problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
How many integers $n$ are there such that $3 \leq n \leq 10$ and $121_n$ (the number written as $121$ in base $n$) is a perfect square? | 8 |
What is the sum of all three-digit and four-digit positive integers up to 2000? | 1996050 |
A triangle has sides of length 888, 925, and $x>0$. Find the value of $x$ that minimizes the area of the circle circumscribed about the triangle. | 259 |
If $(x + y)^2 = 1$ and $xy = -4$, what is the value of $x^2 + y^2$? | 9 |
What is the ones digit of $7^{35}$ when written as an integer? | 3 |
Given the linear function y=kx+b, where k and b are constants, and the table of function values, determine the incorrect function value. | 12 |
James joins a football team and becomes the star. He scores 4 touchdowns per game and each touchdown is worth 6 points. There are 15 games in the season. He also manages to score 2 point conversions 6 times during the season. The old record was 300 points during the season. How many points did James beat the old r... | He scored 4*6=<<4*6=24>>24 points per game
So he scored 15*24=<<15*24=360>>360 points from touchdowns
He also scored 2*6=<<2*6=12>>12 points from the 2 point conversions
So he scored a total of 360+12=<<360+12=372>>372 points
That means he beat the old record by 372-300=<<372-300=72>>72 points
#### 72 |
Determine the value of the expression $\sin 410^{\circ}\sin 550^{\circ}-\sin 680^{\circ}\cos 370^{\circ}$. | \frac{1}{2} |
The square was cut into 25 smaller squares, of which exactly one has a side length different from 1 (each of the others has a side length of 1).
Find the area of the original square. | 49 |
I planned to work 20 hours a week for 12 weeks this summer to earn $3000 to buy a used car. Unfortunately, due to unforeseen events, I wasn't able to work any hours during the first three weeks of the summer. How many hours per week do I need to work for the remaining summer to achieve my financial goal? | 27 |
If I have a $4\times 4$ chess board, in how many ways can I place four distinct pawns on the board such that each column and row of the board contains no more than one pawn? | 576 |
Find the least positive integer $n$ for which $\frac{n-13}{5n+6}$ is a non-zero reducible fraction. | 84 |
Let rectangle $A B C D$ have lengths $A B=20$ and $B C=12$. Extend ray $B C$ to $Z$ such that $C Z=18$. Let $E$ be the point in the interior of $A B C D$ such that the perpendicular distance from $E$ to \overline{A B}$ is 6 and the perpendicular distance from $E$ to \overline{A D}$ is 6 . Let line $E Z$ intersect $A B$... | 72 |
Four different natural numbers, of which one is 1, have the following properties: the sum of any two of them is a multiple of 2, the sum of any three of them is a multiple of 3, and the sum of all four numbers is a multiple of 4. What is the minimum possible sum of these four numbers? | 40 |
On each side of an equilateral triangle with side length $n$ units, where $n$ is an integer, $1 \leq n \leq 100$ , consider $n-1$ points that divide the side into $n$ equal segments. Through these points, draw lines parallel to the sides of the triangle, obtaining a net of equilateral triangles of side length ... | 67 |
Let $N = 34 \cdot 34 \cdot 63 \cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$? | 1 : 14 |
Let $a$, $b$, $c$, $d$, and $e$ be positive integers with $a+b+c+d+e=2020$ and let $M$ be the largest of the sum $a+b$, $b+c$, $c+d$ and $d+e$. What is the smallest possible value of $M$? | 1010 |
For some constants $a$ and $c,$
\[\begin{pmatrix} a \\ -1 \\ c \end{pmatrix} \times \begin{pmatrix} 7 \\ 3 \\ 5 \end{pmatrix} = \begin{pmatrix} -11 \\ -16 \\ 25 \end{pmatrix}.\]Enter the ordered pair $(a,c).$ | (6,2) |
In the diagram, $\triangle ABE$, $\triangle BCE$ and $\triangle CDE$ are right-angled, with $\angle AEB=\angle BEC = \angle CED = 60^\circ$, and $AE=24$. [asy]
pair A, B, C, D, E;
A=(0,20.785);
B=(0,0);
C=(9,-5.196);
D=(13.5,-2.598);
E=(12,0);
draw(A--B--C--D--E--A);
draw(B--E);
draw(C--E);
label("A", A, N);
