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agentica-org/DeepScaleR-Preview-Dataset
|
An [i]animal[/i] with $n$ [i]cells[/i] is a connected figure consisting of $n$ equal-sized cells[1].
A [i]dinosaur[/i] is an animal with at least $2007$ cells. It is said to be [i]primitive[/i] it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.
(1) Animals are also called [i]polyominoes[/i]. They can be defined inductively. Two cells are [i]adjacent[/i] if they share a complete edge. A single cell is an animal, and given an animal with $n$ cells, one with $n+1$ cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.
|
4n-3
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
What is the smallest five-digit positive integer which is congruent to 7 (mod 13)?
|
10,004
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
How many integers 1-9 are divisors of the five-digit number 24,516?
|
6
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices being equally likely - and are sent off to slay a troublesome dragon. Let $P$ be the probability that at least two of the three had been sitting next to each other. If $P$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
|
57
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Points $A,B,C$ and $D$ lie on a line, in that order, with $AB = CD$ and $BC = 12$. Point $E$ is not on the line, and $BE = CE = 10$. The perimeter of $\triangle AED$ is twice the perimeter of $\triangle BEC$. Find $AB$.
|
9
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given a parabola \( y^2 = 6x \) with two variable points \( A(x_1, y_1) \) and \( B(x_2, y_2) \), where \( x_1 \neq x_2 \) and \( x_1 + x_2 = 4 \). The perpendicular bisector of segment \( AB \) intersects the x-axis at point \( C \). Find the maximum area of triangle \( \triangle ABC \).
|
\frac{14}{3}\sqrt{7}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Robot Petya displays three three-digit numbers every minute, which sum up to 2019. Robot Vasya swaps the first and last digits of each of these numbers and then sums the resulting numbers. What is the maximum sum that Vasya can obtain?
|
2118
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given $$\frac {\pi}{2} < \alpha < \pi$$, $$0 < \beta < \frac {\pi}{2}$$, $$\tan\alpha = -\frac {3}{4}$$, and $$\cos(\beta-\alpha) = \frac {5}{13}$$, find the value of $\sin\beta$.
|
\frac {63}{65}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Find all positive integers $ k$ with the following property: There exists an integer $ a$ so that $ (a\plus{}k)^{3}\minus{}a^{3}$ is a multiple of $ 2007$ .
|
669
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let \( x_{1}, x_{2}, \cdots, x_{n} \) and \( a_{1}, a_{2}, \cdots, a_{n} \) be two sets of arbitrary real numbers (where \( n \geqslant 2 \)) that satisfy the following conditions:
1. \( x_{1} + x_{2} + \cdots + x_{n} = 0 \)
2. \( \left| x_{1} \right| + \left| x_{2} \right| + \cdots + \left| x_{n} \right| = 1 \)
3. \( a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{n} \)
Determine the minimum value of the real number \( A \) such that the inequality \( \left| a_{1} x_{1} + a_{2} x_{2} + \cdots + a_{n} x_{n} \right| \leqslant A ( a_{1} - a_{n} ) \) holds, and provide a justification for this value.
|
1/2
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The volume of a hemisphere is $\frac{500}{3}\pi$. What is the total surface area of the hemisphere including its base? Express your answer in terms of $\pi$.
|
3\pi \times 250^{2/3}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Express $0.4\overline5$ as a common fraction.
|
\frac{41}{90}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given in the polar coordinate system, point P moves on the curve $\rho^2\cos\theta-2\rho=0$, the minimum distance from point P to point $Q(1, \frac{\pi}{3})$ is \_\_\_\_\_\_.
|
\frac{3}{2}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A flock of geese is flying, and a lone goose flies towards them and says, "Hello, a hundred geese!" The leader of the flock responds, "No, we are not a hundred geese! If there were as many of us as there are now, plus the same amount, plus half of that amount, plus a quarter of that amount, plus you, goose, then we would be a hundred geese. But as it is..." How many geese were in the flock?
|
36
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In the graph of $\frac{x^2+3x+2}{x^3+x^2-2x}$, let $a$ be the number of holes in the graph, $b$ be the number of vertical asympotes, $c$ be the number of horizontal asymptotes, and $d$ be the number of oblique asymptotes. Find $a+2b+3c+4d$.
