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 |
|---|---|---|---|---|---|---|
The last 5 digits of $99 \times 10101 \times 111 \times 1001001$ are _____. | 88889 | 0.125 | 8,142.25 | 7,794 | 8,192 | |
Let \( P \) be a regular polygon with 2006 sides. A diagonal of \( P \) is called good if its endpoints divide the perimeter of \( P \) into two parts, each having an odd number of sides of \( P \). The sides of \( P \) are also called good. Suppose \( P \) has been subdivided into triangles by 2003 diagonals that do n... | 1003 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Cory has $3$ apples, $2$ oranges and $2$ bananas. If Cory eats one piece of his fruit per day for a week and the pieces of fruit within each category are indistinguishable, in how many orders can Cory eat the fruit? One such order is $AAAOOBB.$ | 210 | 1 | 1,523.625 | 1,523.625 | -1 | |
The American Mathematics College is holding its orientation for incoming freshmen. The incoming freshman class contains fewer than $500$ people. When the freshmen are told to line up in columns of $23$, $22$ people are in the last column. When the freshmen are told to line up in columns of $21$, $14$ people are in the ... | 413 | 1 | 3,085.1875 | 3,085.1875 | -1 | |
Consider the sum $$ S =\sum^{2021}_{j=1} \left|\sin \frac{2\pi j}{2021}\right|. $$ The value of $S$ can be written as $\tan \left( \frac{c\pi}{d} \right)$ for some relatively prime positive integers $c, d$ , satisfying $2c < d$ . Find the value of $c + d$ . | 3031 | 0.3125 | 7,589.75 | 6,264.8 | 8,192 | |
What multiple of 15 is closest to 3500? | 3510 | 0 | 2,561.75 | -1 | 2,561.75 | |
Cut a 12cm long thin iron wire into three segments with lengths a, b, and c,
(1) Find the maximum volume of the rectangular solid with lengths a, b, and c as its dimensions;
(2) If these three segments each form an equilateral triangle, find the minimum sum of the areas of these three equilateral triangles. | \frac {4 \sqrt {3}}{3} | 0 | 4,064.9375 | -1 | 4,064.9375 | |
Express $\frac{214_8}{32_5} + \frac{343_9}{133_4}$ in base 10. | \frac{9134}{527} | 0.1875 | 8,150.9375 | 7,973 | 8,192 | |
The vectors $\mathbf{a} = \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 0 \\ 2 \\ -1 \end{pmatrix}.$ There exist scalars $p,$ $q,$ and $r$ such that
\[\begin{pmatrix} 4 \\ 1 \\ -4 \end{pmatrix} = p \mathbf{a} + q \mathbf{b} + r (\mathbf{a} \times \mathbf{b}).\]Find $r.$ | -\frac{1}{6} | 0.875 | 3,415.875 | 3,421.928571 | 3,373.5 | |
What is the smallest prime factor of 1821? | 3 | 1 | 1,728.9375 | 1,728.9375 | -1 | |
In $\triangle ABC$, $BC= \sqrt {5}$, $AC=3$, $\sin C=2\sin A$. Find:
1. The value of $AB$.
2. The value of $\sin(A- \frac {\pi}{4})$. | - \frac { \sqrt {10}}{10} | 0 | 5,570.4375 | -1 | 5,570.4375 | |
An ordered pair $(a, b)$ of positive integers is called spicy if $\operatorname{gcd}(a+b, ab+1)=1$. Compute the probability that both $(99, n)$ and $(101, n)$ are spicy when $n$ is chosen from $\{1,2, \ldots, 2024\}$ uniformly at random. | \frac{96}{595} | We claim that $(a, b)$ is spicy if and only if both $\operatorname{gcd}(a+1, b-1)=1$ and $\operatorname{gcd}(a-1, b+1)=1$. To prove the claim, we note that $$\operatorname{gcd}(a+b, ab+1)=\operatorname{gcd}(a+b, b(-b)+1)=\operatorname{gcd}(a+b, b^{2}-1)$$ Hence, we have $$\begin{aligned} \operatorname{gcd}(a+b, ab+1)=1... | 0 | 8,192 | -1 | 8,192 |
Let $n$ be a $5$-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by $100$. For how many values of $n$ is $q+r$ divisible by $11$? | 9000 | 1. **Understanding the division of $n$ by $100$:**
When a $5$-digit number $n$ is divided by $100$, the quotient $q$ is formed by the first three digits of $n$, and the remainder $r$ is formed by the last two digits. Thus, $n = 100q + r$.
2. **Range of $q$ and $r$:**
- $q$ can range from $100$ to $999$, as $q$ r... | 0 | 6,865.1875 | -1 | 6,865.1875 |
A sequence is defined over non-negative integral indexes in the following way: $a_{0}=a_{1}=3$, $a_{n+1}a_{n-1}=a_{n}^{2}+2007$.
