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agentica-org/DeepScaleR-Preview-Dataset | A right rectangular prism $P_{}$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\le b\le c.$ A plane parallel to one of the faces of $P_{}$ cuts $P_{}$ into two prisms, one of which is similar to $P_{},$ and both of which have nonzero volume. Given that $b=1995,$ for how many ordered triples $(a, b, c)$ does such a plane exist? | 40 |
|
agentica-org/DeepScaleR-Preview-Dataset | Consider license plates consisting of a sequence of four digits followed by two letters. Assume each arrangement is equally likely for these plates. What is the probability that such a license plate contains at least one palindrome sequence (either the four-digit sequence or the two-letter sequence)? Express your result as a simplified fraction. | \frac{5}{104} |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x) + y) = f(x^2 - y) + 4f(x) y\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(3),$ and let $s$ be the sum of all possible values of $f(3).$ Find $n \times s.$ | 18 |
|
agentica-org/DeepScaleR-Preview-Dataset | An array of integers is arranged in a grid of 7 rows and 1 column with eight additional squares forming a separate column to the right. The sequence of integers in the main column of squares and in each of the two rows form three distinct arithmetic sequences. Find the value of $Q$ if the sequence in the additional columns only has one number given.
[asy]
unitsize(0.35inch);
draw((0,0)--(0,7)--(1,7)--(1,0)--cycle);
draw((0,1)--(1,1));
draw((0,2)--(1,2));
draw((0,3)--(1,3));
draw((0,4)--(1,4));
draw((0,5)--(1,5));
draw((0,6)--(1,6));
draw((1,5)--(2,5)--(2,0)--(1,0)--cycle);
draw((1,1)--(2,1));
draw((1,2)--(2,2));
draw((1,3)--(2,3));
draw((1,4)--(2,4));
label("-9",(0.5,6.5),S);
label("56",(0.5,2.5),S);
label("$Q$",(1.5,4.5),S);
label("16",(1.5,0.5),S);
[/asy] | \frac{-851}{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | A store prices an item so that when 5% sales tax is added to the price in cents, the total cost rounds naturally to the nearest multiple of 5 dollars. What is the smallest possible integer dollar amount $n$ to which the total cost could round?
A) $50$
B) $55$
C) $60$
D) $65$
E) $70$ | 55 |
|
agentica-org/DeepScaleR-Preview-Dataset | For a finite sequence $B = (b_1, b_2, \dots, b_n)$ of numbers, the Cesaro sum is defined as
\[\frac{T_1 + T_2 + \cdots + T_n}{n},\]
where $T_k = b_1 + b_2 + \cdots + b_k$ for $1 \leq k \leq n$.
If the Cesaro sum of the 100-term sequence $(b_1, b_2, \dots, b_{100})$ is 1200, where $b_1 = 2$, calculate the Cesaro sum of the 101-term sequence $(3, b_1, b_2, \dots, b_{100})$. | 1191 |
|
agentica-org/DeepScaleR-Preview-Dataset | The sum of the lengths of the twelve edges of a rectangular box is $140$, and the distance from one corner of the box to the farthest corner is $21$. What is the total surface area of the box? | 784 |
|
agentica-org/DeepScaleR-Preview-Dataset | What is the smallest base-10 integer that can be represented as $XX_6$ and $YY_8$, where $X$ and $Y$ are valid digits in their respective bases? | 63 |
|
agentica-org/DeepScaleR-Preview-Dataset | In the Cartesian coordinate system $(xOy)$, a curve $C_{1}$ is defined by the parametric equations $x=\cos{\theta}$ and $y=\sin{\theta}$, and a line $l$ is defined by the polar equation $\rho(2\cos{\theta} - \sin{\theta}) = 6$.
1. Find the Cartesian equations for the curve $C_{1}$ and the line $l$.
2. Find a point $P$ on the curve $C_{1}$ such that the distance from $P$ to the line $l$ is minimized, and compute this minimum distance. | \frac{6\sqrt{5}}{5} - 1 |
|
agentica-org/DeepScaleR-Preview-Dataset | A shopping mall sells a batch of branded shirts, with an average daily sales volume of $20$ shirts, and a profit of $40$ yuan per shirt. In order to expand sales and increase profits, the mall decides to implement an appropriate price reduction strategy. After investigation, it was found that for every $1$ yuan reduction in price per shirt, the mall can sell an additional $2$ shirts on average.
$(1)$ If the price reduction per shirt is set at $x$ yuan, and the average daily profit is $y$ yuan, find the functional relationship between $y$ and $x$.
$(2)$ At what price reduction per shirt will the mall have the maximum average daily profit?
