problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
Vitya and Masha were born in the same year in June. Find the probability that Vitya is at least one day older than Masha. | 29/60 | 0.875 | 4,712.0625 | 4,575.642857 | 5,667 | |
The expression $\frac{\sqrt{3}\tan 12^{\circ} - 3}{(4\cos^2 12^{\circ} - 2)\sin 12^{\circ}}$ equals \_\_\_\_\_\_. | -4\sqrt{3} | 0.1875 | 7,748.6875 | 5,827.666667 | 8,192 | |
Calculate the surface area of the part of the paraboloid of revolution \( 3y = x^2 + z^2 \) that is located in the first octant and bounded by the plane \( y = 6 \). | \frac{39 \pi}{4} | 0.3125 | 6,600.25 | 5,595.4 | 7,057 | |
What is the smallest positive integer $n$ such that $\frac{n}{n+150}$ is equal to a terminating decimal? | 50 | 0.0625 | 6,660.6875 | 4,877 | 6,779.6 | |
The number $n$ can be written in base $14$ as $\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, can be written in base $15$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{b}$, and can be written in base $6$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }$, whe... | 925 | We have these equations: $196a+14b+c=225a+15c+b=222a+37c$. Taking the last two we get $3a+b=22c$. Because $c \neq 0$ otherwise $a \ngtr 0$, and $a \leq 5$, $c=1$.
Then we know $3a+b=22$. Taking the first two equations we see that $29a+14c=13b$. Combining the two gives $a=4, b=10, c=1$. Then we see that $222 \times 4+3... | 0.8125 | 4,541.125 | 3,698.615385 | 8,192 |
Bob's password consists of a non-negative single-digit number followed by a letter and another non-negative single-digit number (which could be the same as the first one). What is the probability that Bob's password consists of an odd single-digit number followed by a letter and a positive single-digit number? | \frac{9}{20} | 0.9375 | 1,984 | 1,570.133333 | 8,192 | |
A right rectangular prism, with edge lengths $\log_{5}x, \log_{6}x,$ and $\log_{8}x,$ must satisfy the condition that the sum of the squares of its face diagonals is numerically equal to 8 times the volume. What is $x?$
A) $24$
B) $36$
C) $120$
D) $\sqrt{240}$
E) $240$ | \sqrt{240} | 0 | 8,192 | -1 | 8,192 | |
Find the distance between the foci of the ellipse and its eccentricity when defined by the equation:
\[\frac{x^2}{16} + \frac{y^2}{9} = 8.\] | \frac{\sqrt{7}}{4} | 0 | 4,430.5 | -1 | 4,430.5 | |
A school selects 4 teachers from 8 to teach in 4 remote areas at the same time (one person per area), where teacher A and teacher B cannot go together, and teacher A and teacher C can only go together or not go at all. The total number of different dispatch plans is ___. | 600 | 0 | 7,753.5 | -1 | 7,753.5 | |
The points $A=\left(4, \frac{1}{4}\right)$ and $B=\left(-5,-\frac{1}{5}\right)$ lie on the hyperbola $x y=1$. The circle with diameter $A B$ intersects this hyperbola again at points $X$ and $Y$. Compute $X Y$. | \sqrt{\frac{401}{5}} | Let $A=(a, 1 / a), B=(b, \underline{1 / b})$, and $X=(x, 1 / x)$. Since $X$ lies on the circle with diameter $\overline{A B}$, we have $\angle A X B=90^{\circ}$. Thus, $\overline{A X}$ and $\overline{B X}$ are perpendicular, and so the product of their slopes must be -1 . We deduce: $$\frac{a-x}{\frac{1}{a}-\frac{1}{x}... | 0 | 7,859.875 | -1 | 7,859.875 |
Regular pentagon $ABCDE$ and regular hexagon $AEFGHI$ are drawn on opposite sides of line segment $AE$ such that they are coplanar. What is the degree measure of exterior angle $DEF$? [asy]
draw((0,2.5)--(0,7.5)--(4,10)--(8,7.5)--(8,2.5)--(4,0)--cycle,linewidth(1));
draw((8,2.5)--(11.5,-1)--(9,-5)--(5,-4.5)--(4,0),line... | 132 | 0.6875 | 5,038.625 | 4,040.181818 | 7,235.2 | |
Form a five-digit number without repeating digits using the numbers 0, 1, 2, 3, 4, where exactly one even number is sandwiched between two odd numbers. How many such five-digit numbers are there? | 28 | 0.0625 | 8,081.3125 | 6,421 | 8,192 | |
A square with an area of one square unit is inscribed in an isosceles triangle such that one side of the square lies on the base of the triangle. Find the area of the triangle, given that the centers of mass of the triangle and the square coincide (the center of mass of the triangle lies at the intersection of its medi... | 9/4 | 0.5 | 5,927.5 | 3,855.75 | 7,999.25 | |
The average age of 40 sixth-graders is 12. The average age of 30 of their teachers is 45. What is the average age of all these sixth-graders and their teachers? | 26.14 | 0.8125 | 1,155.375 | 554.076923 | 3,761 | |
The degree measures of the angles in a convex 18-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle. | 143 | Each individual angle in a $18$-gon is $\frac {(18-2) \cdot 180^\circ}{18} = 160^\circ$. Since no angle in a convex polygon can be larger than $180^\circ$, the smallest angle possible is in the set $159, 161, 157, 163, 155, 165, 153, 167, 151, 169, 149, 171, 147, 173, 145, 175, 143, 177$.