Find the greatest integer that does not exceed $\frac{a_{2006}^{2}+a_{2007}^{2}}{a_{2006}a_{2007}}.$ | 224 | We are given that
$a_{n+1}a_{n-1}= a_{n}^{2}+2007$, $a_{n-1}^{2}+2007 = a_{n}a_{n-2}$.
Add these two equations to get
$a_{n-1}(a_{n-1}+a_{n+1}) = a_{n}(a_{n}+a_{n-2})$
$\frac{a_{n+1}+a_{n-1}}{a_{n}}= \frac{a_{n}+a_{n-2}}{a_{n-1}}$.
This is an invariant. Defining $b_{i}= \frac{a_{i}}{a_{i-1}}$ for each $i \ge 2$, the ... | 0 | 8,143.6875 | -1 | 8,143.6875 |
Last Thursday, each of the students in M. Fermat's class brought one piece of fruit to school. Each brought an apple, a banana, or an orange. In total, $20\%$ of the students brought an apple and $35\%$ brought a banana. If 9 students brought oranges, how many students were in the class? | 20 | Each student brought exactly one of an apple, a banana, and an orange. Since $20\%$ of the students brought an apple and $35\%$ brought a banana, then the percentage of students who brought an orange is $100\% - 20\% - 35\% = 45\%$. Therefore, the 9 students who brought an orange represent $45\%$ of the class. This mea... | 1 | 1,681.5 | 1,681.5 | -1 |
How many multiples of 7 between $10^{6}$ and $10^{9}$ are perfect squares? | 4375 | $\left[\sqrt{\frac{10^{9}}{7^{2}}}\right]-\left[\sqrt{\frac{10^{6}}{7^{2}}}\right]=4517-142=4375$. | 0.5 | 5,977.625 | 4,600.5 | 7,354.75 |
Solve the equation $|y-6| + 2y = 9$ for $y$. | 3 | 1 | 1,989 | 1,989 | -1 | |
In triangle $PQR$, we have $\angle P = 90^\circ$, $QR = 20$, and $\tan R = 4\sin R$. What is $PR$? | 5 | 0.875 | 2,211.8125 | 1,857.071429 | 4,695 | |
A product of five primes is of the form $A B C, A B C$, where $A, B$, and $C$ represent digits. If one of the primes is 491, find the product $A B C, A B C$. | 982,982 | $491 \cdot 1001 \cdot 2=982,982$. | 0 | 8,192 | -1 | 8,192 |
Let $ABC$ be a triangle where $M$ is the midpoint of $\overline{AC}$, and $\overline{CN}$ is the angle bisector of $\angle{ACB}$ with $N$ on $\overline{AB}$. Let $X$ be the intersection of the median $\overline{BM}$ and the bisector $\overline{CN}$. In addition $\triangle BXN$ is equilateral with $AC=2$. What is $BX^2$... | \frac{10-6\sqrt{2}}{7} | 1. **Dilation of the Triangle**: We dilate triangle $ABC$ such that the sides of equilateral triangle $BXN$ are all equal to $2$. This simplification is done to ease the calculations. We aim to find the length of segment $AM$ so that we can un-dilate triangle $ABC$ by dividing each of its sides by $AM$. This will make ... | 0 | 8,192 | -1 | 8,192 |
Four students, named A, B, C, and D, are divided into two volunteer groups to participate in two off-campus activities. The probability that students B and C participate in the same activity is ________. | \frac{1}{3} | 0.25 | 7,581.75 | 6,203.5 | 8,041.166667 | |
If $\frac{60}{2^3\cdot5^8}$ is expressed as a decimal, how many non-zero digits are to the right of the decimal point? | 3 | 0.5 | 6,582.4375 | 4,972.875 | 8,192 | |
A rectangular prism measuring 20 cm by 14 cm by 12 cm has a small cube of 4 cm on each side removed from each corner. What percent of the original volume is removed? | 15.24\% | 0.75 | 5,996.375 | 5,264.5 | 8,192 | |
Alice chose five positive integers and found that their product was even. What is the maximum number of odd integers she could have chosen? | 4 | 1 | 974.75 | 974.75 | -1 | |
Suppose in a right triangle LMN, where angle M is the right angle, $\cos N = \frac{4}{5}$ and length LM is given by 20 units. What is the length of LN? | 25 | 0.125 | 2,760.5 | 3,171 | 2,701.857143 | |
What is the measure, in degrees, of the acute angle formed by the minute hand and the hour hand on a standard clock when it indicates $9$:$40$? | 50 | 1 | 3,671.0625 | 3,671.0625 | -1 | |
