problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Triangle $ABC$ has right angle at $B$, and contains a point $P$ for which $PA = 10$, $PB = 6$, and $\angle APB = \angle BPC = \angle CPA$. Find $PC$.
[asy] unitsize(0.2 cm); pair A, B, C, P; A = (0,14); B = (0,0); C = (21*sqrt(3),0); P = intersectionpoint(arc(B,6,0,180),arc(C,33,0,180)); draw(A--B--C--cycle); draw(A--P... | 33 |
The ratio of measures of two complementary angles is 4 to 5. The smallest measure is increased by $10\%$. By what percent must the larger measure be decreased so that the two angles remain complementary? | 8\% |
Convert the point $(1,-\sqrt{3})$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\theta),$ where $r > 0$ and $0 \le \theta < 2 \pi.$ | \left( 2, \frac{5 \pi}{3} \right) |
Given the function $f(x)=- \sqrt {3}\sin ^{2}x+\sin x\cos x$.
(1) Find the value of $f( \dfrac {25π}{6})$;
(2) Let $α∈(0,π)$, $f( \dfrac {α}{2})= \dfrac {1}{4}- \dfrac { \sqrt {3}}{2}$, find the value of $\sin α$. | \dfrac {1+3 \sqrt {5}}{8} |
In the Cartesian coordinate system \(xOy\), the set of points \(K=\{(x, y) \mid x, y=-1,0,1\}\). Three points are randomly selected from \(K\). What is the probability that the distance between any two of these three points does not exceed 2? | 5/14 |
In parallelogram \(ABCD\), \(AB = 1\), \(BC = 4\), and \(\angle ABC = 60^\circ\). Suppose that \(AC\) is extended from \(A\) to a point \(E\) beyond \(C\) so that triangle \(ADE\) has the same area as the parallelogram. Find the length of \(DE\). | 2\sqrt{3} |
What is the smallest positive integer that is a multiple of both 30 and 40 but not a multiple of 16? | 120 |
Let $ABCD$ be an isosceles trapezoid with $\overline{AD}||\overline{BC}$ whose angle at the longer base $\overline{AD}$ is $\dfrac{\pi}{3}$. The diagonals have length $10\sqrt {21}$, and point $E$ is at distances $10\sqrt {7}$ and $30\sqrt {7}$ from vertices $A$ and $D$, respectively. Let $F$ be the foot of the altitud... | 32 |
Using the vertices of a cube as vertices, how many triangular pyramids can you form? | 58 |
Insert a square into an isosceles triangle with a lateral side of 10 and a base of 12. | 4.8 |
Humpty Dumpty walks on a straight line, taking either 37 steps to the left or 47 steps to the right per minute.
What is the minimum time it will take for him to be one step to the right of the starting point? | 59 |
In the geometric sequence $\{a_n\}$ with a common ratio greater than $1$, $a_3a_7=72$, $a_2+a_8=27$, calculate $a_{12}$. | 96 |
Suppose $A B C$ is a triangle with circumcenter $O$ and orthocenter $H$ such that $A, B, C, O$, and $H$ are all on distinct points with integer coordinates. What is the second smallest possible value of the circumradius of $A B C$ ? | \sqrt{10} |
Which of the following divisions is not equal to a whole number: $\frac{60}{12}$, $\frac{60}{8}$, $\frac{60}{5}$, $\frac{60}{4}$, $\frac{60}{3}$? | 7.5 |
In $\triangle XYZ$, angle XZY is a right angle. There are three squares constructed such that each side adjacent to angle XZY has a square on it. The sum of the areas of these three squares is 512 square centimeters. Also, XZ is 20% longer than ZY. What's the area of the largest square? | 256 |
