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agentica-org/DeepScaleR-Preview-Dataset
|
Given a right triangle \(ABC\) with a right angle at \(A\). On the leg \(AC\), a point \(D\) is marked such that \(AD:DC = 1:3\). Circles \(\Gamma_1\) and \(\Gamma_2\) are then drawn with centers at \(A\) and \(C\) respectively, both passing through point \(D\). \(\Gamma_2\) intersects the hypotenuse at point \(E\). Another circle \(\Gamma_3\) with center at \(B\) and radius \(BE\) intersects \(\Gamma_1\) inside the triangle at a point \(F\) such that \(\angle AFB\) is a right angle. Find \(BC\), given that \(AB = 5\).
|
13
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Consider a sphere inscribed in a right cone with the base radius of 10 cm and height of 40 cm. The radius of the inscribed sphere can be expressed as $b\sqrt{d} - b$ cm. Determine the value of $b+d$.
|
19.5
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given that out of 8 teams, there are 3 weak teams, these 8 teams are divided into two groups $A$ and $B$ with 4 teams in each group by drawing lots.
1. The probability that one of the groups $A$ or $B$ has exactly two weak teams.
2. The probability that group $A$ has at least two weak teams.
|
\frac{1}{2}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In $\triangle{ABC}, AB=10, \angle{A}=30^\circ$ , and $\angle{C=45^\circ}$. Let $H, D,$ and $M$ be points on the line $BC$ such that $AH\perp{BC}$, $\angle{BAD}=\angle{CAD}$, and $BM=CM$. Point $N$ is the midpoint of the segment $HM$, and point $P$ is on ray $AD$ such that $PN\perp{BC}$. Then $AP^2=\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
|
77
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given the curve $C:\begin{cases}x=2\cos a \\ y= \sqrt{3}\sin a\end{cases} (a$ is the parameter) and the fixed point $A(0,\sqrt{3})$, ${F}_1,{F}_2$ are the left and right foci of this curve, respectively. With the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established.
$(1)$ Find the polar equation of the line $AF_{2}$;
$(2)$ A line passing through point ${F}_1$ and perpendicular to the line $AF_{2}$ intersects this conic curve at points $M$, $N$, find the value of $||MF_{1}|-|NF_{1}||$.
|
\dfrac{12\sqrt{3}}{13}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Define: \( a \oplus b = a \times b \), \( c \bigcirc d = d \times d \times d \times \cdots \times d \) (d multiplied c times). Find \( (5 \oplus 8) \oplus (3 \bigcirc 7) \).
|
13720
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
How many integers are there in $\{0,1, 2,..., 2014\}$ such that $C^x_{2014} \ge C^{999}{2014}$ ?
Note: $C^{m}_{n}$ stands for $\binom {m}{n}$
|
17
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In a right triangle JKL, where $\angle J$ is $90^\circ$, side JL is known to be 12 units, and the hypotenuse KL is 13 units. Calculate $\tan K$ and $\cos L$.
|
\frac{5}{13}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Add $175_{9} + 714_{9} + 61_9$. Express your answer in base $9$.
|
1061_{9}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Two skiers started from the same point one after another with an interval of 9 minutes. The second skier caught up with the first one 9 km from the starting point. After reaching the “27 km” mark, the second skier turned back and met the first skier at a distance of 2 km from the turning point. Find the speed of the second skier.
|
15
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let $ABCD$ be a parallelogram with $\angle{ABC}=120^\circ$, $AB=16$ and $BC=10$. Extend $\overline{CD}$ through $D$ to $E$ so that $DE=4$. If $\overline{BE}$ intersects $\overline{AD}$ at $F$, then $FD$ is closest to
|
3
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The least common multiple of two integers is 36 and 6 is their greatest common divisor. What is the product of the two numbers?
|
216
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Point $E$ is the midpoint of side $\overline{CD}$ in square $ABCD,$ and $\overline{BE}$ meets diagonal $\overline{AC}$ at $F.$ The area of quadrilateral $AFED$ is $45.$ What is the area of $ABCD?$
|
108
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have area $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch?
|
20
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Person A and person B start walking towards each other from locations A and B simultaneously. The speed of person B is $\frac{3}{2}$ times the speed of person A. After meeting for the first time, they continue to their respective destinations, and then immediately return. Given that the second meeting point is 20 kilometers away from the first meeting point, what is the distance between locations A and B?