Segments $\overline{AB}, \overline{AC},$ and $\overline{AD}$ are edges of a cube and $\overline{AG}$ is a diagonal through the center of the cube. Point $P$ satisfies $BP=60\sqrt{10}$, $CP=60\sqrt{5}$, $DP=120\sqrt{2}$, and $GP=36\sqrt{7}$. Find $AP.$ | 192 | Let $E$ be the vertex of the cube such that $ABED$ is a square. By the British Flag Theorem, we can easily we can show that \[PA^2 + PE^2 = PB^2 + PD^2\] and \[PA^2 + PG^2 = PC^2 + PE^2\] Hence, adding the two equations together, we get $2PA^2 + PG^2 = PB^2 + PC^2 + PD^2$. Substituting in the values we know, we get $2P... | 0.0625 | 8,049.125 | 5,906 | 8,192 |
Six congruent circles form a ring with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle $C$ with radius 30. Let $K$ be the area of the region inside circle $C$ and outside of the six circles in the ring. Find $\lfloor K \rfloor$ (the floor function). | 942 | Define the radii of the six congruent circles as $r$. If we draw all of the radii to the points of external tangency, we get a regular hexagon. If we connect the vertices of the hexagon to the center of the circle $C$, we form several equilateral triangles. The length of each side of the triangle is $2r$. Notice that t... | 1 | 3,614.875 | 3,614.875 | -1 |
For any sequence of real numbers $A=(a_1,a_2,a_3,\ldots)$, define $\Delta A^{}_{}$ to be the sequence $(a_2-a_1,a_3-a_2,a_4-a_3,\ldots)$, whose $n^{\mbox{th}}_{}$ term is $a_{n+1}-a_n^{}$. Suppose that all of the terms of the sequence $\Delta(\Delta A^{}_{})$ are $1^{}_{}$, and that $a_{19}=a_{92}^{}=0$. Find $a_1^{}$. | 819 | Note that the $\Delta$s are reminiscent of differentiation; from the condition $\Delta(\Delta{A}) = 1$, we are led to consider the differential equation \[\frac{d^2 A}{dn^2} = 1\] This inspires us to guess a quadratic with leading coefficient 1/2 as the solution; \[a_{n} = \frac{1}{2}(n-19)(n-92)\] as we must have root... | 0.875 | 4,723.25 | 4,625.5 | 5,407.5 |
In the following diagram, \(\angle ACB = 90^\circ\), \(DE \perp BC\), \(BE = AC\), \(BD = \frac{1}{2} \mathrm{~cm}\), and \(DE + BC = 1 \mathrm{~cm}\). Suppose \(\angle ABC = x^\circ\). Find the value of \(x\). | 30 | 0.125 | 7,581.9375 | 3,311.5 | 8,192 | |
Trapezoid $ABCD^{}_{}$ has sides $AB=92^{}_{}$, $BC=50^{}_{}$, $CD=19^{}_{}$, and $AD=70^{}_{}$, with $AB^{}_{}$ parallel to $CD^{}_{}$. A circle with center $P^{}_{}$ on $AB^{}_{}$ is drawn tangent to $BC^{}_{}$ and $AD^{}_{}$. Given that $AP^{}_{}=\frac mn$, where $m^{}_{}$ and $n^{}_{}$ are relatively prime positive... | 164 | From $(1)$ above, $x = \frac{70r}{h}$ and $92-x = \frac{50r}{h}$. Adding these equations yields $92 = \frac{120r}{h}$. Thus, $x = \frac{70r}{h} = \frac{7}{12}\cdot\frac{120r}{h} = \frac{7}{12}\cdot92 = \frac{161}{3}$, and $m+n = \boxed{164}$.
We can use $(1)$ from Solution 1 to find that $h/r = 70/x$ and $h/r = 50/ (9... | 0.125 | 8,075.125 | 8,192 | 8,058.428571 |
For how many ordered triples $(a, b, c)$ of positive integers are the equations $abc+9=ab+bc+ca$ and $a+b+c=10$ satisfied? | 21 | Subtracting the first equation from the second, we obtain $1-a-b-c+ab+bc+ca-abc=(1-a)(1-b)(1-c)=0$. Since $a, b$, and $c$ are positive integers, at least one must equal 1. Note that $a=b=c=1$ is not a valid triple, so it suffices to consider the cases where exactly two or one of $a, b, c$ are equal to 1. If $a=b=1$, we... | 0.1875 | 7,916.5625 | 6,723 | 8,192 |
Using five distinct digits, $1$, $4$, $5$, $8$, and $9$, determine the $51\text{st}$ number in the sequence when arranged in ascending order.
A) $51489$
B) $51498$
C) $51849$
D) $51948$ | 51849 | 0 | 6,433.75 | -1 | 6,433.75 | |
Let $N$ be the largest integer whose square has exactly $3$ digits when written in base 9.