$(3)$ If the mall needs an average daily profit of $1200$ yuan, how much should the price per shirt be reduced? | 20 |
|
agentica-org/DeepScaleR-Preview-Dataset | If $\left|\frac{12}{x}+3\right|=2$, find the product of all possible values of $x$. Express your answer as an improper fraction. | \frac{144}{5} |
|
agentica-org/DeepScaleR-Preview-Dataset | A circle having radius $r_{1}$ centered at point $N$ is tangent to a circle of radius $r_{2}$ centered at $M$. Let $l$ and $j$ be the two common external tangent lines to the two circles. A circle centered at $P$ with radius $r_{2}$ is externally tangent to circle $N$ at the point at which $l$ coincides with circle $N$, and line $k$ is externally tangent to $P$ and $N$ such that points $M, N$, and $P$ all lie on the same side of $k$. For what ratio $r_{1} / r_{2}$ are $j$ and $k$ parallel? | 3 |
|
agentica-org/DeepScaleR-Preview-Dataset | Write $2017$ following numbers on the blackboard: $-\frac{1008}{1008}, -\frac{1007}{1008}, ..., -\frac{1}{1008}, 0,\frac{1}{1008},\frac{2}{1008}, ... ,\frac{1007}{1008},\frac{1008}{1008}$ .
One processes some steps as: erase two arbitrary numbers $x, y$ on the blackboard and then write on it the number $x + 7xy + y$ . After $2016$ steps, there is only one number. The last one on the blackboard is | $-\frac{144}{1008}$ |
|
agentica-org/DeepScaleR-Preview-Dataset | Shuffle the 12 cards of the four suits of A, K, Q in a deck of playing cards.
(1) If a person draws 2 cards at random, what is the probability that both cards are Aces?
(2) If the person has already drawn 2 Kings without replacement, what is the probability that another person draws 2 Aces? | \frac{2}{15} |
|
agentica-org/DeepScaleR-Preview-Dataset | For each positive integer $n$, let $a_{n}$ be the smallest nonnegative integer such that there is only one positive integer at most $n$ that is relatively prime to all of $n, n+1, \ldots, n+a_{n}$. If $n<100$, compute the largest possible value of $n-a_{n}$. | 16 |
|
agentica-org/DeepScaleR-Preview-Dataset | Quadrilateral $EFGH$ is a parallelogram. A line through point $G$ makes a $30^\circ$ angle with side $GH$. Determine the degree measure of angle $E$.
[asy]
size(100);
draw((0,0)--(5,2)--(6,7)--(1,5)--cycle);
draw((5,2)--(7.5,3)); // transversal line
draw(Arc((5,2),1,-60,-20)); // transversal angle
label("$H$",(0,0),SW); label("$G$",(5,2),SE); label("$F$",(6,7),NE); label("$E$",(1,5),NW);
label("$30^\circ$",(6.3,2.8), E);
[/asy] | 150 |
|
agentica-org/DeepScaleR-Preview-Dataset | Dan is walking down the left side of a street in New York City and must cross to the right side at one of 10 crosswalks he will pass. Each time he arrives at a crosswalk, however, he must wait $t$ seconds, where $t$ is selected uniformly at random from the real interval $[0,60](t$ can be different at different crosswalks). Because the wait time is conveniently displayed on the signal across the street, Dan employs the following strategy: if the wait time when he arrives at the crosswalk is no more than $k$ seconds, he crosses. Otherwise, he immediately moves on to the next crosswalk. If he arrives at the last crosswalk and has not crossed yet, then he crosses regardless of the wait time. Find the value of $k$ which minimizes his expected wait time. | 60\left(1-\left(\frac{1}{10}\right)^{\frac{1}{9}}\right) |
|
agentica-org/DeepScaleR-Preview-Dataset | Triangle $ABC$ has $BC=20.$ The incircle of the triangle evenly trisects the median $AD.$ If the area of the triangle is $m \sqrt{n}$ where $m$ and $n$ are integers and $n$ is not divisible by the square of a prime, find $m+n.$ | 38 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given that $\sin{\alpha} = -\frac{3}{5}$, $\sin{\beta} = \frac{12}{13}$, and $\alpha \in (\pi, \frac{3\pi}{2})$, $\beta \in (\frac{\pi}{2}, \pi)$, find the values of $\sin({\alpha - \beta})$, $\cos{2\alpha}$, and $\tan{\frac{\beta}{2}}$. | \frac{3}{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $a_1, a_2, \dots, a_k$ be a finite arithmetic sequence with $a_4 + a_7 + a_{10} = 17$ and $a_4 + a_5 + \dots + a_{13} + a_{14} = 77$. If $a_k = 13$, then $k = $ | 18 |
|
agentica-org/DeepScaleR-Preview-Dataset | If 260 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 10 |
|
agentica-org/DeepScaleR-Preview-Dataset | A marine biologist interested in monitoring a specific fish species population in a coastal area. On January 15, he captures and tags 80 fish, then releases them back into the water. On June 15, he captures another sample of 100 fish, finding that 6 of them are tagged. He assumes that 20% of the tagged fish have died or migrated out of the area by June 15, and also that 50% of the fish in the June sample are recent additions due to birth or migration. How many fish were in the coastal area on January 15, based on his assumptions? | 533 |
|
agentica-org/DeepScaleR-Preview-Dataset | Complete the table below, discover the patterns of square roots and cube roots, and apply the patterns to solve the problem.