|
50
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let $f(x) = \displaystyle \frac{1}{ax+b}$ where $a$ and $b$ are nonzero constants. Find all solutions to $f^{-1}(x) = 0$. Express your answer in terms of $a$ and/or $b$.
|
\frac1b
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A deck of fifty-two cards consists of four $1$'s, four $2$'s, ..., four $13$'s. Two matching pairs (two sets of two cards with the same number) are removed from the deck. After removing these cards, find the probability, represented as a fraction $m/n$ in simplest form, where $m$ and $n$ are relatively prime, that two randomly selected cards from the remaining cards also form a pair. Find $m + n$.
|
299
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given that the function f(x) (x ∈ R) satisfies f(x + π) = f(x) + sin(x), and f(x) = 0 when 0 ≤ x ≤ π. Find f(23π/6).
|
\frac{1}{2}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
How many diagonals does a regular seven-sided polygon contain?
|
14
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A geometric sequence $(a_n)$ has $a_1=\sin x$, $a_2=\cos x$, and $a_3= \tan x$ for some real number $x$. For what value of $n$ does $a_n=1+\cos x$?
|
8
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In the tetrahedron \( ABCD \), \( AC = 8 \), \( AB = CD = 7 \), \( BC = AD = 5 \), and \( BD = 6 \). Given a point \( P \) on \( AC \), find the minimum value of \( BP + PD \).
|
2\sqrt{21}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
If $x^5 - x^4 + x^3 - px^2 + qx + 4$ is divisible by $(x + 2)(x - 1),$ find the ordered pair $(p,q).$
|
(-7,-12)
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
For some constants $x$ and $a$, the third, fourth, and fifth terms in the expansion of $(x + a)^n$ are 84, 280, and 560, respectively. Find $n.$
|
7
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let $p(x) = x^2 + bx + c,$ where $b$ and $c$ are integers. If $p(x)$ is factor of both $x^4 + 6x^2 + 25$ and $3x^4 + 4x^ 2+ 28x + 5,$ what is $p(1)$?
|
4
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Compute the sum of the roots of the equation \[x\sqrt{x} - 6x + 7\sqrt{x} - 1 = 0,\]given that all of the roots are real and nonnegative.
|
22
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Four distinct points are arranged on a plane so that the segments connecting them have lengths $a$, $a$, $a$, $a$, $2a$, and $b$. What is the ratio of $b$ to $a$?
|
\sqrt{3}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Complete the following questions:
$(1)$ Calculate: $(\sqrt{8}-\sqrt{\frac{1}{2}})\div \sqrt{2}$.
$(2)$ Calculate: $2\sqrt{3}\times (\sqrt{12}-3\sqrt{75}+\frac{1}{3}\sqrt{108})$.
$(3)$ Given $a=3+2\sqrt{2}$ and $b=3-2\sqrt{2}$, find the value of the algebraic expression $a^{2}-3ab+b^{2}$.
$(4)$ Solve the equation: $\left(2x-1\right)^{2}=x\left(3x+2\right)-7$.
$(5)$ Solve the equation: $2x^{2}-3x+\frac{1}{2}=0$.
$(6)$ Given that real numbers $a$ and $b$ are the roots of the equation $x^{2}-x-1=0$, find the value of $\frac{b}{a}+\frac{a}{b}$.