What is $N$, expressed in base 9? | 28_9 | 0.125 | 3,359.3125 | 2,143.5 | 3,533 | |
When $1 + 9 + 9^2 + \cdots + 9^{2023}$ is divided by $500$, a remainder of $M$ is obtained. Determine the value of $M$. | 45 | 0 | 7,894.0625 | -1 | 7,894.0625 | |
Compute: $98 \times 102$. | 9996 | 0.875 | 399.125 | 407.285714 | 342 | |
Given vectors $\overrightarrow{a}=(1,-1)$ and $\overrightarrow{b}=(2,-1)$, the projection of $\overrightarrow{a}+\overrightarrow{b}$ in the direction of $\overrightarrow{a}$ is ______. | \frac{5\sqrt{2}}{2} | 0 | 4,335.3125 | -1 | 4,335.3125 | |
The diagram below shows a $4\times4$ rectangular array of points, each of which is $1$ unit away from its nearest neighbors.
[asy] unitsize(0.25inch); defaultpen(linewidth(0.7)); int i, j; for(i = 0; i < 4; ++i) for(j = 0; j < 4; ++j) dot(((real)i, (real)j)); [/asy]
Define a growing path to be a sequence of distinct p... | 240 | We label our points using coordinates $0 \le x,y \le 3$, with the bottom-left point being $(0,0)$. By the Pythagorean Theorem, the distance between two points is $\sqrt{d_x^2 + d_y^2}$ where $0 \le d_x, d_y \le 3$; these yield the possible distances (in decreasing order) \[\sqrt{18},\ \sqrt{13},\ \sqrt{10},\ \sqrt{9},\... | 0 | 8,073.5 | -1 | 8,073.5 |
Given a cone and a cylinder made of rubber, the cone has a base radius of $5$ and a height of $4$, while the cylinder has a base radius of $2$ and a height of $8$. If they are remade into a new cone and a new cylinder with the same base radius, while keeping the total volume and height unchanged, find the new base radi... | r = \sqrt{7} | 0.25 | 6,052.9375 | 3,564.75 | 6,882.333333 | |
Find the number of real solutions to the equation
\[
\frac{1}{x - 1} + \frac{2}{x - 2} + \frac{3}{x - 3} + \dots + \frac{10}{x - 10} = 2x.
\] | 11 | 0 | 7,483.3125 | -1 | 7,483.3125 | |
Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible? | 62 | 0.9375 | 3,514.3125 | 3,347.2 | 6,021 | |
Given that the function $f(x)$ satisfies $f(x+y)=f(x)+f(y)$ for any $x, y \in \mathbb{R}$, and $f(x) < 0$ when $x > 0$, with $f(1)=-2$.
1. Determine the parity (odd or even) of the function $f(x)$.
2. When $x \in [-3, 3]$, does the function $f(x)$ have an extreme value (maximum or minimum)? If so, find the extreme valu... | -6 | 1 | 3,553.75 | 3,553.75 | -1 | |
In triangle $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to angles $A$, $B$, and $C$, respectively. Given that $\frac{{c\sin C}}{{\sin A}} - c = \frac{{b\sin B}}{{\sin A}} - a$ and $b = 2$, find:
$(1)$ The measure of angle $B$;
$(2)$ If $a = \frac{{2\sqrt{6}}}{3}$, find the area of tria... | 1 + \frac{\sqrt{3}}{3} | 0 | 7,940.1875 | -1 | 7,940.1875 | |
The coordinates of the 3 vertices of triangle are \( P(-8, 5) \), \( Q(-15, -19) \), and \( R(1, -7) \). Find the equation of the angle bisector of \(\angle P\) and express it in the form \(ax + by + c = 0\), where \(a, b, c \in \mathbf{Z}^{+}\) and \((a, b, c)=1\), and then calculate \(a + c\). | 89 | 0.75 | 5,457.75 | 4,546.333333 | 8,192 | |
Leah and Jackson run for 45 minutes on a circular track. Leah runs clockwise at 200 m/min in a lane with a radius of 40 meters, while Jackson runs counterclockwise at 280 m/min in a lane with a radius of 55 meters, starting on the same radial line as Leah. Calculate how many times they pass each other after the start. | 72 | 0.5625 | 6,212.375 | 5,143.222222 | 7,587 | |
When the square of four times a positive integer is decreased by twice the integer, the result is $8066$. What is the integer? | 182 | 0 | 8,192 | -1 | 8,192 | |
In a cabinet, there are 3 pairs of different shoes. If 2 shoes are randomly taken out, let event A denote "the taken out shoes do not form a pair"; event B denote "both taken out shoes are for the same foot"; event C denote "one shoe is for the left foot and the other is for the right foot, but they do not form a pair"... | \dfrac{2}{5} | 0.25 | 7,181.9375 | 5,153.5 | 7,858.083333 | |
Let $u,$ $v,$ and $w$ be the roots of the equation $x^3 - 18x^2 + 20x - 8 = 0.$ Find the value of $(2+u)(2+v)(2+w).$ | 128 | 0.875 | 3,768.375 | 3,136.428571 | 8,192 | |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2$, $\overrightarrow{b}=(4\cos \alpha,-4\sin \alpha)$, and $\overrightarrow{a}\perp (\overrightarrow{a}- \overrightarrow{b})$, let the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ be $\theta$, then $\theta$ equals \... | \dfrac {\pi}{3} | 0.3125 | 3,425 | 4,201.4 | 3,072.090909 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $(2c-a)\cos B=b\cos A$.