|
8
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
An infinite sequence of circles is composed such that each circle has a decreasing radius, and each circle touches the subsequent circle and the two sides of a given right angle. The ratio of the area of the first circle to the sum of the areas of all subsequent circles in the sequence is
|
$(16+12 \sqrt{2}): 1$
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The volume of the parallelepiped determined by vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ is 6. Find the volume of the parallelepiped determined by $\mathbf{a} + 2\mathbf{b},$ $\mathbf{b} + \mathbf{c},$ and $2\mathbf{c} - 5\mathbf{a}.$
|
48
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
If $x = 202$ and $x^3y - 4x^2y + 2xy = 808080$, what is the value of $y$?
|
\frac{1}{10}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Find the possible value of $x + y$ given that $x^3 + 6x^2 + 16x = -15$ and $y^3 + 6y^2 + 16y = -17$.
|
-4
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given that the polynomial $x^2 - kx + 24$ has only positive integer roots, find the average of all distinct possibilities for $k$.
|
15
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In triangle \( \triangle ABC \), it is given that \( \angle C=90^\circ \), \( \angle A=60^\circ \), and \( AC=1 \). Points \( D \) and \( E \) are on sides \( BC \) and \( AB \) respectively such that triangle \( \triangle ADE \) is an isosceles right triangle with \( \angle ADE=90^\circ \). Find the length of \( BE \).
|
4-2\sqrt{3}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Javier is excited to visit Disneyland during spring break. He plans on visiting five different attractions, but he is particularly excited about the Space Mountain ride and wants to visit it twice during his tour before lunch. How many different sequences can he arrange his visits to these attractions, considering his double visit to Space Mountain?
|
360
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Rodney is trying to guess a secret number based on the following clues:
1. It is a two-digit integer.
2. The tens digit is odd.
3. The units digit is one of the following: 2, 4, 6, 8.
4. The number is greater than 50.
What is the probability that Rodney will guess the correct number? Express your answer as a common fraction.
|
\frac{1}{12}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The three roots of the equation \[ax^3+bx^2+cx+d=0\]are $1,$ $2,$ and $3.$ Compute $\frac{c}{d}.$
|
-\frac{11}{6}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let \( u, v, w \) be positive real numbers, all different from 1. If
\[
\log_{u}(vw) + \log_{v}(w) = 5 \quad \text{and} \quad \log_{v}(u) + \log_{w}(v) = 3,
\]
find the value of \( \log_{w}(u) \).
|
\frac{4}{5}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Dr. Math's four-digit house number $WXYZ$ contains no zeroes and can be split into two different two-digit primes ``$WX$'' and ``$YZ$'' where the digits $W$, $X$, $Y$, and $Z$ are not necessarily distinct. If each of the two-digit primes is less than 60, how many such house numbers are possible?
|
156
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let $\varphi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Let $S$ be the set of positive integers $n$ such that $\frac{2 n}{\varphi(n)}$ is an integer. Compute the sum $\sum_{n \in S} \frac{1}{n}$.
|
\frac{10}{3}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
From the numbers $1, 2, 3, 4, 5$, 3 numbers are randomly drawn (with replacement) to form a three-digit number. What is the probability that the sum of its digits equals 9?
|
$\frac{19}{125}$
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In each square of an $11\times 11$ board, we are to write one of the numbers $-1$ , $0$ , or $1$ in such a way that the sum of the numbers in each column is nonnegative and the sum of the numbers in each row is nonpositive. What is the smallest number of zeros that can be written on the board? Justify your answer.
|
11
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given the digits $1, 3, 7, 8, 9$, find the smallest difference that can be achieved in the subtraction problem
\[\begin{tabular}[t]{cccc} & \boxed{} & \boxed{} & \boxed{} \\ - & & \boxed{} & \boxed{} \\ \hline \end{tabular}\]
|
39
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In the animal kingdom, tigers always tell the truth, foxes always lie, and monkeys sometimes tell the truth and sometimes lie. There are 100 of each of these three types of animals, divided into 100 groups. Each group has exactly 3 animals, with exactly 2 animals of one type and 1 animal of another type.