|
-3
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Two spheres with radii $36$ and one sphere with radius $13$ are each externally tangent to the other two spheres and to two different planes $\mathcal{P}$ and $\mathcal{Q}$. The intersection of planes $\mathcal{P}$ and $\mathcal{Q}$ is the line $\ell$. The distance from line $\ell$ to the point where the sphere with radius $13$ is tangent to plane $\mathcal{P}$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Diagram
[asy] size(275); import graph3; import solids; currentprojection=orthographic((1,0.9,0)); triple O1, O2, O3, T1, T2, T3, A, L1, L2; O1 = (0,-36,0); O2 = (0,36,0); O3 = (0,0,-sqrt(1105)); T1 = (864*sqrt(1105)/1105,-36,-828*sqrt(1105)/1105); T2 = (864*sqrt(1105)/1105,36,-828*sqrt(1105)/1105); T3 = (24*sqrt(1105)/85,0,-108*sqrt(1105)/85); A = (0,0,-36*sqrt(1105)/23); L1 = shift(0,-80,0)*A; L2 = shift(0,80,0)*A; draw(surface(L1--L2--(-T2.x,L2.y,T2.z)--(-T1.x,L1.y,T1.z)--cycle),pink); draw(shift(O2)*rotate(-90,O1,O2)*scale3(36)*unithemisphere,yellow,light=Viewport); draw(surface(L1--L2--(L2.x,L2.y,40)--(L1.x,L1.y,40)--cycle),gray); draw(shift(O1)*rotate(90,O1,O2)*scale3(36)*unithemisphere,yellow,light=White); draw(shift(O2)*rotate(90,O1,O2)*scale3(36)*unithemisphere,yellow,light=White); draw(shift(O3)*rotate(90,O1,O2)*scale3(13)*unithemisphere,yellow,light=White); draw(surface((-T1.x,L1.y,L1.z-abs(T1.z))--(-T2.x,L2.y,L2.z-abs(T2.z))--(T2.x,L2.y,T2.z)--(T1.x,L1.y,T1.z)--cycle),palegreen); draw(surface(L1--L2--(L2.x,L2.y,L2.z-abs(T1.z))--(L1.x,L1.y,L1.z-abs(T2.z))--cycle),gray); draw(surface(L1--L2--(T2.x,L2.y,L2.z-abs(T1.z))--(T1.x,L1.y,L1.z-abs(T2.z))--cycle),pink); draw(L1--L2,L=Label("$\ell$",position=EndPoint,align=3*E),red); label("$\mathcal{P}$",midpoint(L1--(T1.x,L1.y,T1.z)),(0,-3,0),heavygreen); label("$\mathcal{Q}$",midpoint(L1--(T1.x,L1.y,L1.z-abs(T2.z))),(0,-3,0),heavymagenta); dot(O1,linewidth(4.5)); dot(O2,linewidth(4.5)); dot(O3,linewidth(4.5)); dot(T1,heavygreen+linewidth(4.5)); dot(T2,heavygreen+linewidth(4.5)); dot(T3,heavygreen+linewidth(4.5)); dot(A,red+linewidth(4.5)); [/asy] ~MRENTHUSIASM
|
335
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Find the x-coordinate of point Q, given that point P has coordinates $(\frac{3}{5}, \frac{4}{5})$, point Q is in the third quadrant with $|OQ| = 1$ and $\angle POQ = \frac{3\pi}{4}$.
|
-\frac{7\sqrt{2}}{10}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Xiao Ming observed a faucet continuously dripping water due to damage. To investigate the waste caused by the water leakage, Xiao Ming placed a graduated cylinder under the faucet to collect water. He recorded the total amount of water in the cylinder every minute, but due to a delay in starting the timer, there was already a small amount of water in the cylinder at the beginning. Therefore, he obtained a set of data as shown in the table below:
| Time $t$ (minutes) | 1 | 2 | 3 | 4 | 5 | ... |
|---------------------|---|---|---|---|---|----|
| Total water amount $y$ (milliliters) | 7 | 12 | 17 | 22 | 27 | ... |
$(1)$ Investigation: Based on the data in the table above, determine which one of the functions $y=\frac{k}{t}$ and $y=kt+b$ (where $k$ and $b$ are constants) can correctly reflect the functional relationship between the total water amount $y$ and time $t$. Find the expression of $y$ in terms of $t$.
$(2)$ Application:
① Estimate how many milliliters of water will be in the cylinder when Xiao Ming measures it at the 20th minute.
② A person drinks approximately 1500 milliliters of water per day. Estimate how many days the water leaked from this faucet in a month (30 days) can supply one person.
|
144
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The positive integers are arranged in rows and columns as shown below.
| Row 1 | 1 |
| Row 2 | 2 | 3 |
| Row 3 | 4 | 5 | 6 |
| Row 4 | 7 | 8 | 9 | 10 |
| Row 5 | 11 | 12 | 13 | 14 | 15 |
| Row 6 | 16 | 17 | 18 | 19 | 20 | 21 |
| ... |
More rows continue to list the positive integers in order, with each new row containing one more integer than the previous row. How many integers less than 2000 are in the column that contains the number 2000?