(1) Find angle $B$;
(2) If $b=6$ and $c=2a$, find the area of $\triangle ABC$. | 6 \sqrt{3} | 0.875 | 4,445.375 | 3,910.142857 | 8,192 | |
Find all real values of $x$ which satisfy
\[\frac{1}{x + 1} + \frac{6}{x + 5} \ge 1.\] | (-5,-2] \cup (-1,3] | 0.5 | 6,238.4375 | 5,181.125 | 7,295.75 | |
What is the value of $x + y$ if the sequence $3, ~9, ~15, \ldots, ~x, ~y, ~39$ is an arithmetic sequence? | 60 | 0.1875 | 5,329.625 | 3,123.333333 | 5,838.769231 | |
Given the function $f(x)=4\cos x\sin \left(x+ \dfrac{\pi}{6} \right)$.
$(1)$ Find the smallest positive period of $f(x)$;
$(2)$ Find the maximum and minimum values of $f(x)$ in the interval $\left[- \dfrac{\pi}{6}, \dfrac{\pi}{4} \right]$. | -1 | 0 | 4,395.5 | -1 | 4,395.5 | |
Let $x = (2 + \sqrt{2})^6,$ let $n = \lfloor x \rfloor,$ and let $f = x - n.$ Find
\[
x(1 - f).
\] | 64 | 0.75 | 5,731.5625 | 5,056.583333 | 7,756.5 | |
A cylinder is filled with gas at atmospheric pressure (103.3 kPa). Assuming the gas is ideal, determine the work (in joules) during the isothermal compression of the gas by a piston that has moved inside the cylinder by $h$ meters.
Hint: The equation of state for the gas is given by $\rho V=$ const, where $\rho$ is pr... | 900 | 0.4375 | 7,265.4375 | 7,087.142857 | 7,404.111111 | |
What is the sum of the real roots of the equation $4x^4-3x^2+7x-3=0$? | -1 | 0.8125 | 4,289.4375 | 3,700.538462 | 6,841.333333 | |
Please choose one of the following two sub-questions to answer. If multiple choices are made, the score will be based on the first chosen question.
$(①)$ The sum of the internal angles of a regular hexagon is $ $ degrees.
$(②)$ Xiaohua saw a building with a height of $(137)$ meters at its signboard. From the same hor... | 237 | 0.125 | 3,164.375 | 3,866.5 | 3,064.071429 | |
David has a unit triangular array of 10 points, 4 on each side. A looping path is a sequence $A_{1}, A_{2}, \ldots, A_{10}$ containing each of the 10 points exactly once, such that $A_{i}$ and $A_{i+1}$ are adjacent (exactly 1 unit apart) for $i=1,2, \ldots, 10$. (Here $A_{11}=A_{1}$.) Find the number of looping paths ... | 60 | There are $10 \cdot 2$ times as many loop sequences as loops. To count the number of loops, first focus on the three corners of the array: their edges are uniquely determined. It's now easy to see there are 3 loops (they form " $V$-shapes"), so the answer is $10 \cdot 2 \cdot 3=60$. | 0 | 6,228.5625 | -1 | 6,228.5625 |
Two adjacent faces of a tetrahedron, each being equilateral triangles with side length 1, form a dihedral angle of 45 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing this edge. | \frac{\sqrt{3}}{4} | 0 | 8,192 | -1 | 8,192 | |
Determine the value of $b$, where $b$ is a positive number, such that the terms $10, b, \frac{10}{9}, \frac{10}{81}$ are the first four terms, respectively, of a geometric sequence. | 10 | 0 | 8,192 | -1 | 8,192 | |
A circle intersects the sides $AC$ and $CB$ of an isosceles triangle $ACB$ at points $P$ and $Q$ respectively, and is circumscribed around triangle $ABQ$. The segments $AQ$ and $BP$ intersect at point $D$ such that $AQ: AD = 4:3$. Find the area of triangle $DQB$ if the area of triangle $PQC$ is 3. | 9/2 | 0 | 8,171.9375 | -1 | 8,171.9375 | |
Mr. Thompson's students were asked to add two positive integers. Alex subtracted by mistake and got 4. Bella mistakenly multiplied and got 98. What was the correct answer? | 18 | 0 | 4,152.75 | -1 | 4,152.75 | |