Our smallest possible angle i... | 1 | 3,324.75 | 3,324.75 | -1 |
Let $n>0$ be an integer. Each face of a regular tetrahedron is painted in one of $n$ colors (the faces are not necessarily painted different colors.) Suppose there are $n^{3}$ possible colorings, where rotations, but not reflections, of the same coloring are considered the same. Find all possible values of $n$. | 1,11 | We count the possible number of colorings. If four colors are used, there are two different colorings that are mirror images of each other, for a total of $2\binom{n}{4}$ colorings. If three colors are used, we choose one color to use twice (which determines the coloring), for a total of $3\binom{n}{3}$ colorings. If t... | 0 | 5,301 | -1 | 5,301 |
A liquid $Y$ which does not mix with water spreads out on the surface to form a circular film $0.15$ cm thick. If liquid $Y$ is poured from a rectangular holder measuring $10$ cm by $4$ cm by $8$ cm onto a large water surface, what will be the radius in centimeters of the forned circular film?
A) $\sqrt{\frac{213.33}{\... | \sqrt{\frac{2133.33}{\pi}} | 0 | 3,195.25 | -1 | 3,195.25 | |
The bottoms of two vertical poles are 12 feet apart and are on a region of flat ground. One pole is 6 feet tall and the other is 15 feet tall. How long, in feet, is a wire stretched from the top of one pole to the top of the other pole? | 15 | 1 | 1,500.25 | 1,500.25 | -1 | |
Roll a die twice in succession, observing the number of points facing up each time, and calculate:
(1) The probability that the sum of the two numbers is 5;
(2) The probability that at least one of the two numbers is odd;
(3) The probability that the point (x, y), with x being the number of points facing up on th... | \frac{2}{9} | 1 | 2,954.1875 | 2,954.1875 | -1 | |
For a pair of integers \((a, b)(0 < a < b < 1000)\), a set \(S \subseteq \{1, 2, \cdots, 2003\}\) is called a "jump set" for the pair \((a, b)\) if for any pair of elements \(\left(s_{1}, s_{2}\right)\) in \(S\), \(|s_{1}-s_{2}| \notin\{a, b\}\).
Let \(f(a, b)\) denote the maximum number of elements in a jump set for ... | 668 | 0 | 8,192 | -1 | 8,192 | |
Given \\(\alpha \in (0, \frac{\pi}{2})\\) and \\(\beta \in (\frac{\pi}{2}, \pi)\\) with \\(\sin(\alpha + \beta) = \frac{3}{5}\\) and \\(\cos \beta = -\frac{5}{13}\\), find the value of \\(\sin \alpha\\). | \frac{33}{65} | 0.625 | 7,227.375 | 6,648.6 | 8,192 | |
In an arithmetic sequence $\{a_n\}$, $a_{10} < 0$, $a_{11} > 0$, and $a_{11} > |a_{10}|$. The maximum negative value of the partial sum $S_n$ of the first $n$ terms of the sequence $\{a_n\}$ is the sum of the first ______ terms. | 19 | 0 | 7,659.4375 | -1 | 7,659.4375 | |
For what value of $n$ is $5 \times 8 \times 2 \times n = 7!$? | 63 | 1 | 1,947.875 | 1,947.875 | -1 | |
For what value of $k$ does the line represented by the equation $1-kx = -3y$ contain the point $(4,-3)$? | -2 | 1 | 1,989.125 | 1,989.125 | -1 | |
A number of linked rings, each $1$ cm thick, are hanging on a peg. The top ring has an outside diameter of $20$ cm. The outside diameter of each of the outer rings is $1$ cm less than that of the ring above it. The bottom ring has an outside diameter of $3$ cm. What is the distance, in cm, from the top of the top ring ... | 173 | 1. **Identify the sequence of inside diameters**: The outside diameter of the top ring is $20$ cm, and each subsequent ring has an outside diameter $1$ cm less than the ring above it. Since each ring is $1$ cm thick, the inside diameter of each ring is $2$ cm less than its outside diameter. Therefore, the inside diamet... | 0 | 8,040.375 | -1 | 8,040.375 |
Given that $[x]$ is the greatest integer less than or equal to $x$, calculate $\sum_{N=1}^{1024}\left[\log _{2} N\right]$. | 8204 | 0.5625 | 7,146.875 | 6,454.555556 | 8,037 | |
Determine the smallest natural number $n$ for which there exist distinct nonzero naturals $a, b, c$ , such that $n=a+b+c$ and $(a + b)(b + c)(c + a)$ is a perfect cube. | 10 | 0.4375 | 6,936.75 | 5,322.857143 | 8,192 | |
The vertex of the parabola described by the equation $y=-3x^2-30x-81$ is $(m,n)$. What is $n$? | -6 | 1 | 2,169.25 | 2,169.25 | -1 | |
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 | 0.75 | 5,827 | 5,038.666667 | 8,192 | |
What is the area of the smallest square that can contain a circle of radius 6? | 144 | 0.625 | 4,790.5 | 3,142 | 7,538 | |
Given the values $1256, 2561, 5612, 6125$, calculate the sum. | 15554 | 0.5625 | 481.125 | 447.666667 | 524.142857 | |
Alice sells an item at $10 less than the list price and receives $10\%$ of her selling price as her commission.