After the groups were formed, Kung Fu Panda asked each animal, "Is there a tiger in your group?" and 138 animals responded "yes." Kung Fu Panda then asked each animal, "Is there a fox in your group?" and 188 animals responded "yes."
How many monkeys told the truth both times?
|
76
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
What is the value of $\sqrt{36 \times \sqrt{16}}$?
|
12
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The weight of grain in a sample of 256 grains is 18 grains, and the total weight of rice is 1536 dan. Calculate the amount of mixed grain in the total batch of rice.
|
108
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In America, temperature is measured in degrees Fahrenheit. This is a linear scale where the freezing point of water is $32^{\circ} \mathrm{F}$ and the boiling point is $212^{\circ} \mathrm{F}$.
Someone provides the temperature rounded to whole degrees Fahrenheit, which we then convert to Celsius and afterwards round to whole degrees. What is the maximum possible deviation of the obtained value from the original temperature in Celsius degrees?
|
13/18
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Find $\frac{9}{10}+\frac{5}{6}$. Express your answer as a fraction in simplest form.
|
\frac{26}{15}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Calculate the definite integral:
$$
\int_{0}^{\pi} 2^{4} \cdot \cos ^{8} x \, dx
$$
|
\frac{35 \pi}{8}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given points E and D are on sides AB and BC of triangle ABC, where AE:EB=1:3 and CD:DB=1:2, find the value of EF/FC + AF/FD.
|
\frac{3}{2}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In the 2009 East Asian Games, the Chinese men's table tennis team sent Wang Hao and 5 young players to compete. The team competition requires 3 players to participate. If Wang Hao is not the last player to compete, there are $\boxed{\text{answer}}$ different ways of participation (answer in digits).
|
100
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let $g$ be the function defined by $g(x) = -3 \sin(2\pi x)$. How many values of $x$ such that $-3 \le x \le 3$ satisfy the equation $g(g(g(x))) = g(x)$?
|
48
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Refer to the diagram, $P$ is any point inside the square $O A B C$ and $b$ is the minimum value of $P O + P A + P B + P C$. Find $b$.
|
2\sqrt{2}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The four-digit numeral $3AA1$ is divisible by 9. What digit does $A$ represent?
|
7
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
We inscribe a cone around a sphere of unit radius. What is the minimum surface area of the cone?
|
8\pi
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
If $\sum_{n = 0}^{\infty}\sin^{2n}\theta = 4$, what is the value of $\sin{2\theta}$?
|
\frac{\sqrt{3}}{2}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given that $\alpha$ is an angle in the third quadrant, $f(\alpha) = \frac {\sin(\pi-\alpha)\cdot \cos(2\pi-\alpha)\cdot \tan(-\alpha-\pi)}{\tan(-\alpha )\cdot \sin(-\pi -\alpha)}$.
1. Simplify $f(\alpha)$;
2. If $\cos\left(\alpha- \frac {3}{2}\pi\right) = \frac {1}{5}$, find the value of $f(\alpha)$;
3. If $\alpha=-1860^\circ$, find the value of $f(\alpha)$.
|
\frac {1}{2}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In a selection of $2$ algebra questions and $3$ geometry questions, one question is randomly selected each time without replacement. Let $A=$"selecting an algebra question first" and $B=$"selecting a geometry question second". Find $P\left(AB\right)=\_\_\_\_\_\_$ and $P\left(B|A\right)=\_\_\_\_\_\_$.
|
\frac{3}{4}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Find the times between $8$ and $9$ o'clock, correct to the nearest minute, when the hands of a clock will form an angle of $120^{\circ}$.
|
8:22
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given the sets
$$
\begin{array}{l}
M=\{x, x y, \lg (x y)\} \\
N=\{0,|x|, y\},
\end{array}
$$
and $M=N$, determine the value of:
$$\left(x+\frac{1}{y}\right)+\left(x^{2}+\frac{1}{y^{2}}\right)+\left(x^{3}+\frac{1}{y^{3}}\right)+\cdots+\left(x^{2001}+\frac{1}{y^{2001}}\right).$$
|
-2
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Determine all quadruplets ($x, y, z, t$) of positive integers, such that $12^x + 13^y - 14^z = 2013^t$.