Diagonals $A C$ and $C E$ of a regular hexagon $A B C D E F$ are divided by points $M$ and $N$ such that $A M : A C = C N : C E = \lambda$. Find $\lambda$ if it is known that points $B, M$, and $N$ are collinear. | \frac{1}{\sqrt{3}} |
Gracie and Joe are choosing numbers on the complex plane. Joe chooses the point $1+2i$. Gracie chooses $-1+i$. How far apart are Gracie and Joe's points? | \sqrt{5} |
A two-digit integer $AB$ equals $\frac{1}{9}$ of the three-digit integer $AAB$, where $A$ and $B$ represent distinct digits from 1 to 9. What is the smallest possible value of the three-digit integer $AAB$? | 225 |
A given integer Fahrenheit temperature $F$ is first converted to Kelvin using the formula $K = \frac{5}{9}(F - 32) + 273.15$, rounded to the nearest integer, then converted back to Fahrenheit using the inverse formula $F' = \frac{9}{5}(K - 273.15) + 32$, and rounded to the nearest integer again. Find how many integer F... | 401 |
If $\tan x+\tan y=25$ and $\cot x + \cot y=30$, what is $\tan(x+y)$? | 150 |
The product of two positive consecutive integers is 506. What is their sum? | 45 |
An icosidodecahedron is a convex polyhedron with 20 triangular faces and 12 pentagonal faces. How many vertices does it have? | 30 |
The variables \(a, b, c, d, e\), and \(f\) represent the numbers 4, 12, 15, 27, 31, and 39 in some order. Suppose that
\[
\begin{aligned}
& a + b = c, \\
& b + c = d, \\
& c + e = f,
\end{aligned}
\]
Determine the value of \(a + c + f\). | 73 |
Using the 3 vertices of a triangle and 7 points inside it (a total of 10 points), how many smaller triangles can the original triangle be divided into?
(1985 Shanghai Junior High School Math Competition, China;
1988 Jiangsu Province Junior High School Math Competition, China) | 15 |
Federal guidelines recommend eating at least 2 cups of vegetables per day. From breakfast on Sunday to the end of the day on Thursday, Sarah has eaten 8 cups. How many cups per day does Sarah need to eat of her vegetables in order to meet her daily minimum requirement for the week? | There are 7 days in a week and 2 cups are recommended per day, bringing the total to 7 days * 2 cups/day =<<7*2=14>>14 cups for the week.
Out of the 14 total cups needed for the week, Sarah has already eaten 8, for a total of 14 cups - 8 cups = <<14-8=6>>6 cups left to consume for the week.
Sunday through Thursday equa... |
In how many ways can four people sit in a row of five chairs? | 120 |
Lucas wants to buy a book that costs $28.50. He has two $10 bills, five $1 bills, and six quarters in his wallet. What is the minimum number of nickels that must be in his wallet so he can afford the book? | 40 |
There are 1001 people sitting around a round table, each of whom is either a knight (always tells the truth) or a liar (always lies). It turned out that next to each knight there is exactly one liar, and next to each liar there is exactly one knight. What is the minimum number of knights that can be sitting at the tab... | 501 |
On a map, a 12-centimeter length represents 72 kilometers. How many kilometers does a 17-centimeter length represent? | 102 |
For positive real numbers $x,$ $y,$ and $z,$ compute the maximum value of
\[\frac{xyz(x + y + z)}{(x + y)^2 (y + z)^2}.\] | \frac{1}{4} |
Given the set $A=\{x\in \mathbb{R} | ax^2-3x+2=0, a\in \mathbb{R}\}$.
