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 |
|---|---|---|---|---|---|---|
John has 8 green marbles and 7 purple marbles. He chooses a marble at random, records its color, and then does not put the marble back. He repeats this process 6 times. What is the probability that he chooses exactly three green marbles? | \frac{392}{1001} | 0 | 4,935.125 | -1 | 4,935.125 | |
How many distinct arrangements of the letters in the word "balloon" are there, given that it contains repeating letters? | 1260 | 0.5625 | 1,728.75 | 1,799.444444 | 1,637.857143 | |
Here are two functions: $$\begin{array}{ccc}
f(x) & = & 3x^2-2x+ 4\\
g(x) & = & x^2-kx-6
\end{array}$$ If $f(10) - g(10) = 10,$ what is the value of $k?$ | -18 | 1 | 1,849.125 | 1,849.125 | -1 | |
A four-digit integer $m$ and the four-digit integer obtained by reversing the order of the digits of $m$ are both divisible by 45. If $m$ is divisible by 7, what is the greatest possible value of $m$? | 5985 | 0.375 | 7,936 | 7,509.333333 | 8,192 | |
Every positive integer $k$ has a unique factorial base expansion $(f_1,f_2,f_3,\ldots,f_m)$, meaning that $k=1!\cdot f_1+2!\cdot f_2+3!\cdot f_3+\cdots+m!\cdot f_m$, where each $f_i$ is an integer, $0\le f_i\le i$, and $0<f_m$. Given that $(f_1,f_2,f_3,\ldots,f_j)$ is the factorial base expansion of $16!-32!+48!-64!+\c... | 495 | Note that $1+\sum_{k=1}^{n-1} {k\cdot k!} = 1+\sum_{k=1}^{n-1} {((k+1)\cdot k!- k!)} = 1+\sum_{k=1}^{n-1} {((k+1)!- k!)} = n!$
Thus for all $m\in\mathbb{N}$,
$(32m+16)!-(32m)! = \left(1+\sum_{k=1}^{32m+15} {k\cdot k!}\right)-\left(1+\sum_{k=1}^{32m-1} {k\cdot k!}\right) = \sum_{k=32m}^{32m+15}k\cdot k!.$
So now,
\beg... | 0 | 8,192 | -1 | 8,192 |
In the diagram, square \(PQRS\) has side length 40. Points \(J, K, L,\) and \(M\) are on the sides of \(PQRS\), so that \(JQ = KR = LS = MP = 10\). Line segments \(JZ, KW, LX,\) and \(MY\) are drawn parallel to the diagonals of the square so that \(W\) is on \(JZ\), \(X\) is on \(KW\), \(Y\) is on \(LX\), and \(Z\) is ... | 200 | 0 | 8,192 | -1 | 8,192 | |
How many ways are there to put 7 balls in 4 boxes if the balls are not distinguishable and neither are the boxes? | 11 | 0.375 | 6,855.5625 | 5,663.666667 | 7,570.7 | |
For an arithmetic sequence {a_n} with the sum of the first n terms denoted as S_n, it is known that (a_2 - 1)^3 + 2014(a_2 - 1) = sin \frac{2011\pi}{3} and (a_{2013} - 1)^3 + 2014(a_{2013} - 1) = cos \frac{2011\pi}{6}. Determine the value of S_{2014}. | 2014 | 0.4375 | 7,237.4375 | 6,010.142857 | 8,192 | |
The 200-digit number \( M \) is composed of 200 ones. What is the sum of the digits of the product \( M \times 2013 \)? | 1200 | 0 | 8,192 | -1 | 8,192 | |
A right-angled triangle has an area of \( 36 \mathrm{~m}^2 \). A square is placed inside the triangle such that two sides of the square are on two sides of the triangle, and one vertex of the square is at one-third of the longest side.
Determine the area of this square. | 16 | 0.4375 | 7,220.25 | 5,970.857143 | 8,192 | |
Given the numbers 0, 1, 2, 3, 4, 5, 6, determine the total number of 3-digit numbers that can be formed from these digits without repetition and divided by 5. | 55 | 0.125 | 4,403.875 | 3,666.5 | 4,509.214286 | |
Calculate the definite integral:
$$
\int_{6}^{9} \sqrt{\frac{9-2x}{2x-21}} \, dx
$$ | \pi | 0.5 | 7,378.9375 | 7,385.25 | 7,372.625 | |
Simplify $5 \cdot \frac{12}{7} \cdot \frac{49}{-60}$. | -7 | 1 | 3,978.875 | 3,978.875 | -1 | |
Add $36_7 + 274_7.$ Express your answer in base 7. | 343_7 | 0.6875 | 4,819.0625 | 3,285.909091 | 8,192 | |
Circle $T$ has a circumference of $12\pi$ inches, and segment $XY$ is a diameter. If the measure of angle $TXZ$ is $60^{\circ}$, what is the length, in inches, of segment $XZ$?