| $x$ | $\ldots $ | $0.064$ | $0.64$ | $64$ | $6400$ | $64000$ | $\ldots $ |
|---------|-----------|---------|--------|-------|--------|---------|-----------|
| $\sqrt{x}$ | $\ldots $ | $0.25298$ | $0.8$ | $8$ | $m$ | $252.98$ | $\ldots $ |
| $\sqrt[3]{x}$ | $\ldots $ | $n$ | $0.8618$ | $4$ | $18.566$ | $40$ | $\ldots $ |
$(1)$ $m=$______, $n= \_\_\_\_\_\_.$
$(2)$ From the numbers in the table, it can be observed that when finding the square root, if the decimal point of the radicand moves two places to the left (or right), the decimal point of its square root moves one place to the left (or right). Please describe in words the pattern of cube root: ______.
$(3)$ If $\sqrt{a}≈14.142$, $\sqrt[3]{700}≈b$, find the value of $a+b$. (Reference data: $\sqrt{2}≈1.4142$, $\sqrt{20}≈4.4721$, $\sqrt[3]{7}≈1.9129$, $\sqrt[3]{0.7}≈0.8879$) | 208.879 |
|
agentica-org/DeepScaleR-Preview-Dataset | What is the remainder when $13^{51}$ is divided by 5? | 2 |
|
agentica-org/DeepScaleR-Preview-Dataset | A pentagon is formed by placing an equilateral triangle on top of a square. Calculate the percentage of the pentagon's total area that is made up by the equilateral triangle. | 25.4551\% |
|
agentica-org/DeepScaleR-Preview-Dataset | Find $2 \cdot 5^{-1} + 8 \cdot 11^{-1} \pmod{56}$.
Express your answer as an integer from $0$ to $55$, inclusive. | 50 |
|
agentica-org/DeepScaleR-Preview-Dataset | What is the probability that Claudia gets at least 2 more heads than tails if she flips 12 coins? | \frac{793}{2048} |
|
agentica-org/DeepScaleR-Preview-Dataset | In a cultural performance, there are already 10 programs arranged in the program list. Now, 3 more programs are to be added, with the requirement that the relative order of the originally scheduled 10 programs remains unchanged. How many different arrangements are there for the program list? (Answer with a number). | 1716 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $z$ be a complex number such that
\[|z - 12| + |z - 5i| = 13.\]Find the smallest possible value of $|z|.$ | \frac{60}{13} |
|
agentica-org/DeepScaleR-Preview-Dataset | Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with a focal distance of $2\sqrt{3}$, the line $l_1: y = kx$ ($k \neq 0$) intersects the ellipse at points A and B. A line $l_2$ passing through point B with a slope of $\frac{1}{4}k$ intersects the ellipse at another point D, and $AD \perp AB$.
1. Find the equation of the ellipse.
2. Suppose the line $l_2$ intersects the x-axis and y-axis at points M and N, respectively. Find the maximum value of the area of $\triangle OMN$. | \frac{9}{8} |
|
agentica-org/DeepScaleR-Preview-Dataset | Compute
\[\frac{\lfloor \sqrt[4]{1} \rfloor \cdot \lfloor \sqrt[4]{3} \rfloor \cdot \lfloor \sqrt[4]{5} \rfloor \dotsm \lfloor \sqrt[4]{2015} \rfloor}{\lfloor \sqrt[4]{2} \rfloor \cdot \lfloor \sqrt[4]{4} \rfloor \cdot \lfloor \sqrt[4]{6} \rfloor \dotsm \lfloor \sqrt[4]{2016} \rfloor}.\] | \frac{5}{16} |
|
agentica-org/DeepScaleR-Preview-Dataset | When four positive integers are divided by $11$, the remainders are $2,$ $4,$ $6,$ and $8,$ respectively.