1. If $A$ is an empty set, find the range of values for $a$.
2. If $A$ contains only one element, find the value of $a$ and write down this element. | \frac{4}{3} |
The distance from home to work is $s = 6$ km. At the moment Ivan left work, his favorite dog dashed out of the house and ran to meet him. They met at a distance of one-third of the total route from work. The dog immediately turned back and ran home. Upon reaching home, the dog turned around instantly and ran back towar... | 12 |
Given two four-digit numbers \( M \) and \( N \) which are reverses of each other, and have \( q^{p}-1 \) identical positive divisors, \( M \) and \( N \) can be factorized into prime factors as \( p q^{q} r \) and \( q^{p+q} r \) respectively, where \( p \), \( q \), and \( r \) are prime numbers. Find the value of \(... | 1998 |
Square pyramid $ABCDE$ has base $ABCD$, which measures $3$ cm on a side, and altitude $AE$ perpendicular to the base, which measures $6$ cm. Point $P$ lies on $BE$, one third of the way from $B$ to $E$; point $Q$ lies on $DE$, one third of the way from $D$ to $E$; and point $R$ lies on $CE$, two thirds of the way from ... | 2\sqrt{2} |
Compute the length of the segment tangent from the origin to the circle that passes through the points $(3,4),$ $(6,8),$ and $(5,13).$ | 5 \sqrt{2} |
The circle $2x^2 = -2y^2 + 12x - 4y + 20$ is inscribed inside a square which has a pair of sides parallel to the x-axis. What is the area of the square? | 80 |
One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed there was a positive integer such that \[133^5+110^5+84^5+27^5=n^{5}.\] Find the value of $n$. | 144 |
The function \( f(x) = \max \left\{\sin x, \cos x, \frac{\sin x + \cos x}{\sqrt{2}}\right\} \) (for \( x \in \mathbb{R} \)) has a maximum value and a minimum value. Find the sum of these maximum and minimum values. | 1 - \frac{\sqrt{2}}{2} |
Find the number of sets of composite numbers less than 23 that sum to 23. | 4 |
Given that the side lengths of a convex quadrilateral are $a=4, b=5, c=6, d=7$, find the radius $R$ of the circumscribed circle around this quadrilateral. Provide the integer part of $R^{2}$ as the answer. | 15 |
In Montana, 500 people were asked what they call soft drinks. The results of the survey are shown in the pie chart. The central angle of the ``Soda'' sector of the graph is $200^\circ$, to the nearest whole degree. How many of the people surveyed chose ``Soda''? Express your answer as a whole number. | 278 |
In triangle $ABC$, if $A = \frac{\pi}{3}$, $\tan B = \frac{1}{2}$, and $AB = 2\sqrt{3} + 1$, then find the length of $BC$. | \sqrt{15} |
Find the greatest common divisor of $10293$ and $29384$. | 1 |
According to the classification standard of the Air Pollution Index (API) for city air quality, when the air pollution index is not greater than 100, the air quality is good. The environmental monitoring department of a city randomly selected the air pollution index for 5 days from last month's air quality data, and th... | 440 |
Below is the graph of $y = a \sin bx$ for some constants $a < 0$ and $b > 0.$ Find $a.$
[asy]import TrigMacros;
size(400);
real g(real x)
{
return (-2*sin(x/3));
}
draw(graph(g,-3*pi,3*pi,n=700,join=operator ..),red);
trig_axes(-3*pi,3*pi,-3,3,pi/2,1);
layer();
rm_trig_labels(-5, 5, 2);
label("$1$", (0,1), E);
l... | -2 |
The median of the numbers 3, 7, x, 14, 20 is equal to the mean of those five numbers. Calculate the sum of all real numbers \( x \) for which this is true. | 28 |
Find the number of cubic centimeters in the volume of the cylinder formed by rotating a square with side length 14 centimeters about its vertical line of symmetry. Express your answer in terms of $\pi$. | 686\pi |
Given point $A(1,2)$ and circle $C: x^{2}+y^{2}+2mx+2y+2=0$.
$(1)$ If there are two tangents passing through point $A$, find the range of $m$.
$(2)$ When $m=-2$, a point $P$ on the line $2x-y+3=0$ is chosen to form two tangents $PM$ and $PN$ to the circle. Find the minimum area of quadrilateral $PMCN$. | \frac{7\sqrt{15}}{5} |
The year 2000 is a leap year. The year 2100 is not a leap year. The following are the complete rules for determining a leap year:
(i) Year \(Y\) is not a leap year if \(Y\) is not divisible by 4.
(ii) Year \(Y\) is a leap year if \(Y\) is divisible by 4 but not by 100.