label("B",... | 6 |
Find all numbers of the form $\overline{13 x y 45 z}$ that are divisible by 792. | 1380456 |
Given that $4^{-1} \equiv 57 \pmod{119}$, find $64^{-1} \pmod{119}$, as a residue modulo 119. (Give an answer between 0 and 118, inclusive.) | 29 |
Simplify $\sqrt{180}$. | 6\sqrt5 |
The dilation, centered at $-1 + 4i,$ with scale factor $-2,$ takes $2i$ to which complex number? | -3 + 8i |
The mean, median, and mode of the $7$ data values $60, 100, x, 40, 50, 200, 90$ are all equal to $x$. What is the value of $x$? | 90 |
What is the smallest positive integer that is a multiple of each of 3, 5, 7, and 9? | 315 |
Two distinct positive integers from 1 to 50 inclusive are chosen. Let the sum of the integers equal $S$ and the product equal $P$. What is the probability that $P+S$ is one less than a multiple of 5? | \frac{89}{245} |
Rounded to 2 decimal places, what is $\frac{7}{9}$? | 0.78 |
Solve the following equations:
2x + 62 = 248; x - 12.7 = 2.7; x ÷ 5 = 0.16; 7x + 2x = 6.3. | 0.7 |
A charity sells $140$ benefit tickets for a total of $2001$. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets? | $782 |
What is the sum of all integer values of $n$ such that $\frac{20}{2n - 1}$ is an integer? | 2 |
A circle is inscribed in a square, and within this circle, a smaller square is inscribed such that one of its sides coincides with a side of the larger square and two vertices lie on the circle. Calculate the percentage of the area of the larger square that is covered by the smaller square. | 50\% |
Joey studies for his SAT exams 2 hours per night 5 nights a week. On the weekends, he studies 3 hours a day. If his SAT exam is 6 weeks away, how much time will Joey spend studying? | Joey studies 2 hours a night 5 nights a week so that's 2*5 = <<2*5=10>>10 hours
He studies 3 hours a day on the weekends so that's 3*2 = <<3*2=6>>6 hours
In one week, Joey studies 10+6 = <<10+6=16>>16 hours
He has 6 weeks till his exams and he studies for 16 hours a week so that's 6*16 = <<6*16=96>>96 hours of studying... |
Harper needs to buy teacher appreciation gifts for her children’s teachers. Her son has 3 different teachers and her daughter has 4. If she spent $70 on gifts, how much did each gift cost? | Her son has 3 teachers and her daughter has 4, so together they have 3+4 = <<3+4=7>>7 teachers
Harper has already spent $70 on 7 gifts, so she spent $70/7 = $<<70/7=10>>10 per gift
#### 10 |
A dealer plans to sell a new type of air purifier. After market research, the following pattern was discovered: When the profit per purifier is $x$ (unit: Yuan, $x > 0$), the sales volume $q(x)$ (unit: hundred units) and $x$ satisfy the following relationship: If $x$ does not exceed $20$, then $q(x)=\dfrac{1260}{x+1}$;... | 240000 |
If point \( P \) is the circumcenter of \(\triangle ABC\) and \(\overrightarrow{PA} + \overrightarrow{PB} + \lambda \overrightarrow{PC} = \mathbf{0}\), where \(\angle C = 120^\circ\), then find the value of the real number \(\lambda\). | -1 |
The square root of $x$ is greater than 3 and less than 4. How many integer values of $x$ satisfy this condition? | 6 |
What is the value of the expression $(25 + 8)^2 - (8^2 +25^2)$? | 400 |
Let \( S = \{1, 2, 3, 4, \ldots, 16\} \). Each of the following subsets of \( S \):
\[ \{6\},\{1, 2, 3\}, \{5, 7, 9, 10, 11, 12\}, \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \]
has the property that the sum of all its elements is a multiple of 3. Find the total number of non-empty subsets \( A \) of \( S \) such that the sum of all... | 21855 |
Evaluate the expression \[ \frac{a^2 + 2a}{a^2 + a} \cdot \frac{b^2 - 4}{b^2 - 6b + 8} \cdot \frac{c^2 + 16c + 64}{c^2 + 12c + 36} \]
given that \(c = b - 20\), \(b = a + 4\), \(a = 2\), and ensuring none of the denominators are zero. | \frac{3}{4} |
Square $IJKL$ is contained within square $WXYZ$ such that each side of $IJKL$ can be extended to pass through a vertex of $WXYZ$. The side length of square $WXYZ$ is $\sqrt{98}$, and $WI = 2$. What is the area of the inner square $IJKL$?