Bob sells the same item at $20 less than the list price and receives $20\%$ of his selling price as his commission.
If they both get the same commission, then the list price is | $30 | Let $x$ be the list price of the item.
1. **Alice's Selling Price and Commission:**
- Alice sells the item at $x - 10$ dollars.
- Her commission is $10\%$ of her selling price, which is $0.10(x - 10)$.
2. **Bob's Selling Price and Commission:**
- Bob sells the item at $x - 20$ dollars.
- His commission i... | 0 | 627.375 | -1 | 627.375 |
Find the range of $$f(A)=\frac{(\sin A)\left(3 \cos ^{2} A+\cos ^{4} A+3 \sin ^{2} A+\left(\sin ^{2} A\right)\left(\cos ^{2} A\right)\right)}{(\tan A)(\sec A-(\sin A)(\tan A))}$$ if $A \neq \frac{n \pi}{2}$. | (3,4) | We factor the numerator and write the denominator in terms of fractions to get \(\frac{(\sin A)\left(3+\cos ^{2} A\right)\left(\sin ^{2} A+\cos ^{2} A\right)}{\left(\frac{\sin A}{\cos A}\right)\left(\frac{1}{\cos A}-\frac{\sin ^{2} A}{\cos A}\right)}=\frac{(\sin A)\left(3+\cos ^{2} A\right)\left(\sin ^{2} A+\cos ^{2} A... | 0.5625 | 4,688.625 | 4,566.111111 | 4,846.142857 |
A parabola has focus $F$ and vertex $V$ , where $VF = 1$ 0. Let $AB$ be a chord of length $100$ that passes through $F$ . Determine the area of $\vartriangle VAB$ .
| 100\sqrt{10} | 0.4375 | 7,123.25 | 5,749.142857 | 8,192 | |
On the image, there are several circles connected by segments. Kostya chooses a natural number \( n \) and places different natural numbers not exceeding \( n \) in the circles so that for all the placed numbers the following property is satisfied: if the numbers \( a \) and \( b \) are connected by a segment, then th... | 105 | 0 | 7,922.8125 | -1 | 7,922.8125 | |
A triangle with interior angles $60^{\circ}, 45^{\circ}$ and $75^{\circ}$ is inscribed in a circle of radius 2. What is the area of the triangle? | 3 + \sqrt{3} | 0.875 | 5,949.625 | 5,629.285714 | 8,192 | |
For $x > 0$, the area of the triangle with vertices $(0, 0), (x, 2x)$, and $(x, 0)$ is 64 square units. What is the value of $x$? | 8 | 1 | 2,900.5 | 2,900.5 | -1 | |
Ranu starts with one standard die on a table. At each step, she rolls all the dice on the table: if all of them show a 6 on top, then she places one more die on the table; otherwise, she does nothing more on this step. After 2013 such steps, let $D$ be the number of dice on the table. What is the expected value (a... | 10071 | 0.1875 | 7,295.8125 | 3,412.333333 | 8,192 | |
Given that $BDEF$ is a square and $AB = BC = 1$, find the number of square units in the area of the regular octagon.
[asy]
real x = sqrt(2);
pair A,B,C,D,E,F,G,H;
F=(0,0); E=(2,0); D=(2+x,x); C=(2+x,2+x);
B=(2,2+2x); A=(0,2+2x); H=(-x,2+x); G=(-x,x);
draw(A--B--C--D--E--F--G--H--cycle);
draw((-x,0)--(2+x,0)--(2+x,2+2x... | 4+4\sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
A positive integer $n\geq 4$ is called *interesting* if there exists a complex number $z$ such that $|z|=1$ and \[1+z+z^2+z^{n-1}+z^n=0.\] Find how many interesting numbers are smaller than $2022.$ | 404 | 0.0625 | 8,118.125 | 7,010 | 8,192 | |
In acute triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $a\sin B = \frac{1}{2}b$.
