 4 friends, including Alex, Betty, Carl, and Diane, are planning a trip to 3 different cities: Beijing, Paris, and Tokyo. No two friends can visit the same city together. How many different ways can they arrange their visits so that each city is visited by exactly one friend? | 24 |
Consider the parabola defined by the equation \(y = 2x^2 - 4x + 1\) with its vertex at \((1, -1)\). If this parabola is reflected about the line \(y = -1\), determine the equation of the reflected parabola and find the sum of the coefficients \(a + b + c + d + e + f\) in this new equation. | -1 |
For \(0 \le x \le 30\) and \(0 \le y \le 40\), find the minimum value of \(\sqrt{x^2 + 900} + \sqrt{y^2 + 1600} + \sqrt{x^2 + y^2 - 60x - 80y + 1700}\). | 60 + 40\sqrt{2} |
In regular hexagon $ABCDEF$, extending the sides of the hexagon forms a star. What is the measure of angle \( G \) in the figure? | 60 |
Find all solutions to \(\sin \left( an^{-1} (x) + \cot^{-1} \left( rac{1}{x}
ight)
ight) = rac{\sqrt{2}}{2}\). Enter all the solutions, separated by commas. | No solution |
Beth and Charlie are playing a game. Beth starts first. On Beth's turn, she rolls a fair six-sided die. If she rolls an even number, she wins. If not, it becomes Charlie's turn. On Charlie's turn, he rolls a fair six-sided die. If he rolls an odd number, he wins. If not, it becomes Beth's turn. What is the probability ... | \frac{2}{3} |
Evaluate $\sin 10^\circ \sin 30^\circ \sin 50^\circ \sin 70^\circ \sin 90^\circ \sin 110^\circ \sin 130^\circ \sin 150^\circ.$ | \frac{1}{16} |
Given the parabola with equation $y = px^2 + qx + r$ and vertex $(m, n)$, it is reflected about the line $y = n$. The resulting parabola has the equation $y = sx^2 + tx + u$. Express $p + q + r + s + t + u$ in terms of $n.$ | 2n |
Consider three friends: Navin, Luka, and Ian. They are comparing their weekly earnings after working 5 days a week with 8 hours of work each day. The hourly earnings for each of them are 160 rupees for Navin, 25 kuna for Luka, and 34 pula for Ian. The exchange rates are as follows: 1 USD = 32.35 rupees, 1 USD = 5.18 ku... | Navin |
There is a total of 70 squares of three sizes whose vertices are points on this rectangular 4 × m grid of points. What is the value of m? | 8 |
When rolling an unfair six-sided die, the probability of obtaining face \( F \) is \( rac{1}{4} \). The probability of obtaining the face opposite face \( F \) is \( rac{1}{4} \) as well, and the probability of obtaining each of the other four faces is \( rac{1}{12} \). When two such dice are rolled, the probability... | 5 |
Let \( w \) be a complex number such that \( |w| = 1 \). Find the maximum value of \( |1 + w| + |1 - w + w^2 + w^3| \). | 4 |
For a point \( Q \), let \( e_1, e_2, \) and \( e_3 \) be the distances from \( Q \) to the planes \( x - y = 0, \) \( x - 3y + z = 0, \) and \( x + 2y + z = 0. \) Let \( T \) be the set of points \( Q \) such that \( e_1^2 + e_2^2 + e_3^2 = 49. \) Find the region of the volume enclosed by \( T. \) | \frac{1372\pi}{3} |
The equation \(x^{10} + (12x - 1)^{10} = 0\) has 10 complex roots \(r_1, \overline{r}_1, r_2, \overline{r}_2, r_3, \overline{r}_3, r_4, \overline{r}_4, r_5, \overline{r}_5\), where the bar denotes complex conjugation. Find the value of \( rac{1}{r_1 \overline{r}_1} + rac{1}{r_2 \overline{r}_2} + rac{1}{r_3 \overline{... | 5 |
Six points, \(A\), \(B\), \(C\), \(D\), \(E\), and \(F\), are chosen randomly and independently on the circumference of a circle. What is the probability that segments \(AB\) and \(EF\) intersect? | \frac{1}{3} |
Emma and Frank each have a bag that contains one ball of each of the colors, blue, green, orange, red, and violet. Frank randomly selects one ball from his bag and puts it into Emma's bag. Emma then randomly selects one ball from her bag and puts it into Frank's bag. What is the probability that after this process the ... | \frac{1}{6} |