A) $62$
B) $98 - 4\sqrt{94}$
C) $94 - 4\sqrt{94}$
D) $98$
E) $100$ | 98 - 4\sqrt{94} |
Given that $\alpha, \beta, \gamma$ are all acute angles and $\cos^{2} \alpha + \cos^{2} \beta + \cos^{2} \gamma = 1$, find the minimum value of $\tan \alpha \cdot \tan \beta \cdot \tan \gamma$. | 2\sqrt{2} |
How many lattice points are enclosed by the triangle with vertices $(0,99),(5,100)$, and $(2003,500) ?$ Don't count boundary points. | 0 |
Given the general term of a sequence ${a_n}$ is $a_n = -n^2 + 12n - 32$, and the sum of its first $n$ terms is $S_n$, for any $m, n \in \mathbb{N}^*$ with $m < n$, the maximum value of $S_n - S_m$ is __________________. | 10 |
Given $\delta(x) = 3x + 8$ and $\phi(x) = 8x + 7$, what is $x$ if $\delta(\phi(x)) = 7$? | -\dfrac{11}{12} |
In triangle $ABC,$ if median $\overline{AD}$ makes an angle of $45^\circ$ with side $\overline{BC},$ then find the value of $|\cot B - \cot C|.$ | 2 |
A region is bounded by semicircular arcs constructed on the side of a square whose sides measure $2/\pi$, as shown. What is the perimeter of this region? [asy]
path a=(10,0)..(5,5)--(5,-5)..cycle;
path b=(0,10)..(5,5)--(-5,5)..cycle;
path c=(-10,0)..(-5,5)--(-5,-5)..cycle;
path d=(0,-10)..(-5,-5)--(5,-5)..cycle;
path e... | 4 |
In triangle \(A B C\), side \(B C\) equals 4, and the median drawn to this side equals 3. Find the length of the common chord of two circles, each of which passes through point \(A\) and is tangent to \(B C\), with one tangent at point \(B\) and the other at point \(C\). | \frac{5}{3} |
The product of two positive integers is 18. The positive difference of these two integers is 3. What is the sum of the two integers? | 9 |
Given the function $f(x)$ with the domain $[1, +\infty)$, and $f(x) = \begin{cases} 1-|2x-3|, & 1\leq x<2 \\ \frac{1}{2}f\left(\frac{1}{2}x\right), & x\geq 2 \end{cases}$, then the number of zeros of the function $y=2xf(x)-3$ in the interval $(1, 2017)$ is \_\_\_\_\_\_. | 11 |
Suppose $\triangle ABC$ is a triangle where $AB = 36, AC = 36$, and $\angle B = 60^\circ$. A point $P$ is considered a fold point if the creases formed when vertices $A, B,$ and $C$ are folded onto point $P$ do not intersect inside $\triangle ABC$. Find the area of the set of all fold points of $\triangle ABC$, given i... | 381 |
In a $k \times k$ chessboard, a set $S$ of 25 cells that are in a $5 \times 5$ square is chosen uniformly at random. The probability that there are more black squares than white squares in $S$ is $48 \%$. Find $k$. | 9 |
Given the hyperbola $\frac{x^{2}}{4-m} + \frac{y^{2}}{m-2}=1$, find the value of $m$ if its asymptote equations are $y=± \frac{1}{3}x$. | \frac{7}{4} |
The number of solutions of $2^{2x}-3^{2y}=55$, in which $x$ and $y$ are integers, is:
\[\textbf{(A)} \ 0 \qquad\textbf{(B)} \ 1 \qquad \textbf{(C)} \ 2 \qquad\textbf{(D)} \ 3\qquad \textbf{(E)} \ \text{More than three, but finite}\]
| 1 |
Let $p$, $q$, $r$, $s$, and $t$ be distinct integers such that $(8-p)(8-q)(8-r)(8-s)(8-t) = -120$. Calculate the sum $p+q+r+s+t$. | 27 |
George now has an unfair eight-sided die. The probabilities of rolling each number from 1 to 5 are each $\frac{1}{15}$, the probability of rolling a 6 or a 7 is $\frac{1}{6}$ each, and the probability of rolling an 8 is $\frac{1}{5}$. What is the expected value of the number shown when this die is rolled? Express your ... | 4.7667 |
Given the function \\(f(x) = x^2 + 2ax + 4\\) and the interval \\([-3,5]\\), calculate the probability that the function has no real roots. | \dfrac{1}{2} |
Let $A B C$ be a triangle with $A B=7, B C=9$, and $C A=4$. Let $D$ be the point such that $A B \| C D$ and $C A \| B D$. Let $R$ be a point within triangle $B C D$. Lines $\ell$ and $m$ going through $R$ are parallel to $C A$ and $A B$ respectively. Line $\ell$ meets $A B$ and $B C$ at $P$ and $P^{\prime}$ respectivel... | 180 |
In the final round of a giraffe beauty contest, two giraffes named Tall and Spotted have made it to this stage. There are 105 voters divided into 5 districts, each district divided into 7 sections, with each section having 3 voters. Voters select the winner in their section by majority vote; in a district, the giraffe ... | 24 |
Given a triangular pyramid $D-ABC$ with all four vertices lying on the surface of a sphere $O$, if $DC\bot $ plane $ABC$, $\angle ACB=60^{\circ}$, $AB=3\sqrt{2}$, and $DC=2\sqrt{3}$, calculate the surface area of sphere $O$. | 36\pi |
If $x$, $y$, and $z$ are positive with $xy=20\sqrt[3]{2}$, $xz = 35\sqrt[3]{2}$, and $yz=14\sqrt[3]{2}$, then what is $xyz$? | 140 |
A circle with center $O$ has radius 25. Chord $\overline{AB}$ of length 30 and chord $\overline{CD}$ of length 14 intersect at point $P$. The distance between the midpoints of the two chords is 12. The quantity $OP^2$ can be represented as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find th... | 57 |
Suppose $\cos S = 0.5$ in the diagram below. What is $ST$?