When the sum of the four integers is divided by $11$, what is the remainder? | 9 |
|
agentica-org/DeepScaleR-Preview-Dataset | How many units are in the sum of the lengths of the two longest altitudes in a right triangle with sides $9$, $40$, and $41$? | 49 |
|
agentica-org/DeepScaleR-Preview-Dataset | Points $A(11, 9)$ and $B(2, -3)$ are vertices of $ riangle ABC$ with $AB=AC$. The altitude from $A$ meets the opposite side at $D(-1, 3)$. What are the coordinates of point $C$? | $\left( -4, 9 \right)$ |
|
agentica-org/DeepScaleR-Preview-Dataset | A prism has 15 edges. How many faces does the prism have? | 7 |
|
agentica-org/DeepScaleR-Preview-Dataset | The workers laid a floor of size $n\times n$ ($10 <n <20$) with two types of tiles: $2 \times 2$ and $5\times 1$. It turned out that they were able to completely lay the floor so that the same number of tiles of each type was used. For which $n$ could this happen? (You can’t cut tiles and also put them on top of each other.) | 12, 15, 18 |
|
agentica-org/DeepScaleR-Preview-Dataset | A square with sides 6 inches is shown. If $P$ is a point such that the segment $\overline{PA}$, $\overline{PB}$, $\overline{PC}$ are equal in length, and segment $\overline{PC}$ is perpendicular to segment $\overline{FD}$, what is the area, in square inches, of triangle $APB$? [asy]
pair A, B, C, D, F, P;
A = (0,0); B= (2,0); C = (1,2); D = (2,2); F = (0,2); P = (1,1);
draw(A--B--D--F--cycle);
draw(C--P); draw(P--A); draw(P--B);
label("$A$",A,SW); label("$B$",B,SE);label("$C$",C,N);label("$D$",D,NE);label("$P$",P,NW);label("$F$",F,NW);
label("$6''$",(1,0),S);
[/asy] | \dfrac{27}{4} |
|
agentica-org/DeepScaleR-Preview-Dataset | Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition
\[ f(m \plus{} n) \geq f(m) \plus{} f(f(n)) \minus{} 1
\]
for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$
[i]Author: Nikolai Nikolov, Bulgaria[/i] | 1, 2, \ldots, 2008 |
|
agentica-org/DeepScaleR-Preview-Dataset | When evaluated, the sum of the digits of the integer equal to \(10^{2021} - 2021\) is: | 18185 |
|
agentica-org/DeepScaleR-Preview-Dataset | Evaluate \[\frac{3}{\log_8{5000^4}} + \frac{2}{\log_9{5000^4}},\] giving your answer as a fraction in lowest terms. | \frac{1}{4} |
|
agentica-org/DeepScaleR-Preview-Dataset | There are eight different symbols designed on $n\geq 2$ different T-shirts. Each shirt contains at least one symbol, and no two shirts contain all the same symbols. Suppose that for any $k$ symbols $(1\leq k\leq 7)$ the number of shirts containing at least one of the $k$ symbols is even. Determine the value of $n$ . | 255 |
|
agentica-org/DeepScaleR-Preview-Dataset | Medians $\overline{DP}$ and $\overline{EQ}$ of isosceles $\triangle DEF$, where $DE=EF$, are perpendicular. If $DP= 21$ and $EQ = 28$, then what is ${DE}$? | \frac{70}{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | A basketball team has 20 players. The coach needs to choose a starting lineup consisting of one point guard and 7 other players (these players are interchangeable). How many different starting lineups can the coach choose? | 1007760 |
|
agentica-org/DeepScaleR-Preview-Dataset | For odd primes $p$, let $f(p)$ denote the smallest positive integer $a$ for which there does not exist an integer $n$ satisfying $p \mid n^{2}-a$. Estimate $N$, the sum of $f(p)^{2}$ over the first $10^{5}$ odd primes $p$. An estimate of $E>0$ will receive $\left\lfloor 22 \min (N / E, E / N)^{3}\right\rfloor$ points. | 2266067 |
|
agentica-org/DeepScaleR-Preview-Dataset | In the diagram, the circle has center \( O \) and square \( OPQR \) has vertex \( Q \) on the circle. If the area of the circle is \( 72 \pi \), the area of the square is: | 36 |
|
agentica-org/DeepScaleR-Preview-Dataset | In a unit cube \( ABCD-A_1B_1C_1D_1 \), let \( O \) be the center of the square \( ABCD \). Points \( M \) and \( N \) are located on edges \( A_1D_1 \) and \( CC_1 \) respectively, with \( A_1M = \frac{1}{2} \) and \( CN = \frac{2}{3} \). Find the volume of the tetrahedron \( OMNB_1 \). | 11/72 |
|
agentica-org/DeepScaleR-Preview-Dataset | The wavelength of red light that the human eye can see is $0.000077$ cm. Please round the data $0.000077$ to $0.00001$ and express it in scientific notation as ______. | 8 \times 10^{-5} |
|
agentica-org/DeepScaleR-Preview-Dataset | Express -2013 in base -4. | 200203_{-4} |
|
agentica-org/DeepScaleR-Preview-Dataset | Given a triangle $ABC$ with the sides opposite to angles $A$, $B$, $C$ denoted by $a$, $b$, $c$ respectively, let vectors $\overrightarrow{m}=(1-\cos(A+B), \cos \frac{A-B}{2})$ and $\overrightarrow{n}=(\frac{5}{8}, \cos \frac{A-B}{2})$, and it's known that $\overrightarrow{m} \cdot \overrightarrow{n} = \frac{9}{8}$.