(iii) Year \(Y\) is not a leap year if \(Y\) is... | 244 |
Emily's quiz scores so far are: 92, 95, 87, 89 and 100. What score does she need to get on the sixth quiz to make the arithmetic mean of the six scores equal 93? | 95 |
Given that $\sin \alpha = 2 \cos \alpha$, find the value of $\cos ( \frac {2015\pi}{2}-2\alpha)$. | - \frac {4}{5} |
Let $a$, $b$, and $c$ be the $3$ roots of $x^3-x+1=0$. Find $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}$. | -2 |
A triangle $X Y Z$ and a circle $\omega$ of radius 2 are given in a plane, such that $\omega$ intersects segment $\overline{X Y}$ at the points $A, B$, segment $\overline{Y Z}$ at the points $C, D$, and segment $\overline{Z X}$ at the points $E, F$. Suppose that $X B>X A, Y D>Y C$, and $Z F>Z E$. In addition, $X A=1, Y... | \sqrt{10}-1 |
The center of a balloon is observed by two ground observers at angles of elevation of $45^{\circ}$ and $22.5^{\circ}$, respectively. The first observer is to the south, and the second one is to the northwest of the point directly under the balloon. The distance between the two observers is 1600 meters. How high is the ... | 500 |
Let $F_k(a,b)=(a+b)^k-a^k-b^k$ and let $S={1,2,3,4,5,6,7,8,9,10}$ . For how many ordered pairs $(a,b)$ with $a,b\in S$ and $a\leq b$ is $\frac{F_5(a,b)}{F_3(a,b)}$ an integer? | 22 |
There are 10 mountaineers, divided equally into two groups. Among them, 4 are familiar with the trails. Each group needs 2 people who are familiar with the trails. The number of different ways to distribute them is: | 60 |
A chord PQ of the left branch of the hyperbola $x^2 - y^2 = 4$ passes through its left focus $F_1$, and the length of $|PQ|$ is 7. If $F_2$ is the right focus of the hyperbola, then the perimeter of $\triangle PF_2Q$ is. | 22 |
Texas Integrated School has 15 classes and has 20 students per class. They added five more classes, how many students will they have now? | The school has 15 x 20 = <<15*20=300>>300 students right now.
They have an additional 5 x 20 = <<5*20=100>>100 students after adding 5 more classes.
Therefore, Texas Integrated School will have a total of 300+100= <<300+100=400>>400 students.
#### 400 |
If $a$ and $b$ are randomly selected real numbers between 0 and 1, find the probability that the nearest integer to $\frac{a-b}{a+b}$ is odd. | \frac{1}{3} |
Let $p(x) = x^{2008} + x^{2007} + x^{2006} + \cdots + x + 1,$
and let $r(x)$ be the polynomial remainder when $p(x)$ is divided by $x^4+x^3+2x^2+x+1$. Find the remainder when $|r(2008)|$ is divided by $1000$.
| 64 |
(a) A natural number \( n \) is less than 120. What is the maximum remainder that the number 209 can leave when divided by \( n \)?
(b) A natural number \( n \) is less than 90. What is the maximum remainder that the number 209 can leave when divided by \( n \)? | 69 |
Given the function $f(x)=a\frac{x+1}{x}+\ln{x}$, the equation of the tangent line at the point (1, f(1)) is y=bx+5.
1. Find the real values of a and b.
2. Find the maximum and minimum values of the function $f(x)$ on the interval $[\frac{1}{e}, e]$, where $e$ is the base of the natural logarithm. | 3+\ln{2} |
A parallelogram $ABCD$ is inscribed in the ellipse $\frac{x^{2}}{4}+y^{2}=1$, where the slope of the line $AB$ is $k_{1}=1$. Determine the slope of the line $AD$. | -\frac{1}{4} |
Simplify $\dfrac{111}{9999} \cdot 33.$ | \dfrac{37}{101} |
Let $ABC$ be a triangle, and $K$ and $L$ be points on $AB$ such that $\angle ACK = \angle KCL = \angle LCB$ . Let $M$ be a point in $BC$ such that $\angle MKC = \angle BKM$ . If $ML$ is the angle bisector of $\angle KMB$ , find $\angle MLC$ . | 30 |