|
16
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
How many solutions does the system have: $ \{\begin{matrix}&(3x+2y) *(\frac{3}{x}+\frac{1}{y})=2 & x^2+y^2\leq 2012 \end{matrix} $ where $ x,y $ are non-zero integers
|
102
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. Let $O$ be the circumcenter of $A B C$. Find the distance between the circumcenters of triangles $A O B$ and $A O C$.
|
\frac{91}{6}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The greatest common divisor of 15 and some number between 75 and 90 is 5. What is the number?
|
85
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Solve the equation: $2\left(x-1\right)^{2}=x-1$.
|
\frac{3}{2}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In $\triangle ABC$, medians $\overline{AM}$ and $\overline{BN}$ are perpendicular. If $AM = 15$ and $BN = 20$, find the length of side $AB$.
|
\frac{50}{3}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
For how many positive integers $n$ less than $2013$, does $p^2+p+1$ divide $n$ where $p$ is the least prime divisor of $n$?
|
212
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Each edge of a regular tetrahedron is divided into three equal parts. Through each resulting division point, two planes are drawn, parallel to the two faces of the tetrahedron that do not pass through that point. Into how many parts do the constructed planes divide the tetrahedron?
|
15
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Find the sum of the distinct prime factors of $7^7 - 7^4$.
|
31
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Elon Musk's Starlink project belongs to his company SpaceX. He plans to use tens of thousands of satellites to provide internet services to every corner of the Earth. A domestic company also plans to increase its investment in the development of space satellite networks to develop space internet. It is known that the research and development department of this company originally had 100 people, with an average annual investment of $a$ (where $a \gt 0$) thousand yuan per person. Now the research and development department personnel are divided into two categories: technical personnel and research personnel. There are $x$ technical personnel, and after the adjustment, the annual average investment of technical personnel is adjusted to $a(m-\frac{2x}{25})$ thousand yuan, while the annual average investment of research personnel increases by $4x\%$.
$(1)$ To ensure that the total annual investment of the adjusted research personnel is not less than the total annual investment of the original 100 research personnel, what is the maximum number of technical personnel after the adjustment?
$(2)$ Now it is required that the total annual investment of the adjusted research personnel is always not less than the total annual investment of the adjusted technical personnel. Find the maximum value of $m$ and the number of technical personnel at that time.
|
50
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let's call two positive integers almost neighbors if each of them is divisible (without remainder) by their difference. In a math lesson, Vova was asked to write down in his notebook all the numbers that are almost neighbors with \(2^{10}\). How many numbers will he have to write down?
|
21
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given 3 zeros and 2 ones are randomly arranged in a row, calculate the probability that the 2 ones are not adjacent.
|
\frac{3}{5}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The terms of the sequence $(b_i)$ defined by $b_{n + 2} = \frac {b_n + 2021} {1 + b_{n + 1}}$ for $n \ge 1$ are positive integers. Find the minimum possible value of $b_1 + b_2$.
|
90
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A wooden cube with edge length $n$ units (where $n$ is an integer $>2$) is painted black all over. By slices parallel to its faces, the cube is cut into $n^3$ smaller cubes each of unit length. If the number of smaller cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is $n$?
|
8
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In the expansion of ${(6x+\frac{1}{3\sqrt{x}})}^{9}$, arrange the fourth term in ascending powers of $x$.
|
\frac{224}{9}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Calculate the arc lengths of the curves given by equations in the rectangular coordinate system.
$$
y=1+\arcsin x-\sqrt{1-x^{2}}, 0 \leq x \leq \frac{3}{4}
$$
|
\sqrt{2}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Squares $JKLM$ and $NOPQ$ are congruent, $JM=20$, and $P$ is the midpoint of side $JM$ of square $JKLM$. Calculate the area of the region covered by these two squares in the plane.