|
(1, 3, 2, 1)
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The graphs of two functions, $p(x)$ and $q(x),$ are shown here on one set of axes: [asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {
import graph;
real i;
if(complexplane) {
label("$\textnormal{Re}$",(xright,0),SE);
label("$\textnormal{Im}$",(0,ytop),NW);
} else {
label("$x$",(xright+0.4,-0.5));
label("$y$",(-0.5,ytop+0.2));
}
ylimits(ybottom,ytop);
xlimits( xleft, xright);
real[] TicksArrx,TicksArry;
for(i=xleft+xstep; i<xright; i+=xstep) {
if(abs(i) >0.1) {
TicksArrx.push(i);
}
}
for(i=ybottom+ystep; i<ytop; i+=ystep) {
if(abs(i) >0.1) {
TicksArry.push(i);
}
}
if(usegrid) {
xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);//,above=true);
yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);//,Arrows);
}
if(useticks) {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
} else {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));
}
};
rr_cartesian_axes(-4,4,-4,4);
real f(real x) {return abs(x)-2;}
real g(real x) {return -abs(x);}
draw(graph(f,-4,4,operator ..), blue+1.25);
draw(graph(g,-4,4,operator ..), orange+1.25);
draw((-3,-5)--(-1,-5),blue+1.25); label("$y=p(x)$",(-1,-5),E);
draw((-3,-6)--(-1,-6),orange+1.25); label("$y=q(x)$",(-1,-6),E);
[/asy] Each small box in the grid is $1$ unit by $1$ unit.
If $q(p(x))$ is evaluated at $x=-4,$ $-3,$ $-2,$ $-1,$ $0,$ $1,$ $2,$ $3,$ $4,$ what is the sum of the nine values obtained in this way?
|
-10
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let $q(x)=q^{1}(x)=2x^{2}+2x-1$, and let $q^{n}(x)=q(q^{n-1}(x))$ for $n>1$. How many negative real roots does $q^{2016}(x)$ have?
|
\frac{2017+1}{3}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given a sequence $\{a_{n}\}$ such that $a_{1}+2a_{2}+\cdots +na_{n}=n$, and a sequence $\{b_{n}\}$ such that ${b_{m-1}}+{b_m}=\frac{1}{{{a_m}}}({m∈N,m≥2})$. Find:<br/>
$(1)$ The general formula for $\{a_{n}\}$;<br/>
$(2)$ The sum of the first $20$ terms of $\{b_{n}\}$.
|
110
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Antal and Béla start from home on their motorcycles heading towards Cegléd. After traveling one-fifth of the way, Antal for some reason turns back. As a result, he accelerates and manages to increase his speed by one quarter. He immediately sets off again from home. Béla, continuing alone, decreases his speed by one quarter. They travel the final section of the journey together at $48$ km/h and arrive 10 minutes later than planned. What can we calculate from all this?
|
40
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Find the minimum positive integer $k$ such that $f(n+k) \equiv f(n)(\bmod 23)$ for all integers $n$.
|
2530
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In the diagram below, lines $k$ and $\ell$ are parallel. Find the measure of angle $x$ in degrees.
[asy]
size(200);
import markers;
pair A = dir(-22)*(0,0);
pair B = dir(-22)*(4,0);
pair C = dir(-22)*(4,2);
pair D = dir(-22)*(0,2);
pair F = dir(-22)*(0,1.3);
pair G = dir(-22)*(4,1.3);
pair H = dir(-22)*(2,1);
//markangle(.3,B,H,C);
markangle(Label("$x$",Relative(0.4)),n=1,radius=11,B,H,C);
pair X,Y;
X=A;
Y=B;
draw(1.3*X-.3*Y--1.3*Y-.3*X);
X=A;
Y=C;
draw(1.3*X-.3*Y--1.3*Y-.3*X);
X=C;
Y=B;
draw(1.3*X-.3*Y--1.3*Y-.3*X);
X=B;
Y=D;
draw(1.3*X-.3*Y--1.3*Y-.3*X);
X=G;
Y=F;
draw(1.3*X-.3*Y--1.3*Y-.3*X);
label("$\ell$",1.4*A-.4*B);
label("$k$",1.4*F-.4*G);
//label("$x$",H+(.4,-.15));
label("$30^\circ$",A+(1,-.1));
label("$90^\circ$",B+(.4,.1));
label("$30^\circ$",B+(-1,.7));
[/asy]
|
60^\circ
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let $g(n)$ be the product of the proper positive integer divisors of $n$. (Recall that a proper divisor of $n$ is a divisor other than $n$.) For how many values of $n$ does $n$ not divide $g(n)$, given that $2 \le n \le 50$?