1. Find the value of $\tan A \cdot \tan B$.
2. Find the maximum value of $\frac{a b \sin C}{a^2 + b^2 - c^2}$. | -\frac{3}{8} |
|
agentica-org/DeepScaleR-Preview-Dataset | The line $y=kx$ intersects the graph of the function $y=\tan x$ ($-\frac{π}{2}＜x＜\frac{π}{2}$) at points $M$ and $N$ (not coinciding with the origin $O$). The coordinates of point $A$ are $(-\frac{π}{2},0)$. Find $(\overrightarrow{AM}+\overrightarrow{AN})\cdot\overrightarrow{AO}$. | \frac{\pi^2}{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | Four football teams participate in a round-robin tournament, where each team plays a match against every other team. In each match, the winning team earns 3 points, the losing team earns 0 points, and in the case of a draw, both teams earn 1 point each. After all matches are completed, it is known that the total points of the four teams are exactly four consecutive positive integers. Find the product of these four numbers. | 120 |
|
agentica-org/DeepScaleR-Preview-Dataset | Triangle $\triangle P Q R$, with $P Q=P R=5$ and $Q R=6$, is inscribed in circle $\omega$. Compute the radius of the circle with center on $\overline{Q R}$ which is tangent to both $\omega$ and $\overline{P Q}$. | \frac{20}{9} |
|
agentica-org/DeepScaleR-Preview-Dataset | Given an isosceles right triangle ∆PQR with hypotenuse PQ = 4√2, let S be the midpoint of PR. On segment QR, point T divides it so that QT:TR = 2:1. Calculate the area of ∆PST. | \frac{8}{3} |
|
agentica-org/DeepScaleR-Preview-Dataset | In an \(8 \times 8\) table, some cells are black, and the rest are white. In each white cell, the total number of black cells located in the same row or column is written. Nothing is written in the black cells. What is the maximum possible value of the sum of the numbers in the entire table? | 256 |
|
agentica-org/DeepScaleR-Preview-Dataset | $A, B, C, D, E, F, G$ are seven people sitting around a circular table. If $d$ is the total number of ways that $B$ and $G$ must sit next to $C$, find the value of $d$. | 48 |
|
agentica-org/DeepScaleR-Preview-Dataset | In the triangle $ABC$ it is known that $\angle A = 75^o, \angle C = 45^o$ . On the ray $BC$ beyond the point $C$ the point $T$ is taken so that $BC = CT$ . Let $M$ be the midpoint of the segment $AT$ . Find the measure of the $\angle BMC$ .