[asy]
size(150);
draw(Circle((0,0),13),linewidth(1));
draw((-12,-5)--(-5,-12)--(12,5)--cycle,linewidth(1));
dot((0,0));
label("T",(0,0),N);
l... | 6 | 0.875 | 4,631.6875 | 4,123.071429 | 8,192 | |
Let \(\alpha_{n}\) be a real root of the cubic equation \(n x^{3}+2 x-n=0\), where \(n\) is a positive integer. If \(\beta_{n}=\left\lfloor(n+1) \alpha_{n}\right\rfloor\) for \(n=234 \cdots\), find the value of \(\frac{1}{1006} \sum_{k=2}^{2013} \beta_{k}\). | 2015 | 0.5625 | 7,748.6875 | 7,403.888889 | 8,192 | |
A rectangular metal plate measuring \(10\) cm by \(8\) cm has a circular piece of maximum size cut out, followed by cutting a rectangular piece of maximum size from the circular piece. Calculate the total metal wasted in this process. | 48 | 0.75 | 5,124.125 | 4,101.5 | 8,192 | |
What is the greatest two-digit whole number, the product of whose digits is 8? | 81 | 1 | 2,815.875 | 2,815.875 | -1 | |
Find the greatest possible value of $pq + r$ , where p, q, and r are (not necessarily distinct) prime numbers satisfying $pq + qr + rp = 2016$ .
| 1008 | 0.0625 | 7,967.5 | 4,600 | 8,192 | |
The inscribed circle of triangle $ABC$ is tangent to $\overline{AB}$ at $P_{},$ and its radius is $21$. Given that $AP=23$ and $PB=27,$ find the perimeter of the triangle. | 345 | Let $Q$ be the tangency point on $\overline{AC}$, and $R$ on $\overline{BC}$. By the Two Tangent Theorem, $AP = AQ = 23$, $BP = BR = 27$, and $CQ = CR = x$. Using $rs = A$, where $s = \frac{27 \cdot 2 + 23 \cdot 2 + x \cdot 2}{2} = 50 + x$, we get $(21)(50 + x) = A$. By Heron's formula, $A = \sqrt{s(s-a)(s-b)(s-c)} = \... | 0.8125 | 6,118.625 | 5,640.153846 | 8,192 |
Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra? | \frac{10-7\sqrt{2}}{3} | 1. **Identify the shape of the faces after slicing**: After slicing off the corners of the unit cube, each face becomes a regular octagon. This implies that each edge of the cube is divided into three segments: two segments forming the slanted edges of the octagon and one segment forming a side of the octagon.
2. **De... | 0 | 7,324 | -1 | 7,324 |
Define a $\it{good\ word}$ as a sequence of letters that consists only of the letters $A$, $B$, and $C$ --- some of these letters may not appear in the sequence --- and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, and $C$ is never immediately followed by $A$. Additionally... | 94 | 0.0625 | 7,742.3125 | 5,511 | 7,891.066667 | |
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 | 0 | 8,050.3125 | -1 | 8,050.3125 | |
Circle $C$ has radius 6 cm. How many square centimeters are in the area of the largest possible inscribed triangle having one side as a diameter of circle $C$? | 36 | 1 | 3,179.625 | 3,179.625 | -1 | |
Compute the greatest common divisor of $4^{8}-1$ and $8^{12}-1$. | 15 | Let $d=\operatorname{gcd}(a, b)$ for some $a, b \in \mathbb{Z}^{+}$. Then, we can write $d=a x-b y$, where $x, y \in \mathbb{Z}^{+}$, and $$\begin{align*} & 2^{a}-1 \mid 2^{a x}-1 \tag{1}\\ & 2^{b}-1 \mid 2^{b y}-1 \tag{2} \end{align*}$$ Multiplying the right-hand side of (2) by $2^{d}$, we get, $$2^{b}-1 \mid 2^{a x}... | 0.5 | 6,037.5625 | 3,883.125 | 8,192 |