$(1)$ Find angle $A$;
$(2)$ If $b+c=4\sqrt{2}$ and the area of $\triangle ABC$ is $2$, find $a$. | 2\sqrt{3}-2 | 0.9375 | 4,162.25 | 3,893.6 | 8,192 | |
How many distinct arrangements of the letters in the word "balloon" are there? | 1260 | 0.4375 | 1,773.6875 | 2,051.714286 | 1,557.444444 | |
Suppose that we have a right triangle $ABC$ with the right angle at $B$ such that $AC = \sqrt{61}$ and $AB = 5.$ A circle is drawn with its center on $AB$ such that the circle is tangent to $AC$ and $BC.$ If $P$ is the point where the circle and side $AC$ meet, then what is $CP$? | 6 | 0.5 | 7,263.625 | 6,335.25 | 8,192 | |
Let \[f(x) = \left\{
\begin{array}{cl}
x^2+1 &\text{ if }x>5, \\
2x-3 &\text{ if } -5 \le x \le 5, \\
3 &\text{ if } x <-5.
\end{array}
\right.\]Find $f(-7)+f(0)+f(7)$. | 50 | 1 | 1,597.375 | 1,597.375 | -1 | |
If the program flowchart on the right is executed, determine the value of the output S. | 55 | 0 | 7,365.375 | -1 | 7,365.375 | |
Let $\angle XOY = \frac{\pi}{2}$; $P$ is a point inside $\angle XOY$ and we have $OP = 1; \angle XOP = \frac{\pi}{6}.$ A line passes $P$ intersects the Rays $OX$ and $OY$ at $M$ and $N$. Find the maximum value of $OM + ON - MN.$ | 2 |
Given that \(\angle XOY = \frac{\pi}{2}\), \(P\) is a point inside \(\angle XOY\) with \(OP = 1\) and \(\angle XOP = \frac{\pi}{6}\). We need to find the maximum value of \(OM + ON - MN\) where a line passing through \(P\) intersects the rays \(OX\) and \(OY\) at \(M\) and \(N\), respectively.
To solve this problem, ... | 0 | 8,192 | -1 | 8,192 |
For what positive value of $t$ is $|{-4+ti}| = 2\sqrt{13}$? | 6 | 1 | 1,162.25 | 1,162.25 | -1 | |
A triangle with sides of 5, 12, and 13 has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles? | \frac{\sqrt{65}}{2} | 0 | 4,051.0625 | -1 | 4,051.0625 | |
In rectangle $JKLM$, $P$ is a point on $LM$ so that $\angle JPL=90^{\circ}$. $UV$ is perpendicular to $LM$ with $LU=UP$, as shown. $PL$ intersects $UV$ at $Q$. Point $R$ is on $LM$ such that $RJ$ passes through $Q$. In $\triangle PQL$, $PL=25$, $LQ=20$ and $QP=15$. Find $VD$. [asy]
size(7cm);defaultpen(fontsize(9));
re... | \dfrac{28}{3} | 0 | 8,192 | -1 | 8,192 | |
Each corner cube is removed from this $3\text{ cm}\times 3\text{ cm}\times 3\text{ cm}$ cube. The surface area of the remaining figure is
[asy]
draw((2.7,3.99)--(0,3)--(0,0));
draw((3.7,3.99)--(1,3)--(1,0));
draw((4.7,3.99)--(2,3)--(2,0));
draw((5.7,3.99)--(3,3)--(3,0));
draw((0,0)--(3,0)--(5.7,0.99));
draw((0,1)--(3,1... | 54 | 1. **Calculate the original surface area of the cube**:
The original cube has dimensions $3\text{ cm} \times 3\text{ cm} \times 3\text{ cm}$. Each face of the cube is a square with an area of $3^2 = 9\text{ cm}^2$. Since a cube has 6 faces, the total surface area of the cube is:
\[
6 \times 9 = 54\text{ cm}^2... | 0.5 | 6,560.25 | 5,416.125 | 7,704.375 |
Compute $\cos 72^\circ.$ | \frac{-1 + \sqrt{5}}{4} | 0 | 7,757.25 | -1 | 7,757.25 | |
In a box, there are two red balls, two yellow balls, and two blue balls. If a ball is randomly drawn from the box, at least how many balls need to be drawn to ensure getting balls of the same color? If one ball is drawn at a time without replacement until balls of the same color are obtained, let $X$ be the number of d... | \frac{11}{5} | 0.1875 | 7,348 | 8,078.666667 | 7,179.384615 | |
Ten points are equally spaced on a circle. A graph is a set of segments (possibly empty) drawn between pairs of points, so that every two points are joined by either zero or one segments. Two graphs are considered the same if we can obtain one from the other by rearranging the points. Let $N$ denote the number of graph... | 11716571 | The question asks for the number of isomorphism classes of connected graphs on 10 vertices. This is enumerated in http://oeis.org/A001349 the answer is 11716571. In fact, of the $2^{45} \approx 3.51 \cdot 10^{13} \approx 3 \cdot 10^{13}$ graphs on 10 labelled vertices, virtually all (about $3.45 \cdot 10^{13}$) are con... | 0 | 8,093.1875 | -1 | 8,093.1875 |
A sequence has 101 terms, each of which is a positive integer. If a term, $n$, is even, the next term is equal to $\frac{1}{2}n+1$. If a term, $n$, is odd, the next term is equal to $\frac{1}{2}(n+1)$. If the first term is 16, what is the 101st term? | 2 | The 1st term is 16. Since 16 is even, the 2nd term is $\frac{1}{2} \cdot 16+1=9$. Since 9 is odd, the 3rd term is $\frac{1}{2}(9+1)=5$. Since 5 is odd, the 4th term is $\frac{1}{2}(5+1)=3$. Since 3 is odd, the 5th term is $\frac{1}{2}(3+1)=2$. Since 2 is even, the 6th term is $\frac{1}{2} \cdot 2+1=2$. This previous st... | 1 | 3,022.375 | 3,022.375 | -1 |
Let $a, b, x,$ and $y$ be real numbers with $a>4$ and $b>1$ such that\[\frac{x^2}{a^2}+\frac{y^2}{a^2-16}=\frac{(x-20)^2}{b^2-1}+\frac{(y-11)^2}{b^2}=1.\]Find the least possible value of $a+b.$ | 23 | Denote $P = \left( x , y \right)$.