(Anton Trygub) | 45 |
|
agentica-org/DeepScaleR-Preview-Dataset | What percent of the palindromes between 1000 and 2000 contain at least one 7? | 12\% |
|
agentica-org/DeepScaleR-Preview-Dataset | Given positive integers \(a\), \(b\) \((a \leq b)\), a sequence \(\{ f_{n} \}\) satisfies:
\[ f_{1} = a, \, f_{2} = b, \, f_{n+2} = f_{n+1} + f_{n} \text{ for } n = 1, 2, \ldots \]
If for any positive integer \(n\), it holds that
\[ \left( \sum_{k=1}^{n} f_{k} \right)^2 \leq \lambda \cdot f_{n} f_{n+1}, \]
find the minimum value of the real number \(\lambda\). | 2 + \sqrt{5} |
|
agentica-org/DeepScaleR-Preview-Dataset | Vasya and Petya simultaneously started running from the starting point of a circular track in opposite directions at constant speeds. At some point, they met. Vasya completed a full lap and, continuing to run in the same direction, reached the point of their first meeting at the same moment Petya completed a full lap. Find the ratio of Vasya's speed to Petya's speed. | \frac{1+\sqrt{5}}{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | A triangle has vertices $A=(4,3)$, $B=(-4,-1)$, and $C=(9,-7)$. Calculate the equation of the bisector of $\angle A$ in the form $3x - by + c = 0$. Determine the value of $b+c$. | -6 |
|
agentica-org/DeepScaleR-Preview-Dataset | In the convex pentagon $ABCDE$, $\angle A = \angle B = 120^{\circ}$, $EA = AB = BC = 2$, and $CD = DE = 4$. The area of $ABCDE$ is | $7 \sqrt{3}$ |
|
agentica-org/DeepScaleR-Preview-Dataset | In a right trapezoid \(ABCD\), the sum of the lengths of the bases \(AD\) and \(BC\) is equal to its height \(AB\). In what ratio does the angle bisector of \(\angle ABC\) divide the lateral side \(CD\)? | 1:1 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given the function $f(x) = x^2 - 2\cos{\theta}x + 1$, where $x \in \left[-\frac{\sqrt{3}}{2}, \frac{1}{2}\right]$.
(1) When $\theta = \frac{\pi}{3}$, find the maximum and minimum values of $f(x)$.
(2) If $f(x)$ is a monotonous function on $x \in \left[-\frac{\sqrt{3}}{2}, \frac{1}{2}\right]$ and $\theta \in [0, 2\pi)$, find the range of $\theta$.
(3) If $\sin{\alpha}$ and $\cos{\alpha}$ are the two real roots of the equation $f(x) = \frac{1}{4} + \cos{\theta}$, find the value of $\frac{\tan^2{\alpha} + 1}{\tan{\alpha}}$. | \frac{16 + 4\sqrt{11}}{5} |
|
agentica-org/DeepScaleR-Preview-Dataset | There is a room with four doors. Determine the number of different ways for someone to enter and exit this room. | 16 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given that $\sin 2α - 2 = 2 \cos 2α$, find the value of ${\sin}^{2}α + \sin 2α$. | \frac{8}{5} |
|
agentica-org/DeepScaleR-Preview-Dataset | Compute: $5^2-3(4)+3^2$. | 22 |
|
agentica-org/DeepScaleR-Preview-Dataset | What is the probability that all 4 blue marbles are drawn before all 3 yellow marbles are drawn? | \frac{4}{7} |
|
agentica-org/DeepScaleR-Preview-Dataset | Give an example of a number $x$ for which the equation $\sin 2017 x - \operatorname{tg} 2016 x = \cos 2015 x$ holds. Justify your answer. | \frac{\pi}{4} |
|
agentica-org/DeepScaleR-Preview-Dataset | [asy]
draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle, black+linewidth(.75));
draw((0,-1)--(0,1), black+linewidth(.75));
draw((-1,0)--(1,0), black+linewidth(.75));
draw((-1,-1/sqrt(3))--(1,1/sqrt(3)), black+linewidth(.75));
draw((-1,1/sqrt(3))--(1,-1/sqrt(3)), black+linewidth(.75));
draw((-1/sqrt(3),-1)--(1/sqrt(3),1), black+linewidth(.75));
draw((1/sqrt(3),-1)--(-1/sqrt(3),1), black+linewidth(.75));
[/asy]
Amy painted a dartboard over a square clock face using the "hour positions" as boundaries. If $t$ is the area of one of the eight triangular regions such as that between 12 o'clock and 1 o'clock, and $q$ is the area of one of the four corner quadrilaterals such as that between 1 o'clock and 2 o'clock, then $\frac{q}{t}=$ | 2\sqrt{3}-2 |
|
agentica-org/DeepScaleR-Preview-Dataset | Rectangle $ABCD$ is the base of pyramid $PABCD$. If $AB = 8$, $BC = 4$, $\overline{PA}\perp \overline{AD}$, $\overline{PA}\perp \overline{AB}$, and $PB = 17$, then what is the volume of $PABCD$? | 160 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given that $m$ is an integer and $0<3m<27$, what is the sum of all possible integer values of $m$? | 36 |
|
agentica-org/DeepScaleR-Preview-Dataset | For the consumer, a single discount of $n\%$ is more advantageous than any of the following discounts:
(1) two successive $15\%$ discounts
(2) three successive $10\%$ discounts
(3) a $25\%$ discount followed by a $5\%$ discount
What is the smallest possible positive integer value of $n$? | 29 |
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agentica-org/DeepScaleR-Preview-Dataset | Let $\mathcal{P}$ be the parabola in the plane determined by the equation $y = 2x^2$. Suppose a circle $\mathcal{C}$ intersects $\mathcal{P}$ at four distinct points. If three of these points are $(-7, 98)$, $(-1, 2)$, and $(6, 72)$, find the sum of the distances from the vertex of $\mathcal{P}$ to all four intersection points. | \sqrt{9653} + \sqrt{5} + \sqrt{5220} + \sqrt{68} |
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agentica-org/DeepScaleR-Preview-Dataset | Figure $ABCD$ is a square. Inside this square three smaller squares are drawn with side lengths as labeled. What is the area of the shaded $\text L$-shaped region? [asy]
/* AMC8 2000 #6 Problem */
draw((0,0)--(5,0)--(5,5)--(0,5)--cycle);
draw((1,5)--(1,1)--(5,1));
draw((0,4)--(4,4)--(4,0));
fill((0,4)--(1,4)--(1,1)--(4,1)--(4,0)--(0,0)--cycle);
label("$A$", (5,5), NE);
label("$B$", (5,0), SE);
label("$C$", (0,0), SW);
label("$D$", (0,5), NW);
label("1",(.5,5), N);
label("1",(1,4.5), E);
label("1",(4.5,1), N);
label("1",(4,.5), E);
label("3",(1,2.5), E);
label("3",(2.5,1), N);
[/asy] | 7 |
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agentica-org/DeepScaleR-Preview-Dataset | Add $5.467$ and $3.92$ as a decimal. | 9.387 |
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agentica-org/DeepScaleR-Preview-Dataset | $1000 \times 1993 \times 0.1993 \times 10 =$ | $(1993)^2$ |
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agentica-org/DeepScaleR-Preview-Dataset | A rectangular chessboard of size \( m \times n \) is composed of unit squares (where \( m \) and \( n \) are positive integers not exceeding 10). A piece is placed on the unit square in the lower-left corner. Players A and B take turns moving the piece. The rules are as follows: either move the piece any number of squares upward, or any number of squares to the right, but you cannot move off the board or stay in the same position. The player who cannot make a move loses (i.e., the player who first moves the piece to the upper-right corner wins). How many pairs of integers \( (m, n) \) are there such that the first player A has a winning strategy? | 90 |
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agentica-org/DeepScaleR-Preview-Dataset | In triangle $XYZ$, $XY = 12$, $XZ = 15$, and $YZ = 23$. The medians $XM$, $YN$, and $ZO$ of triangle $XYZ$ intersect at the centroid $G$. Let $Q$ be the foot of the altitude from $G$ to $YZ$. Find $GQ$. | \frac{40}{23} |
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agentica-org/DeepScaleR-Preview-Dataset | Let \(ABC\) be a triangle such that \(\frac{|BC|}{|AB| - |BC|} = \frac{|AB| + |BC|}{|AC|}\). Determine the ratio \(\angle A : \angle C\). | 1 : 2 |
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agentica-org/DeepScaleR-Preview-Dataset | A cylinder with radius 15 and height 16 is inscribed in a sphere. Three congruent smaller spheres of radius $x$ are externally tangent to the base of the cylinder, externally tangent to each other, and internally tangent to the large sphere. What is the value of $x$? | \frac{15 \sqrt{37}-75}{4} |
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agentica-org/DeepScaleR-Preview-Dataset | Points $A,B,C$ and $D$ lie on a line in that order, with $AB = CD$ and $BC = 16$. Point $E$ is not on the line, and $BE = CE = 13$. The perimeter of $\triangle AED$ is three times the perimeter of $\triangle BEC$. Find $AB$.
A) $\frac{32}{3}$
B) $\frac{34}{3}$
C) $\frac{36}{3}$
D) $\frac{38}{3}$ | \frac{34}{3} |
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agentica-org/DeepScaleR-Preview-Dataset | How many degrees are in each interior angle of a regular hexagon? | 120^\circ |
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agentica-org/DeepScaleR-Preview-Dataset | What is the largest quotient that can be formed using two numbers chosen from the set $\{-30, -6, -1, 3, 5, 20\}$, where one of the numbers must be negative? | -0.05 |
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agentica-org/DeepScaleR-Preview-Dataset | Given the function $f(x) = 2\sin(\frac{1}{3}x - \frac{π}{6})$, where $x \in \mathbb{R}$.