In $\vartriangle ABC$ points $D, E$ , and $F$ lie on side $\overline{BC}$ such that $\overline{AD}$ is an angle bisector of $\angle BAC$ , $\overline{AE}$ is a median, and $\overline{AF}$ is an altitude. Given that $AB = 154$ and $AC = 128$ , and $9 \times DE = EF,$ fi nd the side length $BC$ . | 94 | 0.75 | 6,337 | 5,718.666667 | 8,192 | |
Consider a sequence $\{a_n\}$ with property P: if $a_p = a_q$ for $p, q \in \mathbb{N}^{*}$, then it must hold that $a_{p+1} = a_{q+1}$. Suppose the sequence $\{a_n\}$ has property P, and it is given that $a_1=1$, $a_2=2$, $a_3=3$, $a_5=2$, and $a_6+a_7+a_8=21$. Determine the value of $a_{2017}$. | 16 | 0.0625 | 7,891.5 | 3,805 | 8,163.933333 | |
Let $P$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$, and $(0,1)$. The probability that the slope of the line determined by $P$ and the point $\left(\frac58, \frac38 \right)$ is greater than or equal to $\frac12$ can be written as $\frac{m}{n}$, where $... | 171 | The areas bounded by the unit square and alternately bounded by the lines through $\left(\frac{5}{8},\frac{3}{8}\right)$ that are vertical or have a slope of $1/2$ show where $P$ can be placed to satisfy the condition. One of the areas is a trapezoid with bases $1/16$ and $3/8$ and height $5/8$. The other area is a tra... | 0.5 | 6,715.5 | 5,805.75 | 7,625.25 |
The real numbers \( x, y, z \) satisfy \( x + y + z = 2 \) and \( xy + yz + zx = 1 \). Find the maximum possible value of \( x - y \). | \frac{2 \sqrt{3}}{3} | 0 | 6,521.375 | -1 | 6,521.375 | |
Let $f(x)$ be a quotient of two quadratic polynomials. Given that $f(n)=n^{3}$ for all $n \in\{1,2,3,4,5\}$, compute $f(0)$. | \frac{24}{17} | Let $f(x)=p(x) / q(x)$. Then, $x^{3} q(x)-p(x)$ has $1,2,3,4,5$ as roots. Therefore, WLOG, let $$x^{3} q(x)-p(x)=(x-1)(x-2)(x-3)(x-4)(x-5)=x^{5}-15 x^{4}+85 x^{3}-\ldots$$ Thus, $q(x)=x^{2}-15 x+85$, so $q(0)=85$. Plugging $x=0$ in the above equation also gives $-p(0)=-120$. Hence, the answer is $\frac{120}{85}=\frac{2... | 0.0625 | 7,955.5625 | 4,409 | 8,192 |
What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors? | 180 | Somewhat similar to the first solution, we see that the number $n$ has two even factors for every odd factor. Thus, if $x$ is an odd factor of $n$, then $2x$ and $4x$ must be the two corresponding even factors. So, the prime factorization of $n$ is $2^2 3^a 5^b 7^c...$ for some set of integers $a, b, c, ...$
Since the... | 0.5 | 6,639.6875 | 5,402.25 | 7,877.125 |
The function $g(x)$ satisfies the equation
\[xg(y) = 2yg(x)\] for all real numbers $x$ and $y$. If $g(10) = 30$, find $g(2)$. | 12 | 0 | 8,192 | -1 | 8,192 | |
How many natural-number factors does $N$ have if $N = 2^4 \cdot 3^3 \cdot 5^2 \cdot 7^2$? | 180 | 1 | 1,647.6875 | 1,647.6875 | -1 | |
Find the sum of all integers $x$ , $x \ge 3$ , such that $201020112012_x$ (that is, $201020112012$ interpreted as a base $x$ number) is divisible by $x-1$ | 32 | 0.875 | 2,892.9375 | 2,517.571429 | 5,520.5 | |
Let $A$ , $B$ , $C$ , and $P$ be points in the plane such that no three of them are collinear. Suppose that the areas of triangles $BPC$ , $CPA$ , and $APB$ are 13, 14, and 15, respectively. Compute the sum of all possible values for the area of triangle $ABC$ .