|
19
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
What is the value of $c$ if the lines with equations $y = 8x + 2$ and $y = (2c)x - 4$ are parallel?
|
4
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
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}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given the variance of a sample is $$s^{2}= \frac {1}{20}[(x_{1}-3)^{2}+(x_{2}-3)^{2}+\ldots+(x_{n}-3)^{2}]$$, then the sum of this set of data equals \_\_\_\_\_\_.
|
60
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The sides of triangle $PQR$ are in the ratio $3:4:5$. Segment $QS$ is the angle bisector drawn to the longest side, dividing it into segments $PS$ and $SR$. What is the length, in inches, of the shorter subsegment of side $PR$ if the length of side $PR$ is $15$ inches? Express your answer as a common fraction.
|
\frac{45}{7}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let $\mathbf{v}$ and $\mathbf{w}$ be vectors such that
\[\operatorname{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} 1 \\ 0 \\ -3 \end{pmatrix}.\]Compute $\operatorname{proj}_{\mathbf{w}} (-2 \mathbf{v}).$
|
\begin{pmatrix} -2 \\ 0 \\ 6 \end{pmatrix}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
All the complex roots of $(z + 1)^4 = 16z^4,$ when plotted in the complex plane, lie on a circle. Find the radius of this circle.
|
\frac{2}{3}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A bar graph shows the number of hamburgers sold by a fast food chain each season. However, the bar indicating the number sold during the winter is covered by a smudge. If exactly $25\%$ of the chain's hamburgers are sold in the fall, how many million hamburgers are sold in the winter?
|
2.5
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given: $\because 4 \lt 7 \lt 9$, $\therefore 2 \lt \sqrt{7} \lt 3$, $\therefore$ the integer part of $\sqrt{7}$ is $2$, and the decimal part is $\sqrt{7}-2$. The integer part of $\sqrt{51}$ is ______, and the decimal part of $9-\sqrt{51}$ is ______.
|
8-\sqrt{51}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A math competition problem has probabilities of being solved independently by person \( A \), \( B \), and \( C \) as \( \frac{1}{a} \), \( \frac{1}{b} \), and \( \frac{1}{c} \) respectively, where \( a \), \( b \), and \( c \) are positive integers less than 10. When \( A \), \( B \), and \( C \) work on the problem simultaneously and independently, the probability that exactly one of them solves the problem is \( \frac{7}{15} \). Determine the probability that none of the three persons solve the problem.
|
4/15
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Ten numbers are written around a circle with their sum equal to 100. It is known that the sum of each triplet of consecutive numbers is at least 29. Identify the smallest number \( A \) such that, in any such set of numbers, each number does not exceed \( A \).
|
13
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square.
|
3
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given the function $f(x)= \begin{cases} \log_{2}x,x > 0 \\ x^{2}+4x+1,x\leqslant 0\\ \end{cases}$, if the real number $a$ satisfies $f(f(a))=1$, calculate the sum of all possible values of the real number $a$.
|
-\frac{15}{16} - \sqrt{5}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Find the largest value of $n$ such that $3x^2 +nx + 72$ can be factored as the product of two linear factors with integer coefficients.
|
217
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
What is the smallest possible median for the six number set $\{x, 2x, 3, 2, 5, 4x\}$ if $x$ can be any positive integer?
|
2.5
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
What is the largest value of $n$ less than 100,000 for which the expression $10(n-3)^5 - n^2 + 20n - 30$ is a multiple of 7?
|
99999
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Two different natural numbers are selected from the set $\ \allowbreak \{1, 2, 3, \ldots, 6\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction.
|
\frac{11}{15}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
What is the value of $19^2-17^2+15^2-13^2+11^2-9^2+7^2-5^2+3^2-1^2?$
|
200
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Find the largest negative integer $x$ which satisfies the congruence $34x+6\equiv 2\pmod {20}$.
|
-6
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given the lateral area of a cylinder with a square cross-section is $4\pi$, calculate the volume of the cylinder.
|
2\pi
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
How many solutions in natural numbers \( x, y \) does the system of equations have
\[
\left\{\begin{array}{l}
\gcd(x, y)=20! \\
\text{lcm}(x, y)=30!