Given that $\tan A=\frac{12}{5}$ , $\cos B=-\frac{3}{5}$ and that $A$ and $B$ are in the same quadrant, find the value of $\cos (A-B)$ . | $\frac{63}{65}$ | 0 | 2,845.0625 | -1 | 2,845.0625 | |
What is the radius of the circle inscribed in triangle $ABC$ if $AB = AC=7$ and $BC=6$? Express your answer in simplest radical form. | \frac{3\sqrt{10}}{5} | 0 | 2,827 | -1 | 2,827 | |
Calculate the definite integral:
$$
\int_{1}^{e} \sqrt{x} \cdot \ln^{2} x \, dx
$$ | \frac{10e\sqrt{e} - 16}{27} | 0 | 4,691.25 | -1 | 4,691.25 | |
Five students, labeled as A, B, C, D, and E, are standing in a row to participate in a literary performance. If A does not stand at either end, calculate the number of different arrangements where C and D are adjacent. | 24 | 0.125 | 8,023.8125 | 6,846.5 | 8,192 | |
Find the sum of all integer values of \( c \) for which the equation \( 15|p-1| + |3p - |p + c|| = 4 \) has at least one solution for \( p \). | -2 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Manya has a stack of $85=1+4+16+64$ blocks comprised of 4 layers (the $k$ th layer from the top has $4^{k-1}$ blocks). Each block rests on 4 smaller blocks, each with dimensions half those of the larger block. Laura removes blocks one at a time from this stack, removing only blocks that currently have no blocks on top ... | 3384 | Each time Laura removes a block, 4 additional blocks are exposed, increasing the total number of exposed blocks by 3 . She removes 5 blocks, for a total of $1 \cdot 4 \cdot 7 \cdot 10 \cdot 13$ ways. However, the stack originally only has 4 layers, so we must subtract the cases where removing a block on the bottom laye... | 0 | 8,192 | -1 | 8,192 |
A total of 120 is divided into four parts proportional to 3, 2, 4, and 5. What is the second largest part? | \frac{240}{7} | 0.1875 | 665.1875 | 837.333333 | 625.461538 | |
Compute the number of ordered pairs of integers $(x, y)$ such that $x^{2}+y^{2}<2019$ and $$x^{2}+\min (x, y)=y^{2}+\max (x, y)$$ | 127 | We have $$x^{2}-y^{2}=\max (x, y)-\min (x, y)=|x-y|$$ Now if $x \neq y$, we can divide by $x-y$ to obtain $x+y= \pm 1$. Thus $x=y$ or $x+y= \pm 1$. If $x=y$, we see that $2019>x^{2}+y^{2}=2 x^{2}$, so we see that $-31 \leq x \leq 31$. There are 63 ordered pairs in this case. In the second case, note that $|x| \geq|y|$ ... | 0 | 8,192 | -1 | 8,192 |
Let $p$, $q$, $r$, and $s$ be real numbers such that $|p-q|=1$, $|q-r|=5$, and $|r-s|=6$. What is the sum of all possible values of $|p-s|$? | 24 | 0.75 | 6,096.8125 | 5,398.416667 | 8,192 | |
Let $a, b, c, d$ be the four roots of $X^{4}-X^{3}-X^{2}-1$. Calculate $P(a)+P(b)+P(c)+P(d)$, where $P(X) = X^{6}-X^{5}-X^{4}-X^{3}-X$. | -2 | 0.1875 | 7,451.25 | 6,038 | 7,777.384615 | |
The 64 whole numbers from 1 through 64 are written, one per square, on a checkerboard (an 8 by 8 array of 64 squares). The first 8 numbers are written in order across the first row, the next 8 across the second row, and so on. After all 64 numbers are written, the sum of the numbers in the four corners will be | 130 | 1. **Identify the positions of the corner numbers on the checkerboard:**
- The top left corner is the first square, so the number is $1$.
- The top right corner is the last square of the first row, which is the 8th square, so the number is $8$.
- The bottom right corner is the last square of the last row, whic... | 1 | 3,284.6875 | 3,284.6875 | -1 |
The area of the region in the $xy$ -plane satisfying the inequality \[\min_{1 \le n \le 10} \max\left(\frac{x^2+y^2}{4n^2}, \, 2 - \frac{x^2+y^2}{4n^2-4n+1}\right) \le 1\] is $k\pi$ , for some integer $k$ . Find $k$ .