Because $\frac{x^2}{a^2}+\frac{y^2}{a^2-16} = 1$, $P$ is on an ellipse whose center is $\left( 0 , 0 \right)$ and foci are $\left( - 4 , 0 \right)$ and $\left( 4 , 0 \right)$.
Hence, the sum of distance from $P$ to $\left( - 4 , 0 \right)$ and $\left( 4 , 0 \right)$ is equal to twic... | 0 | 8,123.625 | -1 | 8,123.625 |
In the equation $\frac{1}{j} + \frac{1}{k} = \frac{1}{4}$, both $j$ and $k$ are positive integers. What is the sum of all possible values for $j+k$? | 59 | 0.875 | 3,106.3125 | 2,799.642857 | 5,253 | |
Chen, Ruan, Lu, Tao, and Yang did push-ups. It is known that Chen, Lu, and Yang together averaged 40 push-ups per person, Ruan, Tao, and Chen together averaged 28 push-ups per person, and Ruan, Lu, Tao, and Yang together averaged 33 push-ups per person. How many push-ups did Chen do? | 36 | 0.8125 | 2,486.25 | 1,773.692308 | 5,574 | |
A point $P$ is chosen at random in the interior of a unit square $S$. Let $d(P)$ denote the distance from $P$ to the closest side of $S$. The probability that $\frac{1}{5}\le d(P)\le\frac{1}{3}$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 281 | First, let's figure out $d(P) \geq \frac{1}{3}$ which is\[\left(\frac{3}{5}\right)^2=\frac{9}{25}.\]Then, $d(P) \geq \frac{1}{5}$ is a square inside $d(P) \geq \frac{1}{3}$, so\[\left(\frac{1}{3}\right)^2=\frac{1}{9}.\]Therefore, the probability that $\frac{1}{5}\le d(P)\le\frac{1}{3}$ is\[\frac{9}{25}-\frac{1}{9}=\fra... | 0.25 | 7,917.375 | 7,093.5 | 8,192 |
The decimal representation of $m/n,$ where $m$ and $n$ are relatively prime positive integers and $m < n,$ contains the digits $2, 5$, and $1$ consecutively, and in that order. Find the smallest value of $n$ for which this is possible. | 127 | To find the smallest value of $n$, we consider when the first three digits after the decimal point are $0.251\ldots$.
Otherwise, suppose the number is in the form of $\frac{m}{n} = 0.X251 \ldots$, where $X$ is a string of $k$ digits and $n$ is small as possible. Then $10^k \cdot \frac{m}{n} - X = \frac{10^k m - nX}{n}... | 0 | 8,192 | -1 | 8,192 |
In $\triangle PQR, \angle RPQ=90^{\circ}$ and $S$ is on $PQ$. If $SQ=14, SP=18$, and $SR=30$, what is the area of $\triangle QRS$? | 168 | Since $\triangle RPS$ is right-angled at $P$, then by the Pythagorean Theorem, $PR^{2}+PS^{2}=RS^{2}$ or $PR^{2}+18^{2}=30^{2}$. This gives $PR^{2}=900-324=576$, from which $PR=24$. Since $P, S$ and $Q$ lie on a straight line and $RP$ is perpendicular to this line, then $RP$ is actually a height for $\triangle QRS$ cor... | 0.9375 | 3,652.1875 | 3,349.533333 | 8,192 |
Inside triangle \(ABC\), a random point \(M\) is chosen. What is the probability that the area of one of the triangles \(ABM\), \(BCM\), or \(CAM\) is greater than the sum of the areas of the other two? | 0.75 | 0 | 7,626.6875 | -1 | 7,626.6875 | |
Calculate the definite integral:
$$
\int_{0}^{\pi} 2^{4} \cdot \cos ^{8} x \, dx
$$ | \frac{35 \pi}{8} | 0.5 | 6,511.0625 | 5,613.25 | 7,408.875 | |
Let $1$; $5$; $9$; $\ldots$ and $8$; $15$; $22$; $\ldots$ be two arithmetic progressions. The set $S$ is the union of the first $2100$ terms of each sequence. How many distinct numbers are in $S$?