[asy]
pair P,S,T;
P = (0,0);
S = (6,0);
T = (0,6*tan(acos(0.5)));
draw(P--S--T--P);
draw(rightanglemark(S,P,T,18));
label("$P$",P,SW);
label("$S$",S,SE);
label("$T$",T,N);
label("$10$",S/2,S);
[/asy] | 20 |
Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible? | 3120 |
Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be convex 8-gon (no three diagonals concruent).
The intersection of arbitrary two diagonals will be called "button".Consider the convex quadrilaterals formed by four vertices of $A_1A_2A_3A_4A_5A_6A_7A_8$ and such convex quadrilaterals will be called "sub quadrilaterals".Find the sm... | 14 |
Divide all coins into two parts of 20 coins each and weigh them. Since the number of fake coins is odd, one of the piles will be heavier. Thus, there is at most one fake coin in that pile. Divide it into two piles of 10 coins and weigh them. If the balance is even, then all 20 coins weighed are genuine. If one of the p... | 16 |
Given the sequence $1, 4, 9, -16, 25, -36,\ldots$, whose $n$th term is $(-1)^{n+1}\cdot n^2$, find the average of the first $100$ terms of the sequence. | -50.5 |
Determine the least real number $M$ such that the inequality \[|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \leq M(a^{2}+b^{2}+c^{2})^{2}\] holds for all real numbers $a$, $b$ and $c$. | M=\frac 9{16\sqrt 2} |
Form a four-digit number using the digits 0, 1, 2, 3, 4, 5 without repetition.
(I) How many different four-digit numbers can be formed?
(II) How many of these four-digit numbers have a tens digit that is larger than both the units digit and the hundreds digit?
(III) Arrange the four-digit numbers from part (I) in ascen... | 2301 |
Calculate the following product:
$$\frac{1}{3}\times9\times\frac{1}{27}\times81\times\frac{1}{243}\times729\times\frac{1}{2187}\times6561\times\frac{1}{19683}\times59049.$$ | 243 |
The number $121_b$, written in the integral base $b$, is the square of an integer, for | $b > 2$ |
What is the median of the numbers in the list $19^{20}, \frac{20}{19}, 20^{19}, 2019, 20 \times 19$? | 2019 |
Find the quadratic polynomial $p(x)$ such that $p(-3) = 10,$ $p(0) = 1,$ and $p(2) = 5.$ | x^2 + 1 |
How many natural numbers between 200 and 400 are divisible by 8? | 26 |
There is a target on the wall consisting of five zones: a central circle (bullseye) and four colored rings. The width of each ring is equal to the radius of the bullseye. It is known that the number of points for hitting each zone is inversely proportional to the probability of hitting that zone and that hitting the bu... | 45 |
A pair of standard $6$-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference? | \frac{1}{12} |
For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$? | 16 |
In $\triangle ABC$, the sides opposite to angles A, B, and C are a, b, and c, respectively. If $C= \frac {\pi}{3}, b= \sqrt {2}, c= \sqrt {3}$, find the measure of angle A. | \frac{5\pi}{12} |
Given that point $(x, y)$ moves on the circle $x^{2}+(y-1)^{2}=1$.
(1) Find the maximum and minimum values of $\frac{y-1}{x-2}$;
(2) Find the maximum and minimum values of $2x+y$. | 1 - \sqrt{5} |
Compute \[\sum_{k=2}^{31} \log_3\left(1 + \frac{2}{k}\right) \log_k 3 \log_{k+1} 3.\] | \frac{0.5285}{3} |
A cryptographer devises the following method for encoding positive integers. First, the integer is expressed in base $5$.