A) $500$
B) $600$
C) $700$
D) $800$
E) $900$
|
600
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given that $n$ is an integer between $1$ and $60$, inclusive, determine for how many values of $n$ the expression $\frac{((n+1)^2 - 1)!}{(n!)^{n+1}}$ is an integer.
|
59
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
For a particular value of the angle $\theta$ we can take the product of the two complex numbers $(8+i)\sin\theta+(7+4i)\cos\theta$ and $(1+8i)\sin\theta+(4+7i)\cos\theta$ to get a complex number in the form $a+bi$ where $a$ and $b$ are real numbers. Find the largest value for $a+b$ .
|
125
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let $\Gamma$ denote the circumcircle of triangle $A B C$. Point $D$ is on $\overline{A B}$ such that $\overline{C D}$ bisects $\angle A C B$. Points $P$ and $Q$ are on $\Gamma$ such that $\overline{P Q}$ passes through $D$ and is perpendicular to $\overline{C D}$. Compute $P Q$, given that $B C=20, C A=80, A B=65$.
|
4 \sqrt{745}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
If the system of inequalities about $x$ is $\left\{{\begin{array}{l}{-2({x-2})-x＜2}\\{\frac{{k-x}}{2}≥-\frac{1}{2}+x}\end{array}}\right.$ has at most $2$ integer solutions, and the solution to the one-variable linear equation about $y$ is $3\left(y-1\right)-2\left(y-k\right)=7$, determine the sum of all integers $k$ that satisfy the conditions.
|
18
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In $\triangle ABC$, if $BC=4$, $\cos B= \frac{1}{4}$, then $\sin B=$ _______, the minimum value of $\overrightarrow{AB} \cdot \overrightarrow{AC}$ is: _______.
|
-\frac{1}{4}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The integer $n$ is the largest positive multiple of $15$ such that every digit of $n$ is either $8$ or $0$. Compute $\frac{n}{15}$.
|
592
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
An urn contains 101 balls, exactly 3 of which are red. The balls are drawn one by one without replacement. On which draw is it most likely to pull the second red ball?
|
51
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The volume of a regular octagonal prism is $8 \, \mathrm{m}^{3}$, and its height is $2.2 \, \mathrm{m}$. Find the lateral surface area of the prism.
|
16 \sqrt{2.2 \cdot (\sqrt{2}-1)}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let \(x_{1}, x_{2}, \ldots, x_{200}\) be natural numbers greater than 2 (not necessarily distinct). In a \(200 \times 200\) table, the numbers are arranged as follows: at the intersection of the \(i\)-th row and the \(k\)-th column, the number \(\log _{x_{k}} \frac{x_{i}}{9}\) is written. Find the smallest possible value of the sum of all the numbers in the table.
|
-40000
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
There are 100 points located on a line. Mark the midpoints of all possible segments with endpoints at these points. What is the minimum number of marked points that can result?
|
197
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The sequence starts with 800,000; each subsequent term is obtained by dividing the previous term by 3. What is the last integer in this sequence?
|
800000
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A rectangle with dimensions $8 \times 2 \sqrt{2}$ and a circle with a radius of 2 have a common center. Find the area of their overlapping region.
|
2 \pi + 4
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
License plates from different states follow different alpha-numeric formats, which dictate which characters of a plate must be letters and which must be numbers. Florida has license plates with an alpha-numeric format like the one pictured. North Dakota, on the other hand, has a different format, also pictured. Assuming all 10 digits are equally likely to appear in the numeric positions, and all 26 letters are equally likely to appear in the alpha positions, how many more license plates can Florida issue than North Dakota? [asy]
import olympiad; size(240); defaultpen(linewidth(0.8)); dotfactor=4;
draw((0,0)--(3,0)--(3,1)--(0,1)--cycle);
label("\LARGE HJF 94K",(1.5,0.6)); label("Florida",(1.5,0.2));
draw((4,0)--(7,0)--(7,1)--(4,1)--cycle);
label("\LARGE DGT 317",(5.5,0.6)); label("North Dakota",(5.5,0.2));
[/asy]
|
28121600
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Two arithmetic sequences $\{a_{n}\}$ and $\{b_{n}\}$ have the sums of the first $n$ terms as $S_{n}$ and $T_{n}$, respectively. It is known that $\frac{{S}_{n}}{{T}_{n}}=\frac{7n+2}{n+3}$. Find $\frac{{a}_{7}}{{b}_{7}}$.
|
\frac{93}{16}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
If $n$ is a real number, then the simultaneous system
$nx+y = 1$
$ny+z = 1$
$x+nz = 1$
has no solution if and only if $n$ is equal to
|
-1
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let $S_n$ be the sum of the first $n$ terms of a geometric sequence $\{a_n\}$, where $a_n > 0$. If $S_6 - 2S_3 = 5$, then the minimum value of $S_9 - S_6$ is ______.