In the circle with center \( P \), radii \( AP \) and \( BP \) form a right angle. The two smaller regions are tangent semicircles, as shown. The radius of the circle with center \( P \) is 20 inches. What is the radius of the smaller semicircle? Express your answer as a common fraction. | 10 |
A rectangle has 831 ones. A second rectangle has twice that total of 3he | 1662 |
Let \( S \) be the set of points \((a, b)\) with \( 0 \le a, b \le 1 \) such that the equation \( x^4 + ax^3 - bx^2 + ax + 1 + 2 = 0 \) has at least one real root. Determine the area of the graph of \( S \). | 1 |
In hexagon $MNPQRS$, extending the sides of the hexagon forms a star. If each exterior angle of the hexagon measures 60 degrees, what is the measure of angle $B$ in the figure? | 60 |
Find the number of ordered pairs \((x, y)\) of integers such that \(|x + yi| \le 6\). | 113 |
A point (x, y) is randomly selected such that \(0 \le x \le 6\) and \(0 \le y \le 3\). What is the probability that \(x + y \le 3\)? Express your answer as a common fraction. | \frac{1}{4} |
Let \(x, y, z\), and \(w\) be distinct complex numbers on the unit circle such that \(x + y + z + w = 0\). Find the maximum value of \(|(x + y)(x + z)(x + w)(y + z)(y + w)(z + w)|\). | 64 |
Let \( w \) be a complex number such that \( |w| = 1 \). Find the minimum value of \( |1 + w| + |1 - w + w^2| + |1 + w^3| \). | 3 |
Compute the value of \( rac{1}{\cos^2 30^\circ} + rac{1}{\sin^2 60^\circ} + rac{1}{\sin^2 90^\circ}\). | \frac{11}{3} |
If $\alpha^5 = 1$ and $\alpha \neq 1,$ then evaluate
$\frac{1}{1 + \alpha} + \frac{1}{1 + \alpha^2} + \dots + \frac{1}{1 + \alpha^5}.$ | \frac{5}{2} |
Consider the parabola described by the equation y = 2x^2 - 6x + 5 with vertex at (1.5, -2.5). Reflect this parabola about the line y = -2.5. Express the coefficients of the resulting parabola in the form y = dx^2 + ex + f and find the sum d + e + f. | -6 |
Which real values of x are not in the domain of the function \( g(x) = rac{1}{|x^2 - 5x + 6| + |x^2 + 7x + 10|} \)? | \emptyset |
Find the largest integer less than $(\sqrt{10} + \sqrt{2})^6.$ (Do not use a calculator!) | 9215 |
Sam has 450 US dollars in his wallet. He wants to withdraw half of it in Canadian dollars and the other half in Australian dollars. How many more Australian dollars than Canadian dollars will he have? Assume 1 Canadian dollar = 1.36 USD and 1 Australian dollar = 1.58 USD, and round to the nearest whole number. | 23 |
Let \(x\) and \(y\) be positive real numbers such that \(xy^3 = 3\). Find the minimum value of \(x^2 + y^6\). | 6 |
What is the value of $$(2x-3)(5x+2)-(2x-3)5x+3$$ when $$x=3$$? | 9 |
Let \(\mathbf{u}\) and \(\mathbf{v}\) be vectors such that the angle between \(\mathbf{u}\) and \(\mathbf{v}\) is \(30^\circ\), and the angle between \(\mathbf{v}\) and \(\mathbf{u} - \mathbf{v}\) is \(75^\circ\). Find the angle between \(\mathbf{u}\) and \(\mathbf{u} - \mathbf{v}\). | 60^\circ |
Let \(D\) and \(E\) denote the orthocenter and circumcenter of triangle \(XYZ\), respectively. Let \(F\) be the midpoint of \(\overline{DE}\). Express \(XF^2 + YF^2 + ZF^2\) in terms of the side lengths \(x, y, z\) and circumradius \(R\) of triangle \(XYZ\). | 3R^2 |
Let \( P \) be the set of all pairs \((x, y)\) of positive integers for which there exist right triangles with legs \(x\) and \(y\). Compute \[ \sum_{(x, y) \in P} rac{2^x}{3^y}. \] | 1 |
Seven dogs come to a playground that has three separate lanes. Each lane must have at least one dog. How many ways can the dogs be distributed across the lanes? | 15 |
Rectangle \(PQRS\) has center \(T\) and \(PQ/PS=m\). A point is randomly chosen from the interior of rectangle \(PQRS\). What is the probability that it is closer to \(T\) than to any of the four vertices? | \frac{1}{4} |
A strictly increasing sequence of positive integers \(b_1, b_2, b_3, \dots\) has the property that for every positive integer \(k\), the subsequence \(b_{2k-1}, b_{2k}, b_{2k+1}\) is geometric with a common ratio of 2 and the subsequence \(b_{2k}, b_{2k+1}, b_{2k+2}\) is arithmetic with a common difference of \(b_{2k-1... | 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 169, then find the area of triangle \(ENG\). | \frac{169}{9} |