Second, a 1-to-1 correspondence is established between the digits that appear in the expressions in base $5$ and the elements of the set
$\{V, W, X, Y, Z\}$. Using this correspondence, the cryptog... | 108 |
Define the operation $\spadesuit$ as $a\,\spadesuit\,b = |a- b|$ . What is the value of $2\, \spadesuit\,(4\,\spadesuit\,7)$? | 1 |
John buys 5 notebooks that have 40 pages each. He uses 4 pages per day. How many days do the notebooks last? | He got 5*40=<<5*40=200>>200 pages
So they will last 200/4=<<200/4=50>>50 days
#### 50 |
How many degrees are in the measure of the smaller angle formed by the hour and minute hands of a clock when the time is 7 p.m.? | 150^\circ |
The set
$$
A=\{\sqrt[n]{n} \mid n \in \mathbf{N} \text{ and } 1 \leq n \leq 2020\}
$$
has the largest element as $\qquad$ . | \sqrt[3]{3} |
If we express $2x^2 + 6x + 11$ in the form $a(x - h)^2 + k$, then what is $h$? | -\frac{3}{2} |
The height of a right triangle $ABC$ dropped to the hypotenuse is 9.6. From the vertex $C$ of the right angle, a perpendicular $CM$ is raised to the plane of the triangle $ABC$, with $CM = 28$. Find the distance from point $M$ to the hypotenuse $AB$. | 29.6 |
Each side of square $ABCD$ with side length of $4$ is divided into equal parts by three points. Choose one of the three points from each side, and connect the points consecutively to obtain a quadrilateral. Which numbers can be the area of this quadrilateral? Just write the numbers without proof.
[asy]
import graph; si... | {6,7,7.5,8,8.5,9,10} |
The areas of three squares are 16, 49 and 169. What is the average (mean) of their side lengths? | 8 |
When a student used a calculator to find the average of 30 data points, they mistakenly entered one of the data points, 105, as 15. Find the difference between the calculated average and the actual average. | -3 |
Given a cube with an edge length of 1, find the surface area of the smaller sphere that is tangent to the larger sphere and the three faces of the cube. | 7\pi - 4\sqrt{3}\pi |
Given the parametric equation of line $l$:
$$
\begin{cases}
x=t+1 \\
y= \sqrt {3}t
\end{cases}
$$
(where $t$ is the parameter), and the polar equation of curve $C$ is $\rho=2\cos\theta$, then the polar radius (taking the positive value) of the intersection point of line $l$ and curve $C$ is ______. | \sqrt{3} |
Cylinder $B$'s height is equal to the radius of cylinder $A$ and cylinder $B$'s radius is equal to the height $h$ of cylinder $A$. If the volume of cylinder $A$ is twice the volume of cylinder $B$, the volume of cylinder $A$ can be written as $N \pi h^3$ cubic units. What is the value of $N$?
[asy]
size(4cm,4cm);
path... | 4 |
A certain number with a sum of digits equal to 2021 was divided by 7 and resulted in a number composed exclusively of the digit 7. How many digits 7 can this number contain? If there are multiple answers, provide their sum. | 503 |
A right pyramid has a square base with perimeter 24 inches. Its apex is 9 inches from each of the other vertices. What is the height of the pyramid from its peak to the center of its square base, in inches? | 3\sqrt{7} |
Triangle $ABC$ is a right triangle with $AC = 7,$ $BC = 24,$ and right angle at $C.$ Point $M$ is the midpoint of $AB,$ and $D$ is on the same side of line $AB$ as $C$ so that $AD = BD = 15.$ Given that the area of triangle $CDM$ may be expressed as $\frac {m\sqrt {n}}{p},$ where $m,$ $n,$ and $p$ are positive integers... | 578 |
Find the result of $46_8 - 63_8$ and express your answer in base 10. | -13 |
For integers a, b, c, and d the polynomial $p(x) =$ $ax^3 + bx^2 + cx + d$ satisfies $p(5) + p(25) = 1906$ . Find the minimum possible value for $|p(15)|$ . | 47 |
Given the parabola $C: x^{2}=2py\left(p \gt 0\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\left(y+4\right)^{2}=1$ is $4$.<br/>$(1)$ Find $p$;<br/>$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with points $A$ and $B$ as the points of tangency, ... | 20\sqrt{5} |
Let $d$ be a randomly chosen divisor of 2016. Find the expected value of $\frac{d^{2}}{d^{2}+2016}$. | \frac{1}{2} |
Let $a$ and $b$ be integers such that $ab = 100.$ Find the minimum value of $a + b.$ | -101 |
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