problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
If $f(x)=\frac{ax+b}{cx+d}, abcd\not=0$ and $f(f(x))=x$ for all $x$ in the domain of $f$, what is the value of $a+d$? | 0 | 1 | 2,865.8125 | 2,865.8125 | -1 | |
Given that $x$ is the median of the data set $1$, $2$, $3$, $x$, $5$, $6$, $7$, and the average of the data set $1$, $2$, $x^{2}$, $-y$ is $1$, find the minimum value of $y- \frac {1}{x}$. | \frac {23}{3} | 0.6875 | 5,992.125 | 4,992.181818 | 8,192 | |
The graph of the function f(x) = sin(2x) is translated to the right by $\frac{\pi}{6}$ units to obtain the graph of the function g(x). Find the analytical expression for g(x). Also, find the minimum value of $|x_1 - x_2|$ for $x_1$ and $x_2$ that satisfy $|f(x_1) - g(x_2)| = 2$. | \frac{\pi}{2} | 0 | 4,749.6875 | -1 | 4,749.6875 | |
The polynomial $Q(x)=x^3-21x+35$ has three different real roots. Find real numbers $a$ and $b$ such that the polynomial $x^2+ax+b$ cyclically permutes the roots of $Q$, that is, if $r$, $s$ and $t$ are the roots of $Q$ (in some order) then $P(r)=s$, $P(s)=t$ and $P(t)=r$. | a = 2, b = -14 |
Given the polynomial \( Q(x) = x^3 - 21x + 35 \), which has three different real roots, we need to find real numbers \( a \) and \( b \), such that the polynomial \( P(x) = x^2 + ax + b \) cyclically permutes the roots of \( Q \). Let the roots of \( Q \) be \( r, s, \) and \( t \). The cyclic permutation property req... | 0 | 8,192 | -1 | 8,192 |
Given the complex number $z=(2m^{2}+3m-2)+(m^{2}+m-2)i$ where $(m\in\mathbb{R})$, find the value of $m$ under the following conditions:
$(1) z$ is a real number;
$(2) z$ is an imaginary number;
$(3) z$ is a pure imaginary number;
$(4) z=0$. | -2 | 0.9375 | 3,766.3125 | 3,471.266667 | 8,192 | |
A club has 30 members, which includes 12 females and 18 males. In how many ways can a 5-person executive committee be formed such that at least one member must be female? | 133,938 | 0 | 3,369.1875 | -1 | 3,369.1875 | |
Consider a rectangle with dimensions 6 units by 7 units. A triangle is formed with its vertices on the sides of the rectangle. Vertex $A$ is on the left side, 3 units from the bottom. Vertex $B$ is on the bottom side, 5 units from the left. Vertex $C$ is on the top side, 2 units from the right. Calculate the area of tr... | 17.5 | 0 | 4,251.5 | -1 | 4,251.5 | |
In triangle $\triangle ABC$, given that $A=60^{\circ}$ and $BC=4$, the diameter of the circumcircle of $\triangle ABC$ is ____. | \frac{8\sqrt{3}}{3} | 0 | 2,341.625 | -1 | 2,341.625 | |
Four of the six numbers 1867, 1993, 2019, 2025, 2109, and 2121 have a mean (average) of 2008. What is the mean (average) of the other two numbers? | 2051 | 0.8125 | 5,680.875 | 5,318.769231 | 7,250 | |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a=3$, and $\left(a+b\right)\sin B=\left(\sin A+\sin C\right)\left(a+b-c\right)$.<br/>$(1)$ Find angle $A$;<br/>$(2)$ If $acosB+bcosA=\sqrt{3}$, find the area of $\triangle ABC$. | \frac{3\sqrt{3}}{2} | 0 | 6,391.5 | -1 | 6,391.5 | |
Inside a pentagon, 1000 points were marked and the pentagon was divided into triangles such that each of the marked points became a vertex of at least one of them. What is the minimum number of triangles that could be formed? | 1003 | 0 | 5,368.8125 | -1 | 5,368.8125 | |
If 5 points are placed in the plane at lattice points (i.e. points $(x, y)$ where $x$ and $y$ are both integers) such that no three are collinear, then there are 10 triangles whose vertices are among these points. What is the minimum possible number of these triangles that have area greater than $1 / 2$ ? | 4 | By the pigeonhole principle, the 5 points cannot all be distinct modulo 2, so two of them must have a midpoint that is also a lattice point. This midpoint is not one of the 5 since no 3 are collinear. Pick's theorem states that the area of a polygon whose vertices are lattice points is $B / 2+I-1$ where $B$ is the numb... | 0 | 8,192 | -1 | 8,192 |
