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 |
|---|---|---|---|---|---|---|
If the function $f(x)=\tan (\omega x+ \frac {\pi}{4})$ ($\omega > 0$) has a minimum positive period of $2\pi$, then $\omega=$ ______ ; $f( \frac {\pi}{6})=$ ______ . | \sqrt {3} | 0 | 1,726.5625 | -1 | 1,726.5625 | |
$P, A, B, C,$ and $D$ are five distinct points in space such that $\angle APB = \angle BPC = \angle CPD = \angle DPA = \theta$, where $\theta$ is a given acute angle. Determine the greatest and least values of $\angle APC + \angle BPD$. | 0^\circ \text{ and } 360^\circ |
Consider five distinct points \( P, A, B, C, \) and \( D \) in space where the angles formed at \( P \) satisfy \( \angle APB = \angle BPC = \angle CPD = \angle DPA = \theta \). We are tasked with finding the greatest and least possible values of the sum of angles \( \angle APC + \angle BPD \).
### Analyzing the Geom... | 0 | 8,192 | -1 | 8,192 |
A six digit number (base 10) is squarish if it satisfies the following conditions:
(i) none of its digits are zero;
(ii) it is a perfect square; and
(iii) the first of two digits, the middle two digits and the last two digits of the number are all perfect squares when considered as two digit numbers.
How many squarish ... | 2 | To find the number of squarish numbers, we need to satisfy the following conditions:
1. The number is a six-digit number.
2. None of its digits are zero.
3. It is a perfect square.
4. The first two digits, the middle two digits, and the last two digits of the number are all perfect squares when considered as two-digit ... | 0 | 8,192 | -1 | 8,192 |
Let $n$ be the largest integer that is the product of exactly 3 distinct prime numbers $d$, $e$, and $10d+e$, where $d$ and $e$ are single digits. What is the sum of the digits of $n$? | 12 | 1. **Identify the possible values for $d$ and $e$:**
- Since $d$ and $e$ are single-digit prime numbers, the possible values for $d$ are $\{2, 3, 5, 7\}$.
- Since $e$ is also a single-digit prime number and must be the units digit of the prime number $10d + e$, the possible values for $e$ are $\{3, 7\}$ (as these... | 0.9375 | 4,738.125 | 4,507.866667 | 8,192 |
If two lines \( l \) and \( m \) have equations \( y = -2x + 8 \), and \( y = -3x + 9 \), what is the probability that a point randomly selected in the 1st quadrant and below \( l \) will fall between \( l \) and \( m \)? Express your answer as a decimal to the nearest hundredth. | 0.16 | 0 | 6,598.8125 | -1 | 6,598.8125 | |
For what smallest natural $k$ is the number \( 2016 \cdot 20162016 \cdot 201620162016 \cdot \ldots \cdot 20162016\ldots2016 \) (with $k$ factors) divisible by \(3^{67}\)? | 34 | 0.0625 | 7,802.1875 | 3,549 | 8,085.733333 | |
Let $a_1$, $a_2$, $a_3$, $d_1$, $d_2$, and $d_3$ be real numbers such that for every real number $x$, we have
\[
x^8 - x^6 + x^4 - x^2 + 1 = (x^2 + a_1 x + d_1)(x^2 + a_2 x + d_2)(x^2 + a_3 x + d_3)(x^2 + 1).
\]
Compute $a_1 d_1 + a_2 d_2 + a_3 d_3$. | -1 | 0.0625 | 8,192 | 8,192 | 8,192 | |
In a right triangle JKL, where angle J is the right angle, KL measures 20 units, and JL measures 12 units. Calculate $\tan K$. | \frac{4}{3} | 0 | 1,584.5625 | -1 | 1,584.5625 | |
A circle of radius $10$ inches has its center at the vertex $C$ of an equilateral triangle $ABC$ and passes through the other two vertices. The side $AC$ extended through $C$ intersects the circle at $D$. Calculate the measure of angle $ADB$. | 90 | 0 | 4,344.625 | -1 | 4,344.625 | |
Find the minimum value of the function $f(x)=\log_{2}^{\sqrt{x}} \cdot \log_{\sqrt{2}}^{(2x)}$. | -\frac{1}{4} | 0.6875 | 5,825.0625 | 4,749.181818 | 8,192 | |
Let $\mathcal{F}$ be a family of subsets of $\{1,2,\ldots, 2017\}$ with the following property: if $S_1$ and $S_2$ are two elements of $\mathcal{F}$ with $S_1\subsetneq S_2$ , then $|S_2\setminus S_1|$ is odd. Compute the largest number of subsets $\mathcal{F}$ may contain. | 2 \binom{2017}{1008} | 0 | 7,984.125 | -1 | 7,984.125 | |
Find the number of digit of $\sum_{n=0}^{99} 3^n$ .
You may use $\log_{10} 3=0.4771$ .