A) 3800
B) 3900
C) 4000
D) 4100
E) 4200 | 3900 | 0 | 7,773.5625 | -1 | 7,773.5625 | |
The coordinates of vertex \( C(x, y) \) of triangle \( \triangle ABC \) satisfy the inequalities \( x^{2}+y^{2} \leq 8+2y \) and \( y \geq 3 \). The side \( AB \) is on the x-axis. Given that the distances from point \( Q(0,1) \) to the lines \( AC \) and \( BC \) are both 1, find the minimum area of \( \triangle ABC \... | 6 \sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
In a 6 by 6 grid of points, what fraction of the larger rectangle's area is inside the shaded right triangle? The vertices of the shaded triangle correspond to grid points and are located at (1,1), (1,5), and (4,1). | \frac{1}{6} | 0.0625 | 5,320.5 | 8,054 | 5,138.266667 | |
The maximum value of the function $y=\sin x \cos x + \sin x + \cos x$ is __________. | \frac{1}{2} + \sqrt{2} | 0.75 | 7,164.6875 | 7,079.833333 | 7,419.25 | |
The circle centered at point $A$ with radius $19$ and the circle centered at point $B$ with radius $32$ are both internally tangent to a circle centered at point $C$ with radius $100$ such that point $C$ lies on segment $\overline{AB}$ . Point $M$ is on the circle centered at $A$ and point $N$ is o... | 140 | 0.1875 | 7,357.875 | 4,242 | 8,076.923077 | |
What is the three-digit (integer) number which, when either increased or decreased by the sum of its digits, results in a number with all identical digits? | 105 | 0 | 8,192 | -1 | 8,192 | |
If \( a^3 + b^3 + c^3 = 3abc = 6 \) and \( a^2 + b^2 + c^2 = 8 \), find the value of \( \frac{ab}{a+b} + \frac{bc}{b+c} + \frac{ca}{c+a} \). | -8 | 0.3125 | 7,626.3125 | 6,381.8 | 8,192 | |
Six bags contain 18, 19, 21, 23, 25, and 34 marbles, respectively. One of the bags contains marbles with cracks, while the remaining five bags contain marbles without cracks. Jenny took three of the bags, and George took two of the other bags, leaving the bag with the cracked marbles. If the number of marbles Jenny rec... | 23 | 0.875 | 4,234.625 | 3,669.285714 | 8,192 | |
In $\triangle ABC$, $AB = 86$, and $AC = 97$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover $\overline{BX}$ and $\overline{CX}$ have integer lengths. What is $BC$? | 61 |
#### Step 1: Understanding the Problem
We are given a triangle $\triangle ABC$ with $AB = 86$ and $AC = 97$. A circle centered at $A$ with radius $AB$ intersects line segment $BC$ at points $B$ and $X$. We need to find the length of $BC$, denoted as $BX + CX$, where both $BX$ and $CX$ are integers.
#### Step 2: Apply... | 0.6875 | 6,145.3125 | 5,786.454545 | 6,934.8 |
Given an arithmetic sequence $\{a_n\}$ with a common difference of $2$ and an even number of terms, the sum of all odd terms is $15$, and the sum of all even terms is $25$, calculate the number of terms in this sequence. | 10 | 0.625 | 5,758.8125 | 4,298.9 | 8,192 | |
Consider the geometric sequence $5$, $\dfrac{15}{4}$, $\dfrac{45}{16}$, $\dfrac{135}{64}$, $\ldots$. Find the tenth term of the sequence. Express your answer as a common fraction. | \frac{98415}{262144} | 0.8125 | 4,562.875 | 3,738.615385 | 8,134.666667 | |
A $3 \times 3$ table starts with every entry equal to 0 and is modified using the following steps: (i) adding 1 to all three numbers in any row; (ii) adding 2 to all three numbers in any column. After step (i) has been used a total of $a$ times and step (ii) has been used a total of $b$ times, the table appears as \beg... | 11 | Since the second column includes the number 1, then step (ii) was never used on the second column, otherwise each entry would be at least 2 . To generate the 1,3 and 2 in the second column, we thus need to have used step (i) 1 time on row 1,3 times on row 2 , and 2 times on row 3 . This gives: \begin{tabular}{|l|l|l|} ... | 0.75 | 5,949.1875 | 5,344.75 | 7,762.5 |
Given that $3\sin \alpha + 4\cos \alpha = 5$.