|
20
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Side $AB$ of triangle $ABC$ was divided into $n$ equal parts (dividing points $B_0 = A, B_1, B_2, ..., B_n = B$ ), and side $AC$ of this triangle was divided into $(n + 1)$ equal parts (dividing points $C_0 = A, C_1, C_2, ..., C_{n+1} = C$ ). Colored are the triangles $C_iB_iC_{i+1}$ (where $i = 1,2, ..., n$ ). What part of the area of the triangle is painted over?
|
\frac{1}{2}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given that $S_{n}$ is the sum of the first $n$ terms of the sequence $\{a_{n}\}$, if ${a_1}=\frac{5}{2}$, and ${a_{n+1}}({2-{a_n}})=2$ for $n\in\mathbb{N}^*$, then $S_{22}=$____.
|
-\frac{4}{3}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Suppose $a_{1} < a_{2}< \cdots < a_{2024}$ is an arithmetic sequence of positive integers, and $b_{1} <b_{2} < \cdots <b_{2024}$ is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the two sequences.
*Ray Li*
|
11
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Select 5 different letters from the word "equation" to arrange in a row, including the condition that the letters "qu" are together and in the same order.
|
480
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Expanding $(1+0.2)^{1000}$ by the binomial theorem and doing no further manipulation gives
\[{1000 \choose 0}(0.2)^0+{1000 \choose 1}(0.2)^1+{1000 \choose 2}(0.2)^2+\cdots+{1000 \choose 1000}(0.2)^{1000}= A_0 + A_1 + A_2 + \cdots + A_{1000},\]where $A_k = {1000 \choose k}(0.2)^k$ for $k = 0,1,2,\ldots,1000.$ For which $k$ is $A_k$ the largest?
|
166
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The first digit on the left of a six-digit number is 1. If this digit is moved to the last place, the resulting number is 64 times greater than the original number. Find the original number.
|
142857
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Solve the following cryptarithm ensuring that identical letters correspond to identical digits:
$$
\begin{array}{r}
\text { К O Ш К A } \\
+ \text { К O Ш К A } \\
\text { К O Ш К A } \\
\hline \text { С О Б А К А }
\end{array}
$$
|
50350
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Find the integer $n,$ $0 \le n \le 180,$ such that $\cos n^\circ = \cos 259^\circ.$
|
101
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
On the blackboard, Amy writes 2017 in base-$a$ to get $133201_{a}$. Betsy notices she can erase a digit from Amy's number and change the base to base-$b$ such that the value of the number remains the same. Catherine then notices she can erase a digit from Betsy's number and change the base to base-$c$ such that the value still remains the same. Compute, in decimal, $a+b+c$.
|
22
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Triangle $ABC$ is a right isosceles triangle. Points $D$, $E$ and $F$ are the midpoints of the sides of the triangle. Point $G$ is the midpoint of segment $DF$ and point $H$ is the midpoint of segment $FE$. What is the ratio of the shaded area to the non-shaded area in triangle $ABC$? Express your answer as a common fraction.
[asy]
draw((0,0)--(1,0)--(0,1)--(0,0)--cycle,linewidth(1));
filldraw((0,0)--(1/2,0)--(1/2,1/2)--(0,1/2)--(0,0)--cycle,gray, linewidth(1));
filldraw((1/2,0)--(1/2,1/4)--(1/4,1/2)--(0,1/2)--(1/2,0)--cycle,white,linewidth(1));
label("A", (0,1), W);
label("B", (0,0), SW);
label("C", (1,0), E);
label("D", (0,1/2), W);
label("E", (1/2,0), S);
label("F", (1/2,1/2), NE);
label("G", (1/4,1/2), N);
label("H", (1/2,1/4), E);
[/asy]
|
\frac{5}{11}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
How many ways are there to write $2016$ as the sum of twos and threes, ignoring order? (For example, $1008 \cdot 2 + 0 \cdot 3$ and $402 \cdot 2 + 404 \cdot 3$ are two such ways.)