*Proposed by Ankan Bhattacharya* | 84 | 0.25 | 6,598.5 | 5,456.75 | 6,979.083333 | |
In a right triangle \(ABC\) with \(\angle C = 90^{\circ}\), a segment \(BD\) equal to the leg \(BC\) is laid out on the extension of the hypotenuse \(AB\), and point \(D\) is connected to \(C\). Find \(CD\) if \(BC = 7\) and \(AC = 24\). | 8 \sqrt{7} | 0 | 6,343.5 | -1 | 6,343.5 | |
In our daily life, we often use passwords, such as when making payments through Alipay. There is a type of password generated using the "factorization" method, which is easy to remember. The principle is to factorize a polynomial. For example, the polynomial $x^{3}+2x^{2}-x-2$ can be factorized as $\left(x-1\right)\lef... | 24121 | 0.0625 | 5,037.875 | 5,637 | 4,997.933333 | |
Diagonal $DB$ of rectangle $ABCD$ is divided into three segments of length $1$ by parallel lines $L$ and $L'$ that pass through $A$ and $C$ and are perpendicular to $DB$. The area of $ABCD$, rounded to the one decimal place, is | 4.2 | 1. **Identify the Geometry and Key Points**:
Let $ABCD$ be a rectangle with diagonal $DB$. Lines $L$ and $L'$ are parallel to each other, pass through points $A$ and $C$ respectively, and are perpendicular to diagonal $DB$. Let $E$ and $F$ be the points where lines $L$ and $L'$ intersect diagonal $DB$, respectivel... | 0.75 | 5,852.3125 | 5,072.416667 | 8,192 |
Four steel balls, each with a radius of 1, are to be completely packed into a container shaped as a regular tetrahedron. What is the minimum height of the tetrahedron? | 2 + \frac{2 \sqrt{6}}{3} | 0 | 7,895 | -1 | 7,895 | |
A regular octahedron has side length $1$. A plane parallel to two of its opposite faces cuts the octahedron into the two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area $\frac {a\sqrt {b}}{c}$, where $a$, $b$, and $c$ are positive integers, $a$ and $c$ are relatively pr... | 14 |
To solve this problem, we need to understand the geometry of the octahedron and how the plane intersects it. An octahedron can be thought of as two square pyramids base-to-base. Each face of the octahedron is an equilateral triangle.
#### Step 1: Understanding the Intersection
The plane is parallel to two opposite fa... | 0 | 8,192 | -1 | 8,192 |
The partial fraction decomposition of
\[\frac{x^2 - 19}{x^3 - 2x^2 - 5x + 6}\]is
\[\frac{A}{x - 1} + \frac{B}{x + 2} + \frac{C}{x - 3}.\]Find the product $ABC.$ | 3 | 0.9375 | 3,467.3125 | 3,152.333333 | 8,192 | |
Express $1.\overline{27}$ as a common fraction in lowest terms. | \dfrac{14}{11} | 1 | 1,691 | 1,691 | -1 | |
A covered rectangular soccer field of length 90 meters and width 60 meters is being designed. It must be illuminated by four floodlights, each hung at some point on the ceiling. Each floodlight illuminates a circle with a radius equal to the height at which it is hung. Determine the minimum possible height of the ceili... | 27.1 | 0 | 8,192 | -1 | 8,192 | |
The first three terms of an arithmetic progression are $x - 1, x + 1, 2x + 3$, in the order shown. The value of $x$ is: | 0 | 1. **Identify the common difference**:
Given the arithmetic progression (AP) terms are $x - 1$, $x + 1$, and $2x + 3$. In an AP, the difference between consecutive terms is constant. Therefore, the common difference $d$ can be calculated as:
\[
d = (x + 1) - (x - 1)
\]
Simplifying this, we get:
\[
... | 1 | 1,513.8125 | 1,513.8125 | -1 |
Distinct points $P$, $Q$, $R$, $S$ lie on the circle $x^{2}+y^{2}=25$ and have integer coordinates. The distances $PQ$ and $RS$ are irrational numbers. What is the greatest possible value of the ratio $\frac{PQ}{RS}$? | 7 | 1. **Identify Possible Points**: The circle given by the equation $x^2 + y^2 = 25$ has a radius of 5. We need to find points $(x, y)$ with integer coordinates that lie on this circle. Solving $x^2 + y^2 = 25$ for integer solutions, we find the points $(\pm 3, \pm 4), (\pm 4, \pm 3), (0, \pm 5), (\pm 5, 0)$.
2. **Dista... | 0.25 | 7,571.8125 | 7,243.5 | 7,681.25 |
Given: The lengths of the three sides of a triangle, $a$, $b$, and $c$, are integers, and $a \leq b < c$, where $b = 5$. Calculate the number of such triangles. | 10 | 0.875 | 5,118.25 | 4,679.142857 | 8,192 | |
At 11:00 a.m. how many degrees are in the smaller angle formed by the minute hand and the hour hand of the clock? | 30 | 1 | 2,215.375 | 2,215.375 | -1 | |
Compute $\arccos (\cos 3).$ All functions are in radians. | 3 - 2\pi | 0 | 6,762.75 | -1 | 6,762.75 | |
Eyes are the windows of the soul. In order to protect students' eyesight, Qihang High School conducts eye examinations for students every semester. The table below shows the results of the right eye vision examination for 39 students in a certain class at the school. In this set of vision data, the median is ______.