(1) Find the value of $\tan \alpha$;
(2) Find the value of $\cot (\frac{3\pi}{2} - \alpha) \cdot \sin^2 (\frac{3\pi}{2} + \alpha)$. | \frac{12}{25} | 0.9375 | 4,818.375 | 4,750.133333 | 5,842 | |
What integer value will satisfy the equation $$ 14^2 \times 35^2 = 10^2 \times (M - 10)^2 \ ? $$ | 59 | 0.375 | 4,691.6875 | 3,248.166667 | 5,557.8 | |
If $1998$ is written as a product of two positive integers whose difference is as small as possible, then the difference is | 17 | 1. **Objective**: Find two positive integers whose product is $1998$ and whose difference is minimized.
2. **Calculate the approximate square root of $1998$**:
\[
\sqrt{1998} \approx \sqrt{2000} = \sqrt{4 \times 500} = 2 \times \sqrt{500} \approx 2 \times 22.36 \approx 44.72
\]
Thus, the integers should be... | 1 | 4,386.6875 | 4,386.6875 | -1 |
The bases \( AB \) and \( CD \) of the trapezoid \( ABCD \) are 155 and 13 respectively, and its diagonals are mutually perpendicular. Find the dot product of the vectors \( \overrightarrow{AD} \) and \( \overrightarrow{BC} \). | 2015 | 0.625 | 5,568.5 | 4,451.9 | 7,429.5 | |
One line is described by
\[\begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix} + t \begin{pmatrix} 1 \\ 1 \\ -k \end{pmatrix}.\]Another line is described by
\[\begin{pmatrix} 1 \\ 4 \\ 5 \end{pmatrix} + u \begin{pmatrix} k \\ 2 \\ 1 \end{pmatrix}.\]If the lines are coplanar (i.e. there is a plane that contains both lines), then ... | 0,-3 | 0 | 5,319.125 | -1 | 5,319.125 | |
Let $A B C D$ be a convex quadrilateral such that $\angle A B D=\angle B C D=90^{\circ}$, and let $M$ be the midpoint of segment $B D$. Suppose that $C M=2$ and $A M=3$. Compute $A D$. | \sqrt{21} | Since triangle $B C D$ is a right triangle, we have $C M=B M=D M=2$. With $A M=3$ and $\angle A B M=90^{\circ}$, we get $A B=\sqrt{5}$. Now $$A D^{2}=A B^{2}+B D^{2}=5+16=21$$ so $A D=\sqrt{21}$. | 0.4375 | 7,356.8125 | 6,283 | 8,192 |
Consider the expanded hexagonal lattice shown below, where each point is one unit from its nearest neighbor. Determine the number of equilateral triangles whose vertices lie on this lattice.
```asy
size(100);
dot(origin);
dot(dir(30) + dir(90));
dot(dir(90));
dot(dir(90) + dir(150));
dot(dir(150));
dot(dir(150) + dir(2... | 28 | 0 | 8,192 | -1 | 8,192 | |
For any real numbers $x,y$ that satisfies the equation $$ x+y-xy=155 $$ and $$ x^2+y^2=325 $$ , Find $|x^3-y^3|$ | 4375 | 0.1875 | 7,875.125 | 6,502 | 8,192 | |
Let $a,b,c,d$ be real numbers such that $a^2+b^2+c^2+d^2=1$. Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$ and determine all values of $(a,b,c,d)$ such that the minimum value is achived. | -\frac{1}{8} |
Let \(a, b, c, d\) be real numbers such that \(a^2 + b^2 + c^2 + d^2 = 1\). We want to determine the minimum value of the expression \((a-b)(b-c)(c-d)(d-a)\).