Given two fixed points $A(-1,0)$ and $B(1,0)$, and a moving point $P(x,y)$ on the line $l$: $y=x+3$, an ellipse $C$ has foci at $A$ and $B$ and passes through point $P$. Find the maximum value of the eccentricity of ellipse $C$. | \dfrac{\sqrt{5}}{5} | 0 | 6,569.5625 | -1 | 6,569.5625 | |
You are given a $10 \times 2$ grid of unit squares. Two different squares are adjacent if they share a side. How many ways can one mark exactly nine of the squares so that no two marked squares are adjacent? | 36 | Since each row has only two squares, it is impossible for two marked squares to be in the same row. Therefore, exactly nine of the ten rows contain marked squares. Consider two cases: Case 1: The first or last row is empty. These two cases are symmetrical, so assume without loss of generality that the first row is empt... | 0 | 7,974.0625 | -1 | 7,974.0625 |
The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number? | 7 | Let the three numbers be $a$, $b$, and $c$ such that $a < b < c$. We are given the sums of these numbers taken in pairs:
1. $a + b = 12$
2. $a + c = 17$
3. $b + c = 19$
We aim to find the value of $b$, the middle number. To do this, we can manipulate the given equations to isolate $b$. We start by adding the first and... | 1 | 2,959.75 | 2,959.75 | -1 |
In the arithmetic sequence $\{a\_n\}$, $S=10$, $S\_9=45$, find the value of $a\_{10}$. | 10 | 0 | 4,479.0625 | -1 | 4,479.0625 | |
Suppose two arithmetic sequences $\{a\_n\}$ and $\{b\_n\}$ have the sum of their first n terms as $S\_n$ and $T\_n$, respectively. If $a\_3 = 2b\_3$ and $\frac{S\_n}{T\_n} = \frac{3n - t}{2n + 2}$ for any $n \in \mathbb{N}^*$, find the value of $\frac{a\_9}{b\_6 + b\_8} + \frac{a\_5}{b\_3 + b\_11}$. | \frac{12}{7} | 0.375 | 7,032.8125 | 5,253.5 | 8,100.4 | |
The angle of inclination of the line $$\begin{cases} \left.\begin{matrix}x=3- \frac { \sqrt {2}}{2}t \\ y= \sqrt {5}- \frac { \sqrt {2}}{2}t\end{matrix}\right.\end{cases}$$ is ______. | \frac{\pi}{4} | 0.375 | 5,329.875 | 3,506.5 | 6,423.9 | |
Two identical cylindrical sheets are cut open along the dotted lines and glued together to form one bigger cylindrical sheet. The smaller sheets each enclose a volume of 100. What volume is enclosed by the larger sheet? | 400 | 0.0625 | 4,899.5625 | 3,494 | 4,993.266667 | |
A circle with a radius of 3 units has its center at $(0, 0)$. A circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. What is the value of $x$? Express your answer as a common fraction. | \frac{9}{2} | 0.625 | 6,511 | 5,502.4 | 8,192 | |
A palindrome is a number that reads the same forward and backward. How many three-digit perfect squares are palindromes? | 3 | 0.6875 | 5,654.125 | 5,741.363636 | 5,462.2 | |
Compute $\tan 20^\circ + 4 \sin 20^\circ.$ | \sqrt{3} | 0.4375 | 7,288.375 | 6,126.571429 | 8,192 | |
Given $y=f(x)$ is a quadratic function, and $f(0)=-5$, $f(-1)=-4$, $f(2)=-5$,
(1) Find the analytical expression of this quadratic function.
(2) Find the maximum and minimum values of the function $f(x)$ when $x \in [0,5]$. | - \frac {16}{3} | 0.8125 | 2,769.9375 | 2,799.307692 | 2,642.666667 | |
Given the hyperbola \( C: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \) with \( a > 0 \) and \( b > 0 \), the eccentricity is \( \frac{\sqrt{17}}{3} \). Let \( F \) be the right focus, and points \( A \) and \( B \) lie on the right branch of the hyperbola. Let \( D \) be the point symmetric to \( A \) with respect ... | \frac{1}{2} | 0 | 8,192 | -1 | 8,192 | |
Given the sets
$$
\begin{array}{l}
A=\{(x, y) \mid |x| + |y| = a, a > 0\}, \\
B=\{(x, y) \mid |xy| + 1 = |x| + |y|\}
\end{array}
$$
If $A \cap B$ forms the vertices of a regular octagon in the plane, find the value of $a$. | \sqrt{2} | 0.0625 | 7,981.0625 | 4,817 | 8,192 | |
In $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite angles $A$, $B$, and $C$ respectively, and the three interior angles $A$, $B$, $C$ satisfy $A+C=2B$.