|... | 4.6 | 0.8125 | 2,396.4375 | 1,720.153846 | 5,327 | |
The expression $x^2 + 17x + 70$ can be rewritten as $(x + a)(x + b)$, and the expression $x^2 - 18x + 80$ written as $(x - b)(x - c)$, where a, b, and c are integers. Calculate the value of $a + b + c$. | 28 | 0 | 2,551.25 | -1 | 2,551.25 | |
The least positive integer with exactly $2021$ distinct positive divisors can be written in the form $m \cdot 6^k$, where $m$ and $k$ are integers and $6$ is not a divisor of $m$. What is $m+k?$ | 58 | To find the least positive integer with exactly $2021$ distinct positive divisors, we start by understanding the divisor function. If a number $n$ has a prime factorization of the form $n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$, then the number of divisors of $n$ is given by $(a_1+1)(a_2+1)\cdots(a_k+1)$.
Given that t... | 1 | 4,408 | 4,408 | -1 |
A soccer ball rolls at $4 \mathrm{~m} / \mathrm{s}$ towards Marcos in a direct line from Michael. The ball is $15 \mathrm{~m}$ ahead of Michael who is chasing it at $9 \mathrm{~m} / \mathrm{s}$. Marcos is $30 \mathrm{~m}$ away from the ball and is running towards it at $8 \mathrm{~m} / \mathrm{s}$. Calculate the distan... | 2.5 | 0.0625 | 7,040.1875 | 6,014 | 7,108.6 | |
Find the minimum value for \(a, b > 0\) of the expression
$$
\frac{|6a - 4b| + |3(a + b\sqrt{3}) + 2(a\sqrt{3} - b)|}{\sqrt{a^2 + b^2}}
$$ | \sqrt{39} | 0 | 8,087.125 | -1 | 8,087.125 | |
One material particle entered the opening of a pipe, and after 6.8 minutes, a second particle entered the same opening. Upon entering the pipe, each particle immediately began linear motion along the pipe: the first particle moved uniformly at a speed of 5 meters per minute, while the second particle covered 3 meters i... | 17 | 0.3125 | 6,765.5 | 3,981 | 8,031.181818 | |
When Claire divides her cupcakes into groups of 5, she has 3 remaining, and when she divides her cupcakes into groups of 7, she has 4 remaining. If Claire has fewer than 60 cupcakes, what is the sum of all possible quantities of cupcakes that she could have? | 71 | 1 | 2,118.4375 | 2,118.4375 | -1 | |
What is the number of degrees in the acute angle formed by the hands of a clock at 6:44? | 62^\circ | 0.9375 | 3,744.8125 | 3,448.333333 | 8,192 | |
The first digit of a string of 2002 digits is a 1. Any two-digit number formed by consecutive digits within this string is divisible by 19 or 31. What is the largest possible last digit in this string? | 8 | 0 | 8,078.375 | -1 | 8,078.375 | |
Huanhuan and Lele are playing a game together. In the first round, they both gain the same amount of gold coins, and in the second round, they again gain the same amount of gold coins.
At the beginning, Huanhuan says: "The number of my gold coins is 7 times the number of your gold coins."
At the end of the first rou... | 70 | 0.3125 | 6,672.375 | 3,788 | 7,983.454545 | |
Point P moves on the parabola y^2 = 4x with focus F, and point Q moves on the line x-y+5=0. Find the minimum value of ||PF+|PQ||. | 3\sqrt{2} | 0.625 | 7,041.375 | 6,731.3 | 7,558.166667 | |
If the three points $(1,a,b),$ $(a,2,b),$ $(a,b,3)$ are collinear, what is the value of $a + b$? | 4 | 1 | 4,024.6875 | 4,024.6875 | -1 | |
Let's call a number palindromic if it reads the same left to right as it does right to left. For example, the number 12321 is palindromic.
a) Write down any five-digit palindromic number that is divisible by 5.
b) How many five-digit palindromic numbers are there that are divisible by 5? | 100 | 1 | 1,710.75 | 1,710.75 | -1 | |
Simplify $t^3\cdot t^4$. | t^7 | 0.9375 | 1,989.5 | 1,576 | 8,192 | |
For how many integers $n$ is $\frac n{20-n}$ the square of an integer? | 4 | To solve the problem, we need to find the integers $n$ for which $\frac{n}{20-n}$ is a perfect square.
1. **Examine the range of $n$:**
- If $n < 0$ or $n > 20$, the fraction $\frac{n}{20-n}$ is negative, and thus cannot be a perfect square.