In a regular \( n \)-gon, \( A_{1} A_{2} A_{3} \cdots A_{n} \), where \( n > 6 \), sides \( A_{1} A_{2} \) and \( A_{5} A_{4} \) are extended to meet at point \( P \). If \( \angle A_{2} P A_{4}=120^\circ \), determine the value of \( n \). | 18 |
Given the function $f(x)= \begin{cases} ax^{2}-2x-1, & x\geqslant 0\\ x^{2}+bx+c, & x < 0\end{cases}$ is an even function, and the line $y=t$ intersects the graph of $y=f(x)$ from left to right at four distinct points $A$, $B$, $C$, $D$. If $AB=BC$, then the value of the real number $t$ is \_\_\_\_\_\_. | - \dfrac {7}{4} |
Peter needs 80 ounces of soda for his party. He sees that 8 oz cans cost $.5 each. How much does he spend on soda if he buys the exact number of cans he needs? | He needs 10 cans because 80 / 8 = <<80/8=10>>10
He spends $5 because 10 x .5 = 10
#### 5 |
In triangle $ABC$, $M$ is the midpoint of $\overline{BC}$, $AB = 15$, and $AC = 24$. Let $E$ be a point on $\overline{AC}$, and $H$ be a point on $\overline{AB}$, and let $G$ be the intersection of $\overline{EH}$ and $\overline{AM}$. If $AE = 3AH$, find $\frac{EG}{GH}$. | \frac{2}{3} |
How many factors of 8000 are perfect squares? | 8 |
A capacitor with a capacitance of $C_{1} = 20 \mu$F is charged to a voltage $U_{1} = 20$ V. A second capacitor with a capacitance of $C_{2} = 5 \mu$F is charged to a voltage $U_{2} = 5$ V. The capacitors are connected with opposite-charged plates. Determine the voltage that will be established across the plates. | 15 |
Jan is making candy necklaces for herself and a few friends. Everyone receives a candy necklace each and each candy necklace is made up of 10 pieces of candies. The pieces of candies come from blocks of candy, which each produce 30 pieces of candy. If Jan breaks down 3 blocks of candy and every single piece of candy fr... | There are a total of 3 blocks of candy * 30 pieces of candy per candy block = <<3*30=90>>90 pieces of candy.
Using this, Jan can create 90 pieces of candy / 10 pieces of candy per candy necklace = <<90/10=9>>9 candy necklaces.
As Jan is keeping a candy necklace too, there must have been enough candy necklaces for 9 can... |
What is the value of $25_{10}+36_{10}$ in base 3? | 2021_3 |
Let $G$ be the centroid of quadrilateral $ABCD$. If $GA^2 + GB^2 + GC^2 + GD^2 = 116$, find the sum $AB^2 + AC^2 + AD^2 + BC^2 + BD^2 + CD^2$. | 464 |
Find all composite positive integers \(m\) such that, whenever the product of two positive integers \(a\) and \(b\) is \(m\), their sum is a power of $2$ .
*Proposed by Harun Khan* | 15 |
Calculate the value of $\sqrt{\frac{\sqrt{81} + \sqrt{81}}{2}}$. | 3 |
Jodi and Vance are researching on a deserted island and have to stay on the island for a certain number of weeks to carry out their research. On their first expedition, they stayed for three weeks on the island. They spent two weeks more on the second expedition than they spent on their first expedition. They spent twi... | If they spent two more weeks on their second research expedition than they spent on the first week, they spent 2+3 = <<2+3=5>>5 weeks on the island.
In the two expeditions, they stayed on the island for a total of 5+3 = <<5+3=8>>8 weeks.
On their third expedition, they spent twice as many weeks as they spent on the sec... |
Given set $A=\{a-2, 12, 2a^2+5a\}$, and $-3$ belongs to $A$, find the value of $a$. | -\frac{3}{2} |
How many one-thirds are in one-sixth? | \frac{1}{2} |
Solve for $c$: \[\frac{c-23}{2} = \frac{2c +5}{7}.\] | 57 |
My age is five times that of my son. Next year, my son will be eight years old. How old am I now? | If the son is going to be eight years old after a year from now, then currently, he is 8-1 = <<8-1=7>>7
Since the father's age is five times that of the son, currently he is 5*7 = <<5*7=35>>35 years old.