|
337
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
When submitting problems, Steven the troll likes to submit silly names rather than his own. On day $1$ , he gives no
name at all. Every day after that, he alternately adds $2$ words and $4$ words to his name. For example, on day $4$ he
submits an $8\text{-word}$ name. On day $n$ he submits the $44\text{-word name}$ “Steven the AJ Dennis the DJ Menace the Prince of Tennis the Merchant of Venice the Hygienist the Evil Dentist the Major Premise the AJ Lettuce the Novel’s Preface the Core Essence the Young and the Reckless the Many Tenants the Deep, Dark Crevice”. Compute $n$ .
|
16
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Evaluate the expression given by $$2+\cfrac{3}{4+\cfrac{5}{6+\cfrac{7}{8}}}.$$
|
\frac{137}{52}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In a right triangle $ABC$ with equal legs $AC$ and $BC$, a circle is constructed with $AC$ as its diameter, intersecting side $AB$ at point $M$. Find the distance from vertex $B$ to the center of this circle if $BM = \sqrt{2}$.
|
\sqrt{5}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let $a,$ $b,$ $c$ be three distinct positive real numbers such that $a,$ $b,$ $c$ form a geometric sequence, and
\[\log_c a, \ \log_b c, \ \log_a b\]form an arithmetic sequence. Find the common difference of the arithmetic sequence.
|
\frac{3}{2}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
What is the sum of all the odd divisors of $360$?
|
78
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
If two poles $20''$ and $80''$ high are $100''$ apart, then the height of the intersection of the lines joining the top of each pole to the foot of the opposite pole is:
|
16''
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A box contains $12$ ping-pong balls, of which $9$ are new and $3$ are old. Three balls are randomly drawn from the box for use, and then returned to the box. Let $X$ denote the number of old balls in the box after this process. What is the value of $P(X = 4)$?
|
\frac{27}{220}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A high school is holding a speech contest with 10 participants. There are 3 students from Class 1, 2 students from Class 2, and 5 students from other classes. Using a draw to determine the speaking order, what is the probability that the 3 students from Class 1 are placed consecutively (in consecutive speaking slots) and the 2 students from Class 2 are not placed consecutively?
|
$\frac{1}{20}$
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
What is the value of $a$ if the lines $2y - 2a = 6x$ and $y + 1 = (a + 6)x$ are parallel?
|
-3
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
If $n$ is a positive integer, the notation $n$! (read " $n$ factorial") is used to represent the product of the integers from 1 to $n$. That is, $n!=n(n-1)(n-2) \cdots(3)(2)(1)$. For example, $4!=4(3)(2)(1)=24$ and $1!=1$. If $a$ and $b$ are positive integers with $b>a$, what is the ones (units) digit of $b!-a$! that cannot be?
|
7
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given $a+c=8$, $\cos B= \frac{1}{4}$.
(1) If $\overrightarrow{BA}\cdot \overrightarrow{BC}=4$, find the value of $b$;
(2) If $\sin A= \frac{\sqrt{6}}{4}$, find the value of $\sin C$.
|
\frac{3\sqrt{6}}{8}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Find the number of positive integer divisors of 12 ! that leave a remainder of 1 when divided by 3.
|
66
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given the hyperbola $C:\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1(a > 0,b > 0)$, the line $l$ passing through point $P(3,6)$ intersects $C$ at points $A$ and $B$, and the midpoint of $AB$ is $N(12,15)$. Determine the eccentricity of the hyperbola $C$.
|
\frac{3}{2}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The $8 \times 18$ rectangle $ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $y$?
|
6
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given non-negative real numbers $a$, $b$, $c$ satisfy $\frac{a-1}{2}=\frac{b-2}{3}=\frac{3-c}{4}$, let the maximum value of $S=a+2b+c$ be $m$, and the minimum value be $n$. Then the value of $\frac{n}{m}$ is ______.
|
\frac{6}{11}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A natural number plus 13 is a multiple of 5, and its difference with 13 is a multiple of 6. What is the smallest natural number that satisfies these conditions?
|
37
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Price of some item has decreased by $5\%$ . Then price increased by $40\%$ and now it is $1352.06\$ $ cheaper than doubled original price. How much did the item originally cost?
|
2018
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A square flag has a red cross of uniform width with a blue square in the center on a white background as shown. (The cross is symmetric with respect to each of the diagonals of the square.) If the entire cross (both the red arms and the blue center) takes up 36% of the area of the flag, what percent of the area of the flag is blue?