*Proposed by Michael Tang* | 210 | 0.375 | 6,659.375 | 4,421.166667 | 8,002.3 | |
Let $A B C D E F G H$ be an equilateral octagon with $\angle A \cong \angle C \cong \angle E \cong \angle G$ and $\angle B \cong \angle D \cong \angle F \cong$ $\angle H$. If the area of $A B C D E F G H$ is three times the area of $A C E G$, then $\sin B$ can be written as $\frac{m}{n}$, where $m, n$ are positive inte... | 405 | Assume $A C=1$. Note that from symmetry, it can be seen that all angles in $A C E G$ must be equal. Further, by similar logic all sides must be equal which means that $A C E G$ is a square. Additionally, as $A B=B C, A B C$ is an isosceles triangle, which means the octagon consists of a unit square with four isosceles ... | 0 | 8,192 | -1 | 8,192 |
Four consecutive positive integers have a product of 840. What is the largest of the four integers? | 7 | 1 | 2,824.3125 | 2,824.3125 | -1 | |
In the triangle \( \triangle ABC \), if \(\sin^2 A + \sin^2 B + \sin^2 C = 2\), calculate the maximum value of \(\cos A + \cos B + 2 \cos C\). | \sqrt{5} | 0.3125 | 6,056.5625 | 4,851.2 | 6,604.454545 | |
Let $(x,y)$ satisfy the constraints $\begin{cases} 8x - y - 4 \leqslant 0 \\ x + y + 1 \geqslant 0 \\ y - 4x \leqslant 0 \end{cases}$, and the maximum value of the objective function $z = ax + by (a > 0, b > 0)$ is $2$. Find the minimum value of $\frac{1}{a} + \frac{1}{b}$. | \frac{9}{2} | 0.8125 | 6,105.5625 | 5,624.076923 | 8,192 | |
A standard $n$-sided die has $n$ sides labeled 1 to $n$. Luis, Luke, and Sean play a game in which they roll a fair standard 4-sided die, a fair standard 6-sided die, and a fair standard 8-sided die, respectively. They lose the game if Luis's roll is less than Luke's roll, and Luke's roll is less than Sean's roll. Comp... | \frac{1}{4} | We perform casework on Luke's roll. If Luke rolls $n$, with $2 \leq n \leq 5$, then the probability Luis rolls less than Luke is $\frac{n-1}{4}$, and the probability Sean rolls more than Luke is $\frac{8-n}{8}$. If Luke rolls 6 then Luis will definitely roll less than Luke, and Sean rolls more than Luke with probabilit... | 0.125 | 7,730.3125 | 6,816 | 7,860.928571 |
Fill in the blanks with appropriate numbers.
4 liters 25 milliliters = ___ milliliters
6.09 cubic decimeters = ___ cubic centimeters
4.9 cubic decimeters = ___ liters ___ milliliters
2.03 cubic meters = ___ cubic meters ___ cubic decimeters. | 30 | 0.1875 | 949.625 | 1,056.666667 | 924.923077 | |
In $\triangle PQR$, $PR=15$, $QR=20$, and $PQ=25$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, and points $E$ and $F$ lie on $\overline{PR}$, with $PA=QB=QC=RD=RE=PF=5$. Find the area of hexagon $ABCDEF$.
Diagram
[asy] dot((0,0)); dot((15,0)); dot((15,20)); draw((0,0)--(15,0)--... | 120 | Note that $\triangle{PQR}$ has area $150$ and is a $3$-$4$-$5$ right triangle. Then, by similar triangles, the altitude from $B$ to $QC$ has length $3$ and the altitude from $A$ to $FP$ has length $4$, so $[QBC]+[DRE]+[AFP]=\frac{15}{2}+\frac{25}{2}+\frac{20}{2}=30$, meaning that $[ABCDEF]=\boxed{120}$. -Stormersyle | 0.125 | 8,105.6875 | 7,501.5 | 8,192 |
An infinite geometric series has a first term of \( 512 \) and its sum is \( 8000 \). What is its common ratio? | 0.936 | 0 | 2,936.0625 | -1 | 2,936.0625 | |
Consider a board on $2013 \times 2013$ squares, what is the maximum number of chess knights that can be placed so that no $2$ attack each other? | 2026085 | 0.875 | 4,490.1875 | 4,105.071429 | 7,186 | |
The base of a triangle is $30 \text{ cm}$, and the other two sides are $26 \text{ cm}$ and $28 \text{ cm}$. The height of the triangle is divided in the ratio $2:3$ (counting from the vertex), and a line parallel to the base is drawn through the point of division. Determine the area of the resulting trapezoid. | 322.56 | 0 | 7,198.1875 | -1 | 7,198.1875 | |
The coefficient of the $x^2$ term in the expansion of $(1+x)(1+ \sqrt{x})^5$ is \_\_\_\_\_\_. | 15 | 0.9375 | 5,530.9375 | 5,353.533333 | 8,192 | |
A train took $X$ minutes ($0 < X < 60$) to travel from platform A to platform B. Find $X$ if it's known that at both the moment of departure from A and the moment of arrival at B, the angle between the hour and minute hands of the clock was $X$ degrees. | 48 | 0.125 | 7,792.25 | 4,994 | 8,192 | |
A large cube is made up of 27 small cubes. A plane is perpendicular to one of the diagonals of this large cube and bisects the diagonal. How many small cubes are intersected by this plane? | 19 | 0 | 8,129.1875 | -1 | 8,129.1875 | |
In the figure, $ABCD$ is an isosceles trapezoid with side lengths $AD=BC=5$, $AB=4$, and $DC=10$. The point $C$ is on $\overline{DF}$ and $B$ is the midpoint of hypotenuse $\overline{DE}$ in right triangle $DEF$. Then $CF=$ | 4.0 | 1. **Identify the properties of the trapezoid**: Given that $ABCD$ is an isosceles trapezoid, we know that $AD = BC = 5$, $AB = 4$, and $DC = 10$. Since $ABCD$ is isosceles, the non-parallel sides ($AD$ and $BC$) are equal.