\end{array} \quad (\text{where } n! = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot n) ?\right.
\]
|
256
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Find the degree measure of the least positive angle $\theta$ for which
\[\cos 5^\circ = \sin 25^\circ + \sin \theta.\]
|
35^\circ
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let $\mathbf{u},$ $\mathbf{v},$ and $\mathbf{w}$ be vectors such that $\|\mathbf{u}\| = 3,$ $\|\mathbf{v}\| = 4,$ and $\|\mathbf{w}\| = 5,$ and
\[\mathbf{u} + \mathbf{v} + \mathbf{w} = \mathbf{0}.\]Compute $\mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} + \mathbf{v} \cdot \mathbf{w}.$
|
-25
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let $x,$ $y,$ $z$ be real numbers such that $4x^2 + y^2 + 16z^2 = 1.$ Find the maximum value of
\[7x + 2y + 8z.\]
|
\frac{9}{2}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Elena drives 45 miles in the first hour, but realizes that she will be 45 minutes late if she continues at the same speed. She increases her speed by 20 miles per hour for the rest of the journey and arrives 15 minutes early. Determine the total distance from Elena's home to the convention center.
|
191.25
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The average of $a, b$ and $c$ is 16. The average of $c, d$ and $e$ is 26. The average of $a, b, c, d$, and $e$ is 20. What is the value of $c$?
|
26
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given that a recipe calls for \( 4 \frac{1}{2} \) cups of flour, calculate the amount of flour needed if only half of the recipe is made.
|
2 \frac{1}{4}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
What is the diameter of the circle inscribed in triangle $ABC$ if $AB = 11,$ $AC=6,$ and $BC=7$? Express your answer in simplest radical form.
|
\sqrt{10}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let $A B C$ be a triangle with $A B=6, A C=7, B C=8$. Let $I$ be the incenter of $A B C$. Points $Z$ and $Y$ lie on the interior of segments $A B$ and $A C$ respectively such that $Y Z$ is tangent to the incircle. Given point $P$ such that $$\angle Z P C=\angle Y P B=90^{\circ}$$ find the length of $I P$.
|
\frac{\sqrt{30}}{2}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In the diagram, the square has a perimeter of $48$ and the triangle has a height of $48.$ If the square and the triangle have the same area, what is the value of $x?$ [asy]
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw((3,0)--(6,0)--(6,5)--cycle);
draw((5.8,0)--(5.8,.2)--(6,.2));
label("$x$",(4.5,0),S);
label("48",(6,2.5),E);
[/asy]
|
6
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A $33$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $33$ vertices equals $99$. The midpoints of the sides of $P_1$ form a second $33$-gon, $P_2$. Finally, the midpoints of the sides of $P_2$ form a third $33$-gon, $P_3$. Find the sum of the $x$-coordinates of the vertices of $P_3$.
|
99
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
What is the largest $2$-digit prime factor of the integer $n = {200\choose 100}$?
|
61
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The year 2013 has arrived, and Xiao Ming's older brother sighed and said, "This is the first year in my life that has no repeated digits." It is known that Xiao Ming's older brother was born in a year that is a multiple of 19. How old is the older brother in 2013?
|
18
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
If $b$ and $c$ are constants and $(x + 2)(x + b) = x^2 + cx + 6$, then $c$ is
|
$5$
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given that $a > 0$ and $b > 0$, they satisfy the equation $3a + b = a^2 + ab$. Find the minimum value of $2a + b$.
|
3 + 2\sqrt{2}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The square
$\begin{tabular}{|c|c|c|} \hline 50 & \textit{b} & \textit{c} \\ \hline \textit{d} & \textit{e} & \textit{f} \\ \hline \textit{g} & \textit{h} & 2 \\ \hline \end{tabular}$
is a multiplicative magic square. That is, the product of the numbers in each row, column, and diagonal is the same. If all the entries are positive integers, what is the sum of the possible values of $g$?
|
35
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
We define a function $f(x)$ such that $f(11)=34$, and if there exists an integer $a$ such that $f(a)=b$, then $f(b)$ is defined and
$f(b)=3b+1$ if $b$ is odd
$f(b)=\frac{b}{2}$ if $b$ is even.