$\text{(1)}$ If $b=2$, find the maximum value of the area of $\triangle ABC$ and determine the shape of the triangle when the maximum area i... | \dfrac{\sqrt{2}}{2} | 0 | 7,350.4375 | -1 | 7,350.4375 | |
Triangle $PQR$ has vertices $P(0, 10)$, $Q(3, 0)$, $R(9, 0)$. A line through $Q$ bisects the area of $\triangle PQR$. Find the sum of the slope and $y$-intercept of this line. | \frac{-20}{3} | 0 | 6,301.6875 | -1 | 6,301.6875 | |
Assume the quartic $x^{4}-a x^{3}+b x^{2}-a x+d=0$ has four real roots $\frac{1}{2} \leq x_{1}, x_{2}, x_{3}, x_{4} \leq 2$. Find the maximum possible value of $\frac{\left(x_{1}+x_{2}\right)\left(x_{1}+x_{3}\right) x_{4}}{\left(x_{4}+x_{2}\right)\left(x_{4}+x_{3}\right) x_{1}}$ (over all valid choices of $\left.a, b, ... | \frac{5}{4} | We can rewrite the expression as $$\frac{x_{4}^{2}}{x_{1}^{2}} \cdot \frac{\left(x_{1}+x_{1}\right)\left(x_{1}+x_{2}\right)\left(x_{1}+x_{3}\right)\left(x_{1}+x_{4}\right)}{\left(x_{4}+x_{1}\right)\left(x_{4}+x_{2}\right)\left(x_{4}+x_{3}\right)\left(x_{4}+x_{4}\right)} \frac{x_{4}^{2}}{x_{1}^{2}} \cdot \frac{f\left(-x... | 0 | 8,192 | -1 | 8,192 |
Two sectors of a circle of radius $15$ overlap in the same manner as the original problem, with $P$ and $R$ as the centers of the respective circles. The angle at the centers for both sectors is now $45^\circ$. Determine the area of the shaded region. | \frac{225\pi - 450\sqrt{2}}{4} | 0 | 7,623.5 | -1 | 7,623.5 | |
Determine the value of \(x\) if \(x\) is positive and \(x \cdot \lfloor x \rfloor = 90\). Express your answer as a decimal. | 10 | 0 | 8,192 | -1 | 8,192 | |
If $\log 2 = .3010$ and $\log 3 = .4771$, the value of $x$ when $3^{x+3} = 135$ is approximately | 1.47 | 1. **Rewrite the equation using logarithms**: Given the equation $3^{x+3} = 135$, we take the logarithm (base 10) of both sides:
\[
\log(3^{x+3}) = \log(135)
\]
2. **Apply the power rule of logarithms**: The power rule states $\log(a^b) = b \log(a)$, so we can rewrite the left side:
\[
(x+3) \log(3) = \... | 0.3125 | 7,249 | 6,566.4 | 7,559.272727 |
In triangle $XYZ$, $XY=XZ$ and $W$ is on $XZ$ such that $XW=WY=YZ$. What is the measure of $\angle XYW$? | 36^{\circ} | Let $\angle XYW=\theta$. Since $\triangle XYW$ is isosceles with $WX=WY$, then $\angle YXW=\angle XYW=\theta$. Since the sum of the angles in $\triangle XYW$ is $180^{\circ}$, then $\angle XWY=180^{\circ}-2\theta$. Since $\angle XWY+\angle ZWY=180^{\circ}$, then $\angle ZWY=180^{\circ}-(180^{\circ}-2\theta)=2\theta$. S... | 0 | 8,192 | -1 | 8,192 |
A certain rectangle had its dimensions expressed in whole numbers of decimeters. Then, it changed its dimensions three times. First, one of its dimensions was doubled and the other was adjusted so that the area remained the same. Then, one dimension was increased by $1 \mathrm{dm}$ and the other decreased by $4 \mathrm... | 4:1 | 0 | 7,357.25 | -1 | 7,357.25 | |
Let $X = \{-5,-4,-3,-2,-1,0,1,2,3,4,5\}$ and $S = \{(a,b)\in X\times X:x^2+ax+b \text{ and }x^3+bx+a \text{ have at least a common real zero .}\}$ How many elements are there in $S$ ? | 21 | 0.0625 | 8,045.25 | 7,294 | 8,095.333333 | |
If Ravi shortens the length of one side of a $5 \times 7$ index card by $1$ inch, the card would have an area of $24$ square inches. What is the area of the card in square inches if instead he shortens the length of the other side by $1$ inch? | 18 | 0 | 8,151 | -1 | 8,151 | |
For some integers $m$ and $n$, the expression $(x+m)(x+n)$ is equal to a quadratic expression in $x$ with a constant term of -12. Which of the following cannot be a value of $m$? | 5 | Expanding, $(x+m)(x+n)=x^{2}+n x+m x+m n=x^{2}+(m+n) x+m n$. The constant term of this quadratic expression is $m n$, and so $m n=-12$. Since $m$ and $n$ are integers, they are each divisors of -12 and thus of 12. Of the given possibilities, only 5 is not a divisor of 12, and so $m$ cannot equal 5. | 0.625 | 6,244.1875 | 5,673.9 | 7,194.666667 |
Determine the least positive period $q$ of the functions $g$ such that $g(x+2) + g(x-2) = g(x)$ for all real $x$. | 12 | 0.8125 | 6,648.5 | 6,292.307692 | 8,192 | |