- If $n = 20$, the fraction is undefined.
- If $n \in \{1, 2, \dots,... | 0.8125 | 6,212.875 | 5,756.153846 | 8,192 |
On a 4x4 grid (where each unit distance is 1), calculate how many rectangles can be formed where each of the rectangle's vertices is a point on this grid. | 36 | 0 | 6,082.25 | -1 | 6,082.25 | |
Source: 2018 Canadian Open Math Challenge Part B Problem 2
-----
Let ABCD be a square with side length 1. Points $X$ and $Y$ are on sides $BC$ and $CD$ respectively such that the areas of triangels $ABX$ , $XCY$ , and $YDA$ are equal. Find the ratio of the area of $\triangle AXY$ to the area of $\triang... | \sqrt{5} | 0.75 | 6,947.1875 | 6,532.25 | 8,192 | |
The probability of snow for each of the next four days is $\frac{3}{4}$. However, if it snows any day before the last day, the probability of snow on the last day increases to $\frac{4}{5}$. What is the probability that it will snow at least once during these four days? Express your answer as a common fraction. | \dfrac{1023}{1280} | 0.125 | 7,819.1875 | 6,522 | 8,004.5 | |
In how many ways can I choose a 4-person committee from a club of 9 people? | 126 | 1 | 2,078.4375 | 2,078.4375 | -1 | |
Two circles are inscribed in an angle of 60 degrees and they touch each other. The radius of the smaller circle is 24. What is the radius of the larger circle? | 72 | 0.75 | 5,011.9375 | 3,951.916667 | 8,192 | |
Estimate the population of the island of Thalassa in the year 2050, knowing that its population doubles every 20 years and increases by an additional 500 people every decade thereafter, given that the population in the year 2000 was 250. | 1500 | 0 | 6,990.125 | -1 | 6,990.125 | |
Compute the integer $m > 3$ for which
\[\log_{10} (m - 3)! + \log_{10} (m - 1)! + 3 = 2 \log_{10} m!.\] | 10 | 0 | 8,192 | -1 | 8,192 | |
Given that the Green Park Middle School chess team consists of three boys and four girls, and a girl at each end and the three boys and one girl alternating in the middle, determine the number of possible arrangements. | 144 | 0.3125 | 6,118.0625 | 2,523.6 | 7,751.909091 | |
Three circles with radii 2, 3, and 10 units are placed inside a larger circle such that all circles are touching one another. Determine the value of the radius of the larger circle. | 15 | 0.25 | 7,520.3125 | 5,836.75 | 8,081.5 | |
Triangle $ABC$ obeys $AB=2AC$ and $\angle BAC=120^{\circ}$. Points $P$ and $Q$ lie on segment $BC$ such that $$\begin{aligned} AB^{2}+BC \cdot CP & =BC^{2} \\ 3AC^{2}+2BC \cdot CQ & =BC^{2} \end{aligned}$$ Find $\angle PAQ$ in degrees. | 40^{\circ} | We have $AB^{2}=BC(BC-CP)=BC \cdot BP$, so triangle $ABC$ is similar to triangle $PBA$. Also, $AB^{2}=BC(BC-2CQ)+AC^{2}=(BC-CQ)^{2}-CQ^{2}+AC^{2}$, which rewrites as $AB^{2}+CQ^{2}=$ $BQ^{2}+AC^{2}$. We deduce that $Q$ is the foot of the altitude from $A$. Thus, $\angle PAQ=90^{\circ}-\angle QPA=90^{\circ}-$ $\angle AB... | 0 | 5,434.5 | -1 | 5,434.5 |
What is the minimum number of convex pentagons needed to form a convex 2011-gon? | 670 | 0 | 7,921.9375 | -1 | 7,921.9375 | |
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 | 1 | 3,277.625 | 3,277.625 | -1 | |