Find the number of integer values of \( k \) in the closed interval \([-200, 200]\) for which the equation \(\log(kx) = 2\log(x+1)\) has exactly one real solution. | 1 |
Let \(x\) and \(y\) be positive real numbers such that \(xy^2 = 7\). Find the minimum value of \(x^3 + y^6\). | 14\sqrt{7} |
Find the smallest distance between the origin and a point on the graph of \( y = rac{1}{\sqrt{2}}(x^2 - 2) \) can be expressed as \( \sqrt{a}/b \), where \( a \) and \( b \) are positive integers such that \( a \) is not divisible by the square of any integer greater than one. Find \( a + b \). | 8 |
Point \(C\) lies somewhere within or on the square which has opposite corners at \((1,1)\) and \((3,3)\). Point \(D\) lies somewhere within or on the square which has opposite corners at points \((5,3)\) and \((6,4)\). What is the greatest possible value of the slope of the line containing points \(C\) and \(D\)? | \frac{1}{2} |
Quadrilateral $ABCD$ is a square with area 25 square inches. The figure represents the pieces of a Chinese tangram, where all the triangles are isosceles and piece 'e' is a square. What is the area of the gray piece, in square inches? | \frac{25}{8} |
Find the number of ordered pairs \((m, n)\) of integers such that \(|m + ni| \le 6\). | 130 |
Solve the equation \(\sqrt{x + \sqrt{2x + 4}} + \sqrt{x - \sqrt{2x + 4}} = 8\). | 16 |
Determine the number of integer values of \( k \) in the closed interval \([-100, 100]\) for which the equation \(\log(2kx) = \log((x+1)^2)\) has exactly one real solution. | 1 |
Point \(P\) lies within or on the circle centered at \((0,0)\) with radius \(3\). Point \(Q\) lies within or on the circle centered at \((6,0)\) with radius \(2\). What is the greatest possible value of the slope of the line containing points \(P\) and \(Q\)? | \frac{5}{6} |
If \( -4 \leq x \leq -1 \) and \( 2 \leq y \leq 4 \), what is the greatest possible value of \( \left( x + rac{1}{y}
ight) \left( rac{1}{y} - x
ight) \)? Express your answer as a common fraction. | -\frac{3}{4} |
Find the minimum distance between the origin and a point on the graph of \( y = rac{1}{\sqrt{3}}(x^2 - 4) \). Express this minimum distance in the form \( \sqrt{a} / b \), where \( a \) and \( b \) are positive integers such that \( a \) is not divisible by the square of any integer greater than one. Find \( a + b \). | 15 |
Alice and Bob each have a box containing 3 white and 2 black marbles. Alice randomly selects one marble from her box and puts it into Bob's box. Bob then randomly selects one marble from his box and puts it into Alice's box. What is the probability that after this process the contents of the two boxes are the same? | \frac{2}{5} |
For what real values of \(x\) is \(-5 < x^4 + 5x^2 < 20\) satisfied? Express your answer in interval notation. | \left( -\sqrt{\frac{-5 + \sqrt{105}}{2}}, \sqrt{\frac{-5 + \sqrt{105}}{2}} \right) |
In the diagram below, we have $\overline{AB} \parallel \overline{CD}$, $\angle A = 60^\circ$, and $\angle C = 20^\circ$. Find the measure of $\angle BAC$ in degrees. | 60 |
Let \( g \) be a constant, \( 0 \leq g \leq 3 \), and let \( h : [0,1] o [0,1] \) be defined by \( h(x) = gx(1 - x) \). Find the values of \( g \), \( 0 \leq g \leq 3 \), for which there exists an \( x \in [0,1] \) such that \( h(x)
eq x \) but \( h(h(x)) = x \). | 2 |
Calculate the value of \(\sin 30^\circ \sin 70^\circ \sin 110^\circ \sin 150^\circ\). | \frac{1}{16} |
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 256, then find the area of triangle \(ENG\). | 64 |
In the diagram, two circles, each with center D, have radii of 1 and 2. The total area of the shaded region is \( rac{2}{5}\) of the area of the larger circle. How many degrees are in the measure of \(\angle ADC\)? | 120 |