To find the minimum value of \((a-b)(b-c)(c-d)(d-a)\), we first recognize the symmetry and potential simplifications. The key is to find a particular symmetric... | 0 | 8,192 | -1 | 8,192 |
A boss schedules a meeting at a cafe with two of his staff, planning to arrive randomly between 1:00 PM and 4:00 PM. Each staff member also arrives randomly within the same timeframe. If the boss arrives and any staff member isn't there, he leaves immediately. Each staff member will wait for up to 90 minutes for the ot... | \frac{1}{4} | 0 | 7,453.0625 | -1 | 7,453.0625 | |
In a certain college containing 1000 students, students may choose to major in exactly one of math, computer science, finance, or English. The diversity ratio $d(s)$ of a student $s$ is the defined as number of students in a different major from $s$ divided by the number of students in the same major as $s$ (including ... | \{0,1000,2000,3000\} | It is easy to check that if $n$ majors are present, the diversity is $1000(n-1)$. Therefore, taking $n=1,2,3,4$ gives us all possible answers. | 0 | 7,567.75 | -1 | 7,567.75 |
During the military training for new freshman students, after two days of shooting practice, student A can hit the target 9 times out of 10 shots, and student B can hit the target 8 times out of 9 shots. A and B each take a shot at the same target (their shooting attempts do not affect each other). Determine the probab... | \frac{89}{90} | 1 | 2,331.25 | 2,331.25 | -1 | |
Let $A(2,0)$ be a fixed point in the plane, and let $P\left(\sin \left(2 t-60^{\circ}\right), \cos \left(2 t-60^{\circ}\right)\right)$ be a moving point. Find the area swept by the line segment $AP$ as $t$ changes from $15^{\circ}$ to $45^{\circ}$. | \frac{\pi}{6} | 0.125 | 7,943.4375 | 7,310.5 | 8,033.857143 | |
At the signal of the trainer, two ponies simultaneously started running uniformly along the outer circumference of the circus arena in opposite directions. The first pony ran slightly faster than the second and, by the time they met, had run 5 meters more than the second pony. Continuing to run, the first pony reached ... | 6.25 | 0 | 7,229.375 | -1 | 7,229.375 | |
The instructor of a summer math camp brought several shirts, several pairs of trousers, several pairs of shoes, and two jackets for the entire summer. In each lesson, he wore trousers, a shirt, and shoes, and he wore a jacket only on some lessons. On any two lessons, at least one piece of his clothing or shoes was diff... | 126 | 0.1875 | 7,454.25 | 4,257.333333 | 8,192 | |
Thirty clever students from 6th, 7th, 8th, 9th, and 10th grades were tasked with creating forty problems for an olympiad. Any two students from the same grade came up with the same number of problems, while any two students from different grades came up with a different number of problems. How many students came up wit... | 26 | 0.1875 | 8,085.8125 | 7,625.666667 | 8,192 | |
How many numbers of the form $\overline{a b c d a b c d}$ are divisible by 18769? | 65 | 0.6875 | 5,452.5 | 4,280.272727 | 8,031.4 | |
Find the sum of digits of all the numbers in the sequence $1,2,3,4,\cdots ,10000$. | 180001 |
To find the sum of the digits of all the numbers in the sequence $1, 2, 3, \ldots, 10000$, we can break down the problem into manageable parts based on the number of digits in the numbers.
#### Step 1: Sum of digits for numbers from 1 to 9
Each number from 1 to 9 is a single-digit number. The sum of these digits is s... | 0.0625 | 8,067.875 | 6,206 | 8,192 |
Given that $x$ and $y$ are distinct nonzero real numbers such that $x - \tfrac{2}{x} = y - \tfrac{2}{y}$, determine the product $xy$. | -2 | 1 | 1,850.6875 | 1,850.6875 | -1 | |
The eighth grade class at Lincoln Middle School has 93 students. Each student takes a math class or a foreign language class or both. There are 70 eighth graders taking a math class, and there are 54 eighth graders taking a foreign language class. How many eighth graders take only a math class and not a foreign languag... | 39 | 1. **Identify the total number of students and the students in each category**:
- Total number of students = 93
- Students taking math = 70
- Students taking foreign language = 54
2. **Use the principle of inclusion-exclusion**:
- The principle states that for any two sets, the size of their union is give... | 1 | 1,180.375 | 1,180.375 | -1 |
Maria baked 60 cakes, of which one-third contained strawberries, half contained blueberries, three-fifths contained raspberries, and one-tenth contained coconut flakes. What is the largest possible number of cakes that had none of these ingredients? | 24 | 0.125 | 7,111.75 | 6,883 | 7,144.428571 | |
The base of the quadrilateral prism \( A B C D A_{1} B_{1} C_{1} D_{1} \) is a rhombus \( A B C D \) with \( B D = 12 \) and \( \angle B A C = 60^{\circ} \). A sphere passes through the vertices \( D, A, B, B_{1}, C_{1}, D_{1} \).
a) Find the area of the circle obtained in the cross section of the sphere by the plane ... | 192\sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is $18$ . How many passcodes satisfy these conditions? | 36 | 0.125 | 7,228.6875 | 7,104.5 | 7,246.428571 | |
In a debate competition with 4 participants, the rules are as follows: each participant must choose one topic from two options, A and B. For topic A, answering correctly earns 100 points, and answering incorrectly results in a loss of 100 points. For topic B, answering correctly earns 90 points, and answering incorrect... | 36 | 0.1875 | 7,746.625 | 5,816.666667 | 8,192 | |
A ball inside a rectangular container of width 7 and height 12 is launched from the lower-left vertex of the container. It first strikes the right side of the container after traveling a distance of $\sqrt{53}$ (and strikes no other sides between its launch and its impact with the right side). How many times does the b... | 5 | Every segment the ball traverses between bounces takes it 7 units horizontally and 2 units up. Thus, after 5 bounces it has traveled up 10 units, and the final segment traversed takes it directly to the upper right vertex of the rectangle. | 0 | 6,924.6875 | -1 | 6,924.6875 |
Compute $\tan 225^\circ$. | 1 | 1 | 2,468.8125 | 2,468.8125 | -1 |
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