An infinite sequence of real numbers $a_{1}, a_{2}, \ldots$ satisfies the recurrence $$a_{n+3}=a_{n+2}-2 a_{n+1}+a_{n}$$ for every positive integer $n$. Given that $a_{1}=a_{3}=1$ and $a_{98}=a_{99}$, compute $a_{1}+a_{2}+\cdots+a_{100}$. | 3 | A quick telescope gives that $a_{1}+\cdots+a_{n}=2 a_{1}+a_{3}+a_{n-1}-a_{n-2}$ for all $n \geq 3$: $$\begin{aligned} \sum_{k=1}^{n} a_{k} & =a_{1}+a_{2}+a_{3}+\sum_{k=1}^{n-3}\left(a_{k}-2 a_{k+1}+2 a_{k+2}\right) \\ & =a_{1}+a_{2}+a_{3}+\sum_{k=1}^{n-3} a_{k}-2 \sum_{k=2}^{n-2} a_{k}+\sum_{k=3}^{n-1} a_{k} \\ & =2 a_... | 0 | 8,192 | -1 | 8,192 |
On a \(10 \times 10\) grid, there are 11 horizontal grid lines and 11 vertical grid lines. The line segments connecting adjacent nodes on the same line are called "links." What is the minimum number of links that must be removed so that at each node, there are at most 3 remaining links? | 41 | 0 | 7,517.4375 | -1 | 7,517.4375 | |
If $m$ and $n$ are positive integers with $n > 1$ such that $m^{n} = 2^{25} \times 3^{40}$, what is $m + n$? | 209957 | Since $m$ and $n$ are positive integers with $n > 1$ and $m^{n} = 2^{25} \times 3^{40}$, then 2 and 3 are prime factors of $m$ (since they are prime factors of $m^{n}$) and must be the only prime factors of $m$ (since if there were other prime factors of $m$, then there would be other prime factors of $m^{n}$). Therefo... | 1 | 3,593 | 3,593 | -1 |
Compute the product of the sums of the squares and the cubes of the roots of the equation \[x\sqrt{x} - 8x + 9\sqrt{x} - 1 = 0,\] given that all roots are real and nonnegative. | 13754 | 0.125 | 7,375.3125 | 4,974.5 | 7,718.285714 | |
In a triangle, one of the angles is less than $50^{\circ}$, and another is less than $70^{\circ}$. Find the cosine of the third angle if its sine is $\frac{4}{7}$. | -\frac{\sqrt{33}}{7} | 0 | 5,277.5 | -1 | 5,277.5 | |
For a modified toothpick pattern, the first stage is constructed using 5 toothpicks. If each subsequent stage is formed by adding three more toothpicks than the previous stage, what is the total number of toothpicks needed for the $15^{th}$ stage? | 47 | 0.125 | 5,414.3125 | 3,659.5 | 5,665 | |
What is the smallest positive integer $n$ such that $\frac{1}{n}$ is a terminating decimal and $n$ contains the digit '3'? | 3125 | 0 | 5,879.8125 | -1 | 5,879.8125 | |
Given \( m > n \geqslant 1 \), find the smallest value of \( m + n \) such that
\[ 1000 \mid 1978^{m} - 1978^{n} . \ | 106 | 0.375 | 7,607.875 | 7,301.333333 | 7,791.8 | |
A five-digit number is called irreducible if it cannot be expressed as a product of two three-digit numbers.
What is the greatest number of consecutive irreducible five-digit numbers? | 99 | 0 | 8,192 | -1 | 8,192 | |
In the diagram, $ABCD$ and $EFGD$ are squares each with side lengths of 5 and 3 respectively, and $H$ is the midpoint of both $BC$ and $EF$. Calculate the total area of the polygon $ABHFGD$. | 25.5 | 0 | 8,192 | -1 | 8,192 | |
Given that the 32-digit integer 64312311692944269609355712372657 is the product of 6 consecutive primes, compute the sum of these 6 primes. | 1200974 | Because the product is approximately $64 \cdot 10^{30}$, we know the primes are all around 200000. Say they are $200000+x_{i}$ for $i=1, \ldots, 6$. By expanding $\prod_{i=1}^{6}\left(200000+x_{i}\right)$ as a polynomial in 200000, we see that $$31231 \cdot 10^{25}=200000^{5}\left(x_{1}+\cdots+x_{6}\right)$$ plus the c... | 0 | 8,192 | -1 | 8,192 |
Consider the paper triangle whose vertices are $(0,0), (34,0),$ and $(16,24).$ The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the triangle along the sides of its midpoint triangle. What is the volume of this pyramid?