Numbers $1,2,3,4,5,6,7,$ and $8$ are placed at the vertices of a cube such that the sum of any three numbers belonging to any face of the cube is not less than 10. Find the minimum possible sum of four numbers belonging to one face. | 16 | 0 | 8,192 | -1 | 8,192 | |
Let $A$ be an acute angle such that $\tan A = 2 \cos A.$ Find the number of positive integers $n$ less than or equal to $1000$ such that $\sec^n A + \tan^n A$ is a positive integer whose units digit is $9.$ | 167 | \[\tan A = 2 \cos A \implies \sin A = 2 \cos^2 A \implies \sin^2 A + \cos^2 A = 4 \cos^4 A + \cos^2 A = 1\] \[\implies \cos^2 A = \frac {\sqrt {17} - 1}{8}.\] \[c_n = \sec^n A + \tan^n A = \frac {1}{\cos^n A} + 2^n \cos^n A = (4\cos^2 A +1)^{\frac {n}{2}}+(4 \cos^2 A)^{\frac {n}{2}} =\] \[= \left(\frac {\sqrt {17} + 1}... | 0 | 8,192 | -1 | 8,192 |
If $a @ b$ is defined as $a @ b$ = $3a - 3b$, what is the value of $3 @ 5$? | -6 | 1 | 1,688 | 1,688 | -1 | |
In a convex quadrilateral $ABCD$, $M$ and $N$ are the midpoints of sides $AD$ and $BC$, respectively. Given that $|\overrightarrow{AB}|=2$, $|\overrightarrow{MN}|=\frac{3}{2}$, and $\overrightarrow{MN} \cdot (\overrightarrow{AD} - \overrightarrow{BC}) = \frac{3}{2}$, find $\overrightarrow{AB} \cdot \overrightarrow{CD}$... | -2 | 0.1875 | 7,001.25 | 4,786.333333 | 7,512.384615 | |
Let $a$ and $b$ be real numbers so that the roots of
\[z^2 + (10 + ai) z + (27 + bi) = 0\]are complex conjugates. Enter the ordered pair $(a,b).$ | (0,0) | 1 | 2,797.8125 | 2,797.8125 | -1 | |
Let $A$ be the set of positive integers that have no prime factors other than $2$, $3$, or $5$. The infinite sum \[\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \cd... | 19 | To solve the problem, we need to calculate the sum of the reciprocals of all positive integers that have no prime factors other than $2$, $3$, or $5$. These integers can be expressed in the form $2^a 3^b 5^c$ where $a$, $b$, and $c$ are non-negative integers.
1. **Express the sum as a product of geometric series**:
... | 1 | 2,691.6875 | 2,691.6875 | -1 |
A green point and a purple point are chosen at random on the number line between 0 and 1. What is the probability that the purple point is greater than the green point but less than three times the green point? | \frac{1}{3} | 0.6875 | 6,757.5625 | 6,330 | 7,698.2 | |
If \( x, y, \) and \( k \) are positive real numbers such that
\[
5 = k^2\left(\dfrac{x^2}{y^2}+\dfrac{y^2}{x^2}\right) + k\left(\dfrac{x}{y}+\dfrac{y}{x}\right),
\]
find the maximum possible value of \( k \). | \frac{-1+\sqrt{22}}{2} | 0 | 5,673.375 | -1 | 5,673.375 | |
Jody has 6 distinguishable balls and 6 distinguishable sticks, all of the same length. How many ways are there to use the sticks to connect the balls so that two disjoint non-interlocking triangles are formed? Consider rotations and reflections of the same arrangement to be indistinguishable. | 7200 | For two disjoint triangles to be formed, three of the balls must be connected into a triangle by three of the sticks, and the three remaining balls must be connected by the three remaining sticks. There are $\binom{6}{3}$ ways to pick the 3 balls for the first triangle. Note that once we choose the 3 balls for the firs... | 0 | 7,723.6875 | -1 | 7,723.6875 |
A positive integer \overline{A B C}, where $A, B, C$ are digits, satisfies $\overline{A B C}=B^{C}-A$. Find $\overline{A B C}$. | 127 | The equation is equivalent to $100 A+10 B+C=B^{C}-A$. Suppose $A=0$, so that we get $10 B+C=B^{C}$. Reducing $\bmod B$, we find that $C$ must be divisible by $B$. $C \neq 0$, since otherwise $10 B=1$, contradiction, so $C \geq B$. Thus $10 B+C \geq B^{B}$ for digits $B, C$. For $B \geq 4$, we have $100>10 B+C \geq B^{B... | 0.5625 | 6,993.625 | 6,061.555556 | 8,192 |
Given two circles C<sub>1</sub>: $x^{2}+y^{2}-x+y-2=0$ and C<sub>2</sub>: $x^{2}+y^{2}=5$, determine the positional relationship between the two circles; if they intersect, find the equation of the common chord and the length of the common chord. | \sqrt{2} | 0.9375 | 4,134.75 | 3,864.266667 | 8,192 | |
In the rectangular coordinate system $(xOy)$, if the initial side of angle $\alpha$ is the non-negative semi-axis of $x$, and the terminal side is the ray $l$: $y=2x(x\leqslant 0)$.