2. **Drop perpendiculars from $A$ and $B$ to $DC$**: Let's denote the feet of these perpendicul... | 0 | 6,742.6875 | -1 | 6,742.6875 |
The polynomial \( x^8 - 4x^7 + 7x^6 + \cdots + a_0 \) has all its roots positive and real numbers. Find the possible values for \( a_0 \). | \frac{1}{256} | 0.5 | 6,271.75 | 4,351.5 | 8,192 | |
What is the smallest positive odd number that has the same number of divisors as 360? | 3465 | 0.1875 | 8,081.375 | 7,602 | 8,192 | |
The line $y=kx$ intersects the graph of the function $y=\tan x$ ($-\frac{π}{2}＜x＜\frac{π}{2}$) at points $M$ and $N$ (not coinciding with the origin $O$). The coordinates of point $A$ are $(-\frac{π}{2},0)$. Find $(\overrightarrow{AM}+\overrightarrow{AN})\cdot\overrightarrow{AO}$. | \frac{\pi^2}{2} | 0.875 | 4,480.5 | 3,950.285714 | 8,192 | |
If $a,b,c,d$ are Distinct Real no. such that $a = \sqrt{4+\sqrt{5+a}}$ $b = \sqrt{4-\sqrt{5+b}}$ $c = \sqrt{4+\sqrt{5-c}}$ $d = \sqrt{4-\sqrt{5-d}}$ Then $abcd = $ | 11 | 0.375 | 6,924.4375 | 4,811.833333 | 8,192 | |
Find the numbers $\mathbf{1 5 3 , 3 7 0 , 3 7 1 , 4 0 7}$. | 153, 370, 371, 407 | The numbers are $\mathbf{1 5 3 , 3 7 0 , 3 7 1 , 4 0 7}$. | 0.0625 | 2,542.5 | 3,247 | 2,495.533333 |
For a positive integer $n,$ let
\[H_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}.\]Compute
\[\sum_{n = 1}^\infty \frac{1}{(n + 1) H_n H_{n + 1}}.\] | 1 | 0.6875 | 4,895.875 | 3,397.636364 | 8,192 | |
Given a positive number $x$ has two square roots, which are $2a-3$ and $5-a$, find the values of $a$ and $x$. | 49 | 0.6875 | 1,652.1875 | 2,189.090909 | 471 | |
A cube has six faces, and each face has two diagonals. From these diagonals, choose two to form a pair. Among these pairs, how many form an angle of $60^\circ$? | 48 | 0 | 8,064.125 | -1 | 8,064.125 | |
For what value of $k$ is the line through the points $(3,2k+1)$ and $(8,4k-5)$ parallel to the $x$-axis? | 3 | A line segment joining two points is parallel to the $x$-axis exactly when the $y$-coordinates of the two points are equal. Here, this means that $2k+1=4k-5$ and so $6=2k$ or $k=3$. (We can check that when $k=3$, the coordinates of the points are $(3,7)$ and $(8,7)$.) | 1 | 1,479.0625 | 1,479.0625 | -1 |
Given $m, n \in \mathbb{R}$, if the line $(m+1)x + (n+1)y - 2 = 0$ is tangent to the circle $x^2 + y^2 = 1$, find the maximum value of $m - n$. | 2\sqrt{2} | 1 | 3,444.375 | 3,444.375 | -1 | |
Add $45.23$ to $78.569$ and round your answer to the nearest tenth. | 123.8 | 1 | 2,340.1875 | 2,340.1875 | -1 | |
When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $$(a-1)(b-1)(c-1) \neq 0$$ | \frac{125}{216} | 1 | 2,663.6875 | 2,663.6875 | -1 |
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