If the domain of the function \( \log(x^2 - 1) \) is \( x < a \) or \( x > b \), for some \( a \) and \( b \), find \( a + b \). | 0 |
Find the greatest integer less than \((\sqrt{10} + \sqrt{6})^6\). (Do not use a calculator!) | 31231 |
In triangle \(PQR\), the medians \(PD\), \(QE\), and \(RF\) intersect at the centroid \(O\). The line through \(O\) that is parallel to \(QR\) intersects \(PQ\) and \(PR\) at points \(M\) and \(N\), respectively. If the area of triangle \(PQR\) is 24, find the area of triangle \(OEN\). | 4 |
If the domain of the function \(\sqrt{x^2}\) is \(x < a\) or \(x > b\), for some \(a\) and \(b\), find \(a + b\). | 0 |
Find the smallest positive real number \( k \) for which the inequality \(\left\| egin{pmatrix} 3 & 4 \ 0 & -3 \end{pmatrix} \mathbf{w}
ight\| \le k \|\mathbf{w}\|\) holds for all two-dimensional vectors \(\mathbf{w}\). | 5 |
The parabola with equation \(y = ax^2 + bx + c\) and vertex \((h, k)\) is reflected about the line \(y = k\). This results in the parabola with equation \(y = dx^2 + ex + f\). If the original parabola has a y-intercept at \((0, c)\), and the reflected parabola has a y-intercept at \((0, f)\), express \(a + b + c + d + ... | 2k |
What real values of $x$ are not in the domain of \( g(x) = rac{1}{|x^2 - 5x + 6| + |x^2 - 7x + 10|} \)? | 2 |
A toy car's wheel has a radius of 5 inches. If the wheel makes 3 revolutions every 2 seconds, what is the car's speed in inches per second? | 15\pi |
Let $g(x) = \left\lfloor \left( - rac{3}{7}
ight)^x
ight
floor$ 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 |
Four cities A, B, C, and D are located on a flat plane. City A is directly north of City O, city B is directly west of City O, city C is directly south of City O, and city D is directly east of City O. The distance between C and D is 100 m. A radio tower is located at point H above City O. Three cables HA, HB, HC, and ... | 110 |
Determine the number of integers \(a\) such that \(|a| \le 3\). | 7 |
The coordinates of a parallelogram are (4, 2), (5, 7), (6, 3) and \((x, y)\) with \(x > 6\). What is the value of \(x + y\)? | 15 |
A certain unfair six-sided die has faces numbered 1, 2, 3, 4, 5, and 6. The probability of obtaining face \( F \) is greater than \( rac{1}{6} \), the probability of obtaining the face opposite face \( F \) is less than \( rac{1}{6} \), and the probability of obtaining each of the other faces is \( rac{1}{6} \). The... | 5 |
Forty-five students attend a science club meeting. The number of girls at the meeting is a multiple of 14, and there are more girls than boys attending the meeting. How many boys are at the meeting? | 17 |
In right triangle $PQR$, $PQ = 5$, $PR = 12$, and $QR = 13$. Let $S$ be the foot of the altitude from $P$ to $QR$. Find the area of triangle $PSR$. | \frac{144}{13} |
Let \( S \) be the set of all triples \( (x, y, z) \) of positive integers for which there exist triangles with side lengths \( x \), \( y \), and \( z \). Calculate the sum \(\sum_{(x,y,z) \in S} rac{2^x}{3^y 5^z}.\) | \frac{1}{2} |
In triangle \(XYZ\), \(XY = 15\), \(XZ = 20\), and \(YZ = 25\). Let \(W\) be the foot of the altitude from \(Z\) to \(XY\). Find the area of triangle \(XZW\). | 72 |
Let \( F_1 \) and \( F_2 \) be the foci of the hyperbola \( x^2 - ky^2 = 1 \), where \( k > 1 \) is a constant. Suppose that there is a circle which passes through \( F_1 \) and \( F_2 \) and which lies tangent to the hyperbola at two points on the \( x \)-axis. Compute \( k \). | 2 |
Find the value of the expression \( rac{2+3i}{1+i}\). Use the correct reasoning and avoid missing any signs. | \frac{5}{2} + \frac{1}{2}i |
What is the units digit of \(3\$?\) where \(n\$ is defined as \(n!^{{n}^{{\cdot}^{{\cdot}^{{\cdot}^{n}}}}}\) with \(n\) levels of exponentiation. | 6 |
Find the sum of all complex solutions of the equation
\[ rac{1}{x^2 - 1} + rac{2}{x^2 - 4} + rac{3}{x^2 - 9} = 2010x - 3.\] | 0 |
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