[asy] unitsize(2.5 cm); pair[] A, B, C; real t = 0.2; A[1] = (0,0); A[2] = (1,0); A[3] = (1,1); A[4] = (0,1); B[1] = (t,0); B[2] = (1 - t,0); B[3] = (1,t); B[4] = (1,1 - t); B[5] = (1 - t,1); B[6] = (t,1); B[7] = (0,1 - t); B[8] = (0,t); C[1] = extension(B[1],B[4],B[7],B[2]); C[2] = extension(B[3],B[6],B[1],B[4]); C[3] = extension(B[5],B[8],B[3],B[6]); C[4] = extension(B[7],B[2],B[5],B[8]); fill(C[1]--C[2]--C[3]--C[4]--cycle,blue); fill(A[1]--B[1]--C[1]--C[4]--B[8]--cycle,red); fill(A[2]--B[3]--C[2]--C[1]--B[2]--cycle,red); fill(A[3]--B[5]--C[3]--C[2]--B[4]--cycle,red); fill(A[4]--B[7]--C[4]--C[3]--B[6]--cycle,red); draw(A[1]--A[2]--A[3]--A[4]--cycle); draw(B[1]--B[4]); draw(B[2]--B[7]); draw(B[3]--B[6]); draw(B[5]--B[8]); [/asy]
$\text{(A)}\ 0.5\qquad\text{(B)}\ 1\qquad\text{(C)}\ 2\qquad\text{(D)}\ 3\qquad\text{(E)}\ 6$
|
2
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given that \( 2^{a} \times 3^{b} \times 5^{c} \times 7^{d} = 252000 \), what is the probability that a three-digit number formed by any 3 of the natural numbers \( a, b, c, d \) is divisible by 3 and less than 250?
|
1/4
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Find the sum of the $2007$ roots of $(x-1)^{2007}+2(x-2)^{2006}+3(x-3)^{2005}+\cdots+2006(x-2006)^2+2007(x-2007)$.
|
2005
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$.
[i]
|
n=1,2,3,4
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The sides of a triangle are all integers, and the longest side is 11. Calculate the number of such triangles.
|
36
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Triangle $ABC$ has $AB = 15, BC = 16$, and $AC = 17$. The points $D, E$, and $F$ are the midpoints of $\overline{AB}, \overline{BC}$, and $\overline{AC}$ respectively. Let $X \neq E$ be the intersection of the circumcircles of $\triangle BDE$ and $\triangle CEF$. Determine $XA + XB + XC$.
A) $\frac{480 \sqrt{39}}{7}$
B) $\frac{960 \sqrt{39}}{14}$
C) $\frac{1200 \sqrt{39}}{17}$
D) $\frac{1020 \sqrt{39}}{15}$
|
\frac{960 \sqrt{39}}{14}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Suppose we flip four coins simultaneously: a penny, a nickel, a dime, and a quarter. What is the probability that at least 15 cents worth of coins come up heads?
|
\dfrac{5}{8}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In the diagram, $O$ is the center of a circle with radii $OP=OQ=5$. What is the perimeter of the shaded region?
[asy]
size(100);
import graph;
label("$P$",(-1,0),W); label("$O$",(0,0),NE); label("$Q$",(0,-1),S);
fill(Arc((0,0),1,-90,180)--cycle,mediumgray);
draw(Arc((0,0),1,-90,180));
fill((0,0)--(-1,0)--(0,-1)--cycle,white);
draw((-1,0)--(0,0)--(0,-1));
draw((-.1,0)--(-.1,-.1)--(0,-.1));
[/asy]
|
10 + \frac{15}{2}\pi
|
|
MatrixStudio/Codeforces-Python-Submissions
|
Imp likes his plush toy a lot.
Recently, he found a machine that can clone plush toys. Imp knows that if he applies the machine to an original toy, he additionally gets one more original toy and one copy, and if he applies the machine to a copied toy, he gets two additional copies.
Initially, Imp has only one original toy. He wants to know if it is possible to use machine to get exactly *x* copied toys and *y* original toys? He can't throw toys away, and he can't apply the machine to a copy if he doesn't currently have any copies.
|
```python
x, y = map(int, input().split())
if y == x + 1:
print("Yes")
elif y >= x:
print("No")
else:
c = y-1
if (x-c)%2 == 0:
print("Yes")
else:
print("No")
```
|
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