(1) Find the value of $\tan \alpha$;
(2) Find the value of $\frac{\cos \left(\alpha-\pi\right)-2\cos \left( \frac{\pi}{2}+\alpha\right)}... | -3 | 0.8125 | 4,420.9375 | 3,551.153846 | 8,190 | |
Suppose $f(x)=\frac{3}{2-x}$. If $g(x)=\frac{1}{f^{-1}(x)}+9$, find $g(3)$. | 10 | 0.9375 | 3,208 | 2,875.733333 | 8,192 | |
Let $k$ be a positive real number. In the $X-Y$ coordinate plane, let $S$ be the set of all points of the form $(x,x^2+k)$ where $x\in\mathbb{R}$ . Let $C$ be the set of all circles whose center lies in $S$ , and which are tangent to $X$ -axis. Find the minimum value of $k$ such that any two circles in ... | \frac{1}{4} | 0.375 | 7,418.9375 | 6,594 | 7,913.9 | |
Given the sets $M={x|m\leqslant x\leqslant m+ \frac {7}{10}}$ and $N={x|n- \frac {2}{5}\leqslant x\leqslant n}$, both of which are subsets of ${x|0\leqslant x\leqslant 1}$, find the minimum value of the "length" of the set $M\cap N$. (Note: The "length" of a set ${x|a\leqslant x\leqslant b}$ is defined as $b-a$.) | \frac{1}{10} | 0.1875 | 7,985.3125 | 7,089.666667 | 8,192 | |
A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(4,0)$, $(4,1)$, and $(0,1)$. What is the probability that $x < y$? | \frac{1}{8} | 0.875 | 3,939.625 | 3,332.142857 | 8,192 | |
The new PUMaC tournament hosts $2020$ students, numbered by the following set of labels $1, 2, . . . , 2020$ . The students are initially divided up into $20$ groups of $101$ , with each division into groups equally likely. In each of the groups, the contestant with the lowest label wins, and the winners advance ... | 2123 | 0.25 | 6,928.875 | 6,584 | 7,043.833333 | |
Given condition p: $|5x - 1| > a$ and condition q: $x^2 - \frac{3}{2}x + \frac{1}{2} > 0$, please choose an appropriate real number value for $a$, and use the given two conditions as A and B to construct the proposition: If A, then B. Make sure the constructed original proposition is true, while its converse is false, ... | a = 4 | 0.6875 | 6,354.9375 | 5,519.909091 | 8,192 | |
What is the greatest prime factor of $3^7+6^6$? | 67 | 0.9375 | 3,003.5625 | 3,012.733333 | 2,866 | |
Meg started with the number 100. She increased this number by $20\%$ and then increased the resulting number by $50\%$. What was her final result? | 180 | $20\%$ of the number 100 is 20, so when 100 is increased by $20\%$, it becomes $100 + 20 = 120$. $50\%$ of a number is half of that number, so $50\%$ of 120 is 60. Thus, when 120 is increased by $50\%$, it becomes $120 + 60 = 180$. Therefore, Meg's final result is 180. | 1 | 795.125 | 795.125 | -1 |
In how many ways can 10 people be seated in a row of chairs if four of the people, Alice, Bob, Charlie, and Dave, refuse to sit in four consecutive seats? | 3507840 | 0.5625 | 5,472.125 | 3,356.666667 | 8,192 | |
For real numbers \( x \) and \( y \) within the interval \([0, 12]\):
$$
xy = (12 - x)^2 (12 - y)^2
$$
What is the maximum value of the product \( xy \)? | 81 | 0.25 | 7,883.1875 | 7,504 | 8,009.583333 | |
In the arithmetic sequence $\{a_{n}\}$, $d=-2$, $a_{1}+a_{4}+a_{7}+…+a_{31}=50$. Find the value of $a_{2}+a_{6}+a_{10}+…+a_{42}$. | -82 | 0.3125 | 6,791.625 | 4,612.4 | 7,782.181818 | |