#### 35 |
A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by the square and the circle? Express your answer in terms of $\pi$. | 100+75\pi |
In acute triangle $ABC$ points $P$ and $Q$ are the feet of the perpendiculars from $C$ to $\overline{AB}$ and from $B$ to $\overline{AC}$, respectively. Line $PQ$ intersects the circumcircle of $\triangle ABC$ in two distinct points, $X$ and $Y$. Suppose $XP=10$, $PQ=25$, and $QY=15$. The value of $AB\cdot AC$ can be w... | 574 |
Albert is wondering how much pizza he can eat in one day. He buys 2 large pizzas and 2 small pizzas. A large pizza has 16 slices and a small pizza has 8 slices. If he eats it all, how many pieces does he eat that day? | He eats 32 from the largest pizzas because 2 x 16 = <<2*16=32>>32
He eats 16 from the small pizza because 2 x 8 = <<2*8=16>>16
He eats 48 pieces because 32 + 16 = <<32+16=48>>48
#### 48 |
Samantha has 10 different colored marbles in her bag. In how many ways can she choose five different marbles such that at least one of them is red? | 126 |
Given sets $A=\{1,3,5,7,9\}$ and $B=\{0,3,6,9,12\}$, then $A\cap (\complement_{\mathbb{N}} B) = \_\_\_\_\_\_$; the number of proper subsets of $A\cup B$ is $\_\_\_\_\_\_$. | 255 |
What is the minimum number of equilateral triangles, of side length 1 unit, needed to cover an equilateral triangle of side length 10 units? | 100 |
On the last night that roller skate rink was open, 40 people showed up to roller skate one last time. When all 40 people skated at one time, how many wheels were on the floor? | 40 people were roller skating and they all had 2 feet which means there were 40*2 = <<40*2=80>>80 feet
Each foot needs a roller skate and each roller skate has 4 wheels so 80*4 = <<80*4=320>>320 wheels when everyone skated
#### 320 |
Calculate the number of seven-digit palindromes where the second digit cannot be odd. | 4500 |
Porche has 3 hours to get all her homework done. Her math homework takes her 45 minutes. Her English homework takes her 30 minutes. Her science homework takes her 50 minutes. Her history homework takes her 25 minutes. She also has a special project due the next day. How much time does she have left to get that project ... | She has 180 minutes to do work because 3 times 60 equals <<3*60=180>>180
She has already spent 150 minutes on homework because 45 plus 30 plus 50 plus 25 equals <<45+30+50+25=150>>150
She has 30 minutes to finish her project because 180 minus 150 equals <<180-150=30>>30.
#### 30 |
In polar coordinates, the point $\left( -2, \frac{3 \pi}{8} \right)$ is equivalent to what other point, in the standard polar coordinate representation? Enter your answer in the form $(r,\theta),$ where $r > 0$ and $0 \le \theta < 2 \pi.$ | \left( 2, \frac{11 \pi}{8} \right) |
Given the function $f(x) = x^3 - ax^2 + 3x$, and $x=3$ is an extremum of $f(x)$.
(Ⅰ) Determine the value of the real number $a$;
(Ⅱ) Find the equation of the tangent line $l$ to the graph of $y=f(x)$ at point $P(1, f(1))$;
(Ⅲ) Find the minimum and maximum values of $f(x)$ on the interval $[1, 5]$. | 15 |
There are 203 students in the third grade, which is 125 fewer than the fourth grade. How many students are there in total in the third and fourth grades? | 531 |
Simplify $2 \cos ^{2}(\ln (2009) i)+i \sin (\ln (4036081) i)$. | \frac{4036082}{4036081} |
A classroom is paved with cubic bricks that have an edge length of 0.3 meters, requiring 600 bricks. If changed to cubic bricks with an edge length of 0.5 meters, how many bricks are needed? (Solve using proportions.) | 216 |
500 × 3986 × 0.3986 × 5 = ? | 0.25 \times 3986^2 |
"My phone number," said the trip leader to the kids, "is a five-digit number. The first digit is a prime number, and the last two digits are obtained from the previous pair (which represents a prime number) by rearrangement, forming a perfect square. The number formed by reversing this phone number is even." What is th... | 26116 |
Seven students count from 1 to 1000 as follows:
Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says 1, 3, 4, 6, 7, 9, . . ., 997, 999, 1000.
Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each ... | 365 |
By joining four identical trapezoids, each with equal non-parallel sides and bases measuring 50 cm and 30 cm, we form a square with an area of 2500 cm² that has a square hole in the middle. What is the area, in cm², of each of the four trapezoids? | 400 |
Let $g(x) = |3\{x\} - 1.5|$ where $\{x\}$ denotes the fractional part of $x$. Determine the smallest positive integer $m$ such that the equation \[m g(x g(x)) = x\] has at least $3000$ real solutions. | 23 |
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