(1) Find the value of $f(\frac{5π}{4})$;
(2) Let $\alpha, \beta \in [0, \frac{π}{2}], f(3\alpha + \frac{π}{2}) = \frac{10}{13}, f(3\beta + 2π) = \frac{6}{5}$, find the value of $\cos(\alpha + \beta)$. | \frac{16}{65} |
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agentica-org/DeepScaleR-Preview-Dataset | Given a checkerboard with 31 rows and 29 columns, where each corner square is black and the squares alternate between red and black, determine the number of black squares on this checkerboard. | 465 |
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agentica-org/DeepScaleR-Preview-Dataset | A set $\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{S},$ the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that 1 belongs to $\mathcal{S}$ and that 2002 is the largest element of $\mathcal{S},$ what is the greatest number of elements that $\mathcal{S}$ can have? | 30 |
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agentica-org/DeepScaleR-Preview-Dataset | Given a point P $(x, y)$ moves on the circle $x^2 + (y-1)^2 = 1$, the maximum value of $\frac{y-1}{x-2}$ is \_\_\_\_\_\_, and the minimum value is \_\_\_\_\_\_. | -\frac{\sqrt{3}}{3} |
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agentica-org/DeepScaleR-Preview-Dataset | Given that $\sin\alpha = \frac{3}{5}$, and $\alpha \in \left(\frac{\pi}{2}, \pi \right)$.
(1) Find the value of $\tan\left(\alpha+\frac{\pi}{4}\right)$;
(2) If $\beta \in (0, \frac{\pi}{2})$, and $\cos(\alpha-\beta) = \frac{1}{3}$, find the value of $\cos\beta$. | \frac{6\sqrt{2} - 4}{15} |
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agentica-org/DeepScaleR-Preview-Dataset | The surface of a 3 x 3 x 3 Rubik's Cube consists of 54 cells. What is the maximum number of cells you can mark such that the marked cells do not share any vertices? | 14 |
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agentica-org/DeepScaleR-Preview-Dataset | Find the positive integer $n$ such that \[ \underbrace{f(f(\cdots f}_{2013 \ f\text{'s}}(n)\cdots ))=2014^2+1 \] where $f(n)$ denotes the $n$ th positive integer which is not a perfect square.
*Proposed by David Stoner* | 6077248 |
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agentica-org/DeepScaleR-Preview-Dataset | What is the largest value of $n$ less than 100,000 for which the expression $8(n-2)^5-n^2+14n-24$ is a multiple of 5? | 99997 |
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agentica-org/DeepScaleR-Preview-Dataset | Define \[P(x) =(x-1^2)(x-2^2)\cdots(x-50^2).\] How many integers $n$ are there such that $P(n)\leq 0$? | 1300 |
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agentica-org/DeepScaleR-Preview-Dataset | The diagonal of a regular 2006-gon \(P\) is called good if its ends divide the boundary of \(P\) into two parts, each containing an odd number of sides. The sides of \(P\) are also called good. Let \(P\) be divided into triangles by 2003 diagonals, none of which have common points inside \(P\). What is the maximum number of isosceles triangles, each of which has two good sides, that such a division can have? | 1003 |
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agentica-org/DeepScaleR-Preview-Dataset | Square $P Q R S$ has an area of 900. $M$ is the midpoint of $P Q$ and $N$ is the midpoint of $P S$. What is the area of triangle $P M N$? | 112.5 |
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agentica-org/DeepScaleR-Preview-Dataset | Let $\{a_{n}\}$ be a geometric sequence, and let $S_{n}$ be the sum of the first n terms of $\{a_{n}\}$. Given that $S_{2}=2$ and $S_{6}=4$, calculate the value of $S_{4}$. | 1+\sqrt{5} |
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agentica-org/DeepScaleR-Preview-Dataset | Compute the number of functions $f:\{1,2, \ldots, 9\} \rightarrow\{1,2, \ldots, 9\}$ which satisfy $f(f(f(f(f(x)))))=$ $x$ for each $x \in\{1,2, \ldots, 9\}$. | 3025 |
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agentica-org/DeepScaleR-Preview-Dataset | Using the numbers 0, 1, 2, 3, 4, 5 to form unique three-digit numbers, determine the total number of even numbers that can be formed. | 52 |
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agentica-org/DeepScaleR-Preview-Dataset | What is the largest quotient that can be obtained using two numbers from the set $\{ -30, -4, 0, 3, 5, 10 \}$? | 7.5 |
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agentica-org/DeepScaleR-Preview-Dataset | The ages of Jo, her daughter, and her grandson are all even numbers. The product of their three ages is 2024. How old is Jo? | 46 |
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agentica-org/DeepScaleR-Preview-Dataset | In 500 kg of ore, there is a certain amount of iron. After removing 200 kg of impurities, which contain on average 12.5% iron, the iron content in the remaining ore increased by 20%. What amount of iron remains in the ore? | 187.5 |
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