What is the smallest possible number of integers in the domain of $f$?
|
15
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Simplify the expression: $(a-3+\frac{1}{a-1})÷\frac{{a}^{2}-4}{{a}^{2}+2a}⋅\frac{1}{a-2}$, then choose a suitable number from $-2$, $-1$, $0$, $1$, $2$ to substitute and evaluate.
|
\frac{1}{2}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
What is the sum of all positive integers $n$ such that $\text{lcm}(2n, n^2) = 14n - 24$ ?
|
17
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let $\triangle ABC$ be an isosceles triangle with $\angle A = 90^\circ.$ There exists a point $P$ inside $\triangle ABC$ such that $\angle PAB = \angle PBC = \angle PCA$ and $AP = 10.$ Find the area of $\triangle ABC.$
Diagram
[asy] /* Made by MRENTHUSIASM */ size(200); pair A, B, C, P; A = origin; B = (0,10*sqrt(5)); C = (10*sqrt(5),0); P = intersectionpoints(Circle(A,10),Circle(C,20))[0]; dot("$A$",A,1.5*SW,linewidth(4)); dot("$B$",B,1.5*NW,linewidth(4)); dot("$C$",C,1.5*SE,linewidth(4)); dot("$P$",P,1.5*NE,linewidth(4)); markscalefactor=0.125; draw(rightanglemark(B,A,C,10),red); draw(anglemark(P,A,B,25),red); draw(anglemark(P,B,C,25),red); draw(anglemark(P,C,A,25),red); add(pathticks(anglemark(P,A,B,25), n = 1, r = 0.1, s = 10, red)); add(pathticks(anglemark(P,B,C,25), n = 1, r = 0.1, s = 10, red)); add(pathticks(anglemark(P,C,A,25), n = 1, r = 0.1, s = 10, red)); draw(A--B--C--cycle^^P--A^^P--B^^P--C); label("$10$",midpoint(A--P),dir(-30),blue); [/asy] ~MRENTHUSIASM
|
250
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given a 4-inch cube constructed from 64 smaller 1-inch cubes, with 50 red and 14 white cubes, arrange these cubes such that the white surface area exposed on the larger cube is minimized, and calculate the fraction of the total surface area of the 4-inch cube that is white.
|
\frac{1}{16}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Joel selected an acute angle $x$ (strictly between 0 and 90 degrees) and wrote the values of $\sin x$, $\cos x$, and $\tan x$ on three different cards. Then he gave those cards to three students, Malvina, Paulina, and Georgina, one card to each, and asked them to figure out which trigonometric function (sin, cos, or tan) produced their cards. Even after sharing the values on their cards with each other, only Malvina was able to surely identify which function produced the value on her card. Compute the sum of all possible values that Joel wrote on Malvina's card.
|
\frac{1 + \sqrt{5}}{2}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Sophie has written three tests. Her marks were $73\%$, $82\%$, and $85\%$. She still has two tests to write. All tests are equally weighted. Her goal is an average of $80\%$ or higher. With which of the following pairs of marks on the remaining tests will Sophie not reach her goal: $79\%$ and $82\%$, $70\%$ and $91\%$, $76\%$ and $86\%$, $73\%$ and $83\%$, $61\%$ and $99\%$?
|
73\% and 83\%
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
An integer $N$ is selected at random in the range $1 \leq N \leq 2030$. Calculate the probability that the remainder when $N^{12}$ is divided by $7$ is $1$.
|
\frac{6}{7}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Fix a sequence $a_1,a_2,a_3\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$ ,we have $a_{pk+1}=pa_k-3a_p+13$ .Determine all possible values of $a_{2013}$ .
|
13
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
What is the largest possible remainder that is obtained when a two-digit number is divided by the sum of its digits?
|
15
|
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