The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base $10$, where digit $a$ is not zero. Find the three-digit number $abc$. | 937 | We have that $N^2 - N = N(N - 1)\equiv 0\mod{10000}$
Thus, $N(N-1)$ must be divisible by both $5^4$ and $2^4$. Note, however, that if either $N$ or $N-1$ has both a $5$ and a $2$ in its factorization, the other must end in either $1$ or $9$, which is impossible for a number that is divisible by either $2$ or $5$. Thus... | 0.875 | 5,054.25 | 4,606 | 8,192 |
Mohan is selling cookies at the economics fair. As he decides how to package the cookies, he finds that when he bags them in groups of 4, he has 3 left over. When he bags them in groups of 5, he has 2 left over. When he bags them in groups of 7, he has 4 left over. What is the least number of cookies that Mohan could h... | 67 | 0.8125 | 3,758.625 | 2,735.538462 | 8,192 | |
Calculate the value of $$1+\cfrac{2}{3+\cfrac{6}{7}}$$ and express it as a simplified fraction. | \frac{41}{27} | 1 | 1,955.0625 | 1,955.0625 | -1 | |
How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations? \[\left\{ \begin{aligned} x+3y&=3 \\ \left| |x| - |y| \right| &= 1 \end{aligned}\right.\] | 3 | 0.625 | 7,412.25 | 6,944.4 | 8,192 | |
In triangle $PQR$, we have $\angle P = 90^\circ$, $QR = 15$, and $\tan R = 5\cos Q$. What is $PQ$? | 6\sqrt{6} | 0.9375 | 3,572.5 | 3,264.533333 | 8,192 | |
Let $\mathcal{T}$ be the set $\lbrace1,2,3,\ldots,12\rbrace$. Let $m$ be the number of sets of two non-empty disjoint subsets of $\mathcal{T}$. Calculate the remainder when $m$ is divided by $1000$. | 625 | 0.8125 | 4,968.8125 | 4,225 | 8,192 | |
In the polygon shown, each side is perpendicular to its adjacent sides, and all 24 of the sides are congruent. The perimeter of the polygon is 48. Find the area of the polygon. | 48 | 0.0625 | 8,146.5 | 7,464 | 8,192 | |
A coin is tossed. If heads appear, point \( P \) moves +1 on the number line; if tails appear, point \( P \) does not move. The coin is tossed no more than 12 times, and if point \( P \) reaches coordinate +10, the coin is no longer tossed. In how many different ways can point \( P \) reach coordinate +10? | 66 | 0.4375 | 6,703.4375 | 4,789.571429 | 8,192 | |
What is the area, in square units, of a triangle that has sides of $4,3$ and $3$ units? Express your answer in simplest radical form. | 2\sqrt{5} | 1 | 2,072.625 | 2,072.625 | -1 | |
Let \(x, y \in \mathbf{R}\). Define \( M \) as the maximum value among \( x^2 + xy + y^2 \), \( x^2 + x(y-1) + (y-1)^2 \), \( (x-1)^2 + (x-1)y + y^2 \), and \( (x-1)^2 + (x-1)(y-1) + (y-1)^2 \). Determine the minimum value of \( M \). | \frac{3}{4} | 0 | 8,192 | -1 | 8,192 | |
Given that $f'(x_0)=2$, find the value of $\lim_{k\rightarrow 0} \frac{f(x_0-k)-f(x_0)}{2k}$. | -1 | 1 | 3,195.4375 | 3,195.4375 | -1 | |
Let $f(x)$ be a function defined on $R$ such that $f(x+3) + f(x+1) = f(2) = 1$. Find $\sum_{k=1}^{2023} f(k) =$ ____. | 1012 | 0.5625 | 6,811.25 | 5,817.111111 | 8,089.428571 | |
A region is bounded by semicircular arcs constructed on the side of a square whose sides measure $2/\pi$, as shown. What is the perimeter of this region? [asy]
path a=(10,0)..(5,5)--(5,-5)..cycle;
path b=(0,10)..(5,5)--(-5,5)..cycle;
path c=(-10,0)..(-5,5)--(-5,-5)..cycle;