| 408 | 0.4375 | 7,365.4375 | 6,302.714286 | 8,192 | |
Given \( x_{i} \in \mathbf{R}, x_{i} \geq 0 \) for \( i=1,2,3,4,5 \), and \( \sum_{i=1}^{5} x_{i} = 1 \), find the minimum value of \(\max \left\{ x_{1} + x_{2}, x_{2} + x_{3}, x_{3} + x_{4}, x_{4} + x_{5} \right\} \). | \frac{1}{2} | 0 | 7,123.1875 | -1 | 7,123.1875 | |
If a line segment joins the points $(-9,-2)$ and $(6,8)$, how many points on the line segment have coordinates that are both integers? | 6 | The line segment with endpoints $(-9,-2)$ and $(6,8)$ has slope $\frac{8-(-2)}{6-(-9)}=\frac{10}{15}=\frac{2}{3}$. This means that starting at $(-9,-2)$ and moving 'up 2 and right 3' repeatedly will give other points on the line that have coordinates which are both integers. These points are $(-9,-2),(-6,0),(-3,2),(0,4... | 1 | 2,650.75 | 2,650.75 | -1 |
The secant \( ABC \) intercepts an arc \( BC \), which contains \( 112^\circ \); the tangent \( AD \) at point \( D \) divides this arc in the ratio \( 7:9 \). Find \(\angle BAD\). | 31.5 | 0.125 | 6,385.625 | 5,693 | 6,484.571429 | |
The area of the largest regular hexagon that can fit inside of a rectangle with side lengths 20 and 22 can be expressed as $a \sqrt{b}-c$, for positive integers $a, b$, and $c$, where $b$ is squarefree. Compute $100 a+10 b+c$. | 134610 | Let $s$ be the sidelength of the hexagon. We can view this problem as finding the maximal rectangle of with sides $s$ and $s \sqrt{3}$ that can fit inside this rectangle. Let $A B C D$ be a rectangle with $A B=20$ and $B C=22$ and let $X Y Z W$ be an inscribed rectangle with $X$ on $A B$ and $Y$ on $B C$ with $X Y=s$ a... | 0 | 8,192 | -1 | 8,192 |
Using two red, three blue, and four green small cubes, calculate the number of different towers that can be built using eight of these cubes. | 1260 | 0.625 | 5,964 | 4,627.2 | 8,192 | |
Let $A B C D$ be a convex quadrilateral so that all of its sides and diagonals have integer lengths. Given that $\angle A B C=\angle A D C=90^{\circ}, A B=B D$, and $C D=41$, find the length of $B C$. | 580 | Let the midpoint of $A C$ be $O$ which is the center of the circumcircle of $A B C D . A D C$ is a right triangle with a leg of length 41 , and $41^{2}=A C^{2}-A D^{2}=(A C-A D)(A C+A D)$. As $A C, A D$ are integers and 41 is prime, we must have $A C=840, A D=841$. Let $M$ be the midpoint of $A D . \triangle A O M \sim... | 0 | 8,192 | -1 | 8,192 |
Let $\triangle ABC$ be equilateral with integer side length. Point $X$ lies on $\overline{BC}$ strictly between $B$ and $C$ such that $BX<CX$ . Let $C'$ denote the reflection of $C$ over the midpoint of $\overline{AX}$ . If $BC'=30$ , find the sum of all possible side lengths of $\triangle ABC$ .
*Pr... | 130 | 0.9375 | 5,493.1875 | 5,313.266667 | 8,192 | |
For some constants $x$ and $a$, the third, fourth, and fifth terms in the expansion of $(x + a)^n$ are 84, 280, and 560, respectively. Find $n.$ | 7 | 0.8125 | 5,225.375 | 4,540.769231 | 8,192 | |
Call a three-digit number $\overline{ABC}$ $\textit{spicy}$ if it satisfies $\overline{ABC}=A^3+B^3+C^3$ . Compute the unique $n$ for which both $n$ and $n+1$ are $\textit{spicy}$ . | 370 | 0.75 | 4,626.4375 | 3,437.916667 | 8,192 | |
Compute the value of $p$ such that the equation
\[\frac{2x + 3}{px - 2} = x\]
has exactly one solution. | -\frac{4}{3} | 0 | 8,056.375 | -1 | 8,056.375 | |
Tyrone had $97$ marbles and Eric had $11$ marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric? | 18 | 1. **Identify the total number of marbles**: Initially, Tyrone has 97 marbles and Eric has 11 marbles. The total number of marbles is:
\[
97 + 11 = 108
\]
2. **Set up the equation after redistribution**: Let's denote the number of marbles Tyrone gives to Eric as $x$. After giving $x$ marbles to Eric, Tyrone h... | 0 | 1,368.125 | -1 | 1,368.125 |
A triangle has vertices at coordinates (2,2), (5,6) and (6,2). What is the number of units in the length of the longest side of the triangle? | 5 | 1 | 2,093.1875 | 2,093.1875 | -1 | |
By expanding the expression \((1+\sqrt{11})^{214}\) using the binomial theorem, we obtain terms of the form \(C_{214}^{k}(\sqrt{11})^{k}\). Find the value of \( k \) for which this term has the greatest value. | 165 | 0.6875 | 6,922 | 6,344.727273 | 8,192 |
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