path d=(0,-10)..(-5,-5)--(5,-5)..cycle;
path e... | 4 | 0.9375 | 5,423.875 | 5,239.333333 | 8,192 | |
Molly flips a fair coin five times, and she is very surprised to flip a head each time. What is the probability she will flip a tail on her next flip of the coin? Express your answer as a common fraction. | \frac{1}{2} | 1 | 1,020.9375 | 1,020.9375 | -1 | |
Given a $6 \times 6$ square of $36$ square blocks, find the number of different combinations of $4$ blocks that can be selected so that no two are in the same row or column. | 5400 | 0.75 | 5,525.5 | 4,636.666667 | 8,192 | |
Suppose $A B C$ is a triangle with circumcenter $O$ and orthocenter $H$ such that $A, B, C, O$, and $H$ are all on distinct points with integer coordinates. What is the second smallest possible value of the circumradius of $A B C$ ? | \sqrt{10} | Assume without loss of generality that the circumcenter is at the origin. By well known properties of the Euler line, the centroid $G$ is such that $O, G$, and $H$ are collinear, with $G$ in between $O$ and $H$, such that $G H=2 G O$. Thus, since $G=\frac{1}{3}(A+B+C)$, and we are assuming $O$ is the origin, we have $H... | 0 | 8,192 | -1 | 8,192 |
Given a square with a side length of $1$, if two points are randomly selected with equal probability from the center and the vertices of the square, what is the probability that the distance between these two points is $\frac{\sqrt{2}}{2}$? | \frac{2}{5} | 0.3125 | 6,340.625 | 4,215.2 | 7,306.727273 | |
Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ be vectors, and let $D$ be the determinant of the matrix whose column vectors are $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}.$ Then the determinant of the matrix whose column vectors are $\mathbf{a} \times \mathbf{b},$ $\mathbf{b} \times \mathbf{c},$ and $\mathbf{c} \times... | (1,2) | 0.375 | 6,813.125 | 4,515 | 8,192 | |
Compute the following expression:
\[ 2(1+2(1+2(1+2(1+2(1+2(1+2(1+2))))))) \] | 510 | 0.6875 | 4,936.875 | 4,095.272727 | 6,788.4 | |
In the diagram, all rows, columns, and diagonals have the sum 12. Find the sum of the four corner numbers. | 16 | 0.8125 | 5,116 | 5,081.615385 | 5,265 | |
The 5 a.m. temperatures for seven consecutive days were $-7^{\circ}$, $-4^{\circ}$, $-4^{\circ}$, $-5^{\circ}$, $1^{\circ}$, $3^{\circ}$ and $2^{\circ}$ Celsius. What is the mean 5 a.m. temperature for the week in degrees Celsius? | -2 | 0.9375 | 1,194.6875 | 1,250.333333 | 360 | |
Observe the following set of equations:
$S_{1}=1$,
$S_{2}=2+3+4=9$,
$S_{3}=3+4+5+6+7=25$,
$S_{4}=4+5+6+7+8+9+10=49$,
...
Based on the equations above, guess that $S_{2n-1}=(4n-3)(an+b)$, then $a^{2}+b^{2}=$_______. | 25 | 0.5625 | 5,483.25 | 3,376.444444 | 8,192 | |
$ABCD$ is a square of side length 1. $X$ and $Y$ are points on $BC$ and $CD$ respectively such that $CX = CY = m$. When extended, $AB$ meets $DX$ at $P$, $AD$ meets $BY$ at $Q$, $AX$ meets $DC$ at $R$, and $AY$ meets $BC$ at $S$. If $P, Q, R, S$ are collinear, find $m$. | \frac{3 - \sqrt{5}}{2} | 0 | 6,682.75 | -1 | 6,682.75 | |
Given the function $f(x) = \sqrt{3}\sin 2x - 2\cos^2x + 1$.
(1) Find the range of the function $f(x)$ in the interval $\left[-\frac{\pi}{12}, \frac{\pi}{2}\right)$.
(2) Let $\alpha, \beta \in \left(0, \frac{\pi}{2}\right)$, $f\left(\frac{1}{2}\alpha + \frac{\pi}{12}\right) = \frac{10}{13}$, $f\left(\frac{1}{2}\beta... | -\frac{33}{65} | 0.625 | 6,711.8125 | 5,823.7 | 8,192 | |
Given the complex numbers \( z = \cos \alpha + i \sin \alpha \) and \( u = \cos \beta + i \sin \beta \), and that \( z + u = \frac{4}{5} + \frac{3}{5} i \), find \( \tan(\alpha + \beta) \). | \frac{24}{7} | 0.8125 | 5,834.5625 | 5,290.538462 | 8,192 | |
Let $t(x) = 3x-8$ and $s(t(x)) = x^2 + 3x - 2$. Find $s(1)$. | 16 | 0.9375 | 2,942.0625 | 2,592.066667 | 8,192 | |
Given three positive numbers \( a, b, \mathrm{and} c \) satisfying \( a \leq b+c \leq 3a \) and \( 3b^2 \leq a(a+c) \leq 5b^2 \), what is the minimum value of \(\frac{b-2c}{a}\)? | -\frac{18}{5} | 0.125 | 8,066.5625 | 7,188.5 | 8,192 | |
An 18 inch by 24 inch painting is mounted in a wooden frame where the width of the wood at the top and bottom of the frame is twice the width of the wood at the sides. If the area of the frame is equal to the area of the painting, what is the ratio of the shorter side to the longer side of this frame? | 2:3 | 0 | 5,689.375 | -1 | 5,689.375 | |
Given that Chloe's telephone numbers have the form $555-ab-cdef$, where $a$, $b$, $c$, $d$, $e$, and $f$ are distinct digits, in descending order, and are chosen between $1$ and $8$, calculate the total number of possible telephone numbers that Chloe can have. | 28 | 0.8125 | 3,643.875 | 3,030.230769 | 6,303 | |
The equation $x^3 - 9x^2 + 8x +2 = 0$ has three real roots $p$, $q$, $r$. Find $\frac{1}{p^2} + \frac{1}{q^2} + \frac{1}{r^2}$. | 25 | 0.9375 | 4,479 | 4,231.466667 | 8,192 | |
How many distinct four-digit numbers composed of the digits $1$, $2$, $3$, and $4$ are even? | 12 | 0.875 | 3,024.5625 | 2,692 | 5,352.5 | |
Inside a cube with edge length 1, an inscribed sphere \( O_1 \) is drawn. Another smaller sphere \( O_2 \) is drawn inside the cube such that it is externally tangent to the larger sphere and simultaneously tangent to three faces of the cube. What is the surface area of the smaller sphere \( O_2 \)? | (7-4\sqrt{3})\pi | 0.5625 | 5,735.25 | 5,649 | 5,846.142857 | |
The side lengths of a triangle are distinct positive integers. One of the side lengths is a multiple of 42, and another is a multiple of 72. What is the minimum possible length of the third side? | 7 | Suppose that two of the side lengths are $42 a$ and $72 b$, for some positive integers $a$ and $b$. Let $c$ be the third side length. We know that $42 a$ is not equal to $72 b$, since the side lengths are distinct. Also, $6 \mid 42 a-72 b$. Therefore, by the triangle inequality, we get $c>|42 a-72 b| \geq 6$ and thus $... | 0.0625 | 7,688.125 | 7,779 | 7,682.066667 |
If P and Q are points on the line y = 1 - x and the curve y = -e^x, respectively, find the minimum value of |PQ|. | \sqrt{2} | 0.75 | 5,913.875 | 5,154.5 | 8,192 | |
In the number \(2016 * * * * 02 * *\), each of the 6 asterisks needs to be replaced with any of the digits \(0, 2, 4, 5, 7, 9\) (digits can repeat) so that the resulting 12-digit number is divisible by 15. How many ways can this be done? | 5184 | 0.1875 | 7,658.5 | 6,155.666667 | 8,005.307692 | |
How many pairs of positive integers $(m,n)$ satisfy $m^2 + n < 22?$ | 54 | 0.8125 | 4,513.25 | 3,664.307692 | 8,192 | |
Given triangle \( \triangle ABC \) with circumcenter \( O \) and orthocenter \( H \), and \( O \neq H \). Let \( D \) and \( E \) be the midpoints of sides \( BC \) and \( CA \) respectively. Let \( D' \) and \( E' \) be the reflections of \( D \) and \( E \) with respect to \( H \). If lines \( AD' \) and \( BE' \) in... | 3/2 | 0.1875 | 7,669.25 | 6,034.333333 | 8,046.538462 | |
Define $a \Delta b = a^2 -b $. What is the value of $ (2^{4 \Delta13})\Delta(3^{3\Delta5})$ | -17 | 0.9375 | 1,837.9375 | 1,813 | 2,212 | |
What is the probability that a randomly drawn positive factor of $60$ is less than $7$? | \frac{1}{2} |
To solve this problem, we need to determine the total number of positive factors of $60$ and how many of these factors are less than $7$. We then calculate the probability by dividing the number of favorable outcomes (factors less than $7$) by the total number of outcomes (total factors).
1. **Find the prime factoriz... | 1 | 2,967.125 | 2,967.125 | -1 |
In a game, two wheels are present. The first wheel has six segments with numbers 1 through 6. The second wheel has four segments, numbered 1, 1, 2, and 2. The game is to spin both wheels and add the resulting numbers. The player wins if the sum is a number less than 5. What is the probability of winning the game?
A) $\... | \frac{1}{3} | 0 | 8,014.25 | -1 | 8,014.25 |
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