2012 Tokyo Institute of Technology entrance exam, problem 2-A | 48 | 0.8125 | 6,122.625 | 5,645.076923 | 8,192 | |
Light of a blue laser (wavelength $\lambda=475 \, \text{nm}$ ) goes through a narrow slit which has width $d$ . After the light emerges from the slit, it is visible on a screen that is $ \text {2.013 m} $ away from the slit. The distance between the center of the screen and the first minimum band is $ \text {765 m... | 1250 | 0.75 | 5,758.625 | 4,947.5 | 8,192 | |
Let $n$ be the maximum number of bishops that can be placed on the squares of a $6 \times 6$ chessboard such that no two bishops are attacking each other. Let $k$ be the number of ways to put $n$ bishops on an $6 \times 6$ chessboard such that no two bishops are attacking each other. Find $n+k$. (Two bishops are consid... | 74 | Color the square with coordinates $(i, j)$ black if $i+j$ is odd and white otherwise, for all $1 \leq i, j \leq 6$. Looking at the black squares only, we note that there are six distinct diagonals which run upward and to the right, but that two of them consist only of a corner square; we cannot simultaneously place bis... | 0 | 7,911.375 | -1 | 7,911.375 |
It is known that the numbers \(x, y, z\) form an arithmetic progression in the given order with a common difference \(\alpha = \arccos \frac{5}{9}\), and the numbers \(1 + \cos x, 1 + \cos y, 1 + \cos z\) form a non-constant geometric progression in the given order. Find \(\cos y\). | -\frac{7}{9} | 0 | 8,192 | -1 | 8,192 | |
Let $A$ and $B$ be two opposite vertices of a cube with side length 1. What is the radius of the sphere centered inside the cube, tangent to the three faces that meet at $A$ and to the three edges that meet at $B$? | 2 - \sqrt{2} | 0.5625 | 6,792.1875 | 5,703.444444 | 8,192 | |
Compute $\dbinom{14}{11}$. | 364 | 0.9375 | 4,467.125 | 4,218.8 | 8,192 | |
A polynomial $P(x)$ with integer coefficients possesses the properties
$$
P(1)=2019, \quad P(2019)=1, \quad P(k)=k,
$$
where $k$ is an integer. Find this integer $k$. | 1010 | 0.9375 | 5,995.375 | 5,848.933333 | 8,192 | |
Alice Czarina is bored and is playing a game with a pile of rocks. The pile initially contains 2015 rocks. At each round, if the pile has $N$ rocks, she removes $k$ of them, where $1 \leq k \leq N$, with each possible $k$ having equal probability. Alice Czarina continues until there are no more rocks in the pile. Let $... | -501 | We claim that $p=\frac{1}{5} \frac{6}{10} \frac{11}{15} \frac{16}{20} \cdots \frac{2006}{2010} \frac{2011}{2015}$. Let $p_{n}$ be the probability that, starting with $n$ rocks, the number of rocks left after each round is a multiple of 5. Indeed, using recursions we have $$p_{5 k}=\frac{p_{5 k-5}+p_{5 k-10}+\cdots+p_{5... | 0 | 8,192 | -1 | 8,192 |
Two joggers each run at their own constant speed and in opposite directions from one another around an oval track. They meet every 36 seconds. The first jogger completes one lap of the track in a time that, when measured in seconds, is a number (not necessarily an integer) between 80 and 100. The second jogger complete... | 3705 | 0.5 | 5,063.625 | 4,309 | 5,818.25 | |
If point $P$ is the golden section point of segment $AB$, and $AP < BP$, $BP=10$, then $AP=\_\_\_\_\_\_$. | 5\sqrt{5} - 5 | 0 | 4,329.3125 | -1 | 4,329.3125 | |
Determine the height of a tower from a 20-meter distant building, given that the angle of elevation to the top of the tower is 30° and the angle of depression to the base of the tower is 45°. | 20 \left(1 + \frac {\sqrt {3}}{3}\right) | 0 | 6,855.1875 | -1 | 6,855.1875 | |
Let \(A B C D E\) be a square pyramid of height \(\frac{1}{2}\) with a square base \(A B C D\) of side length \(A B = 12\) (so \(E\) is the vertex of the pyramid, and the foot of the altitude from \(E\) to \(A B C D\) is the center of square \(A B C D\)). The faces \(A D E\) and \(C D E\) meet at an acute angle of meas... | \frac{17}{144} | 0.5 | 7,003.4375 | 5,814.875 | 8,192 | |
A circle is inscribed in a triangle, and the points of tangency with the sides of the triangle are connected to each other. A new circle is inscribed in the resulting triangle, and the points of tangency with its sides are the vertices of a third triangle, which has the same angles as the original triangle. Find these ... | 60 | 0.6875 | 6,209.3125 | 5,308.090909 | 8,192 | |
In a round-robin tournament, every team played exactly once against each other. Each team won 9 games and lost 9 games; there were no ties. Find the number of sets of three teams {A, B, C} such that team A beat team B, team B beat team C, and team C beat team A. | 969 | 0 | 6,648.125 | -1 | 6,648.125 | |
A cube has edges of length 1 cm and has a dot marked in the centre of the top face. The cube is sitting on a flat table. The cube is rolled, without lifting or slipping, in one direction so that at least two of its vertices are always touching the table. The cube is rolled until the dot is again on the top face. Th... | \dfrac{1+\sqrt{5}}{2} | 0 | 7,748.5 | -1 | 7,748.5 | |
Let $\mathbf{a}$ and $\mathbf{b}$ be orthogonal vectors. If $\operatorname{proj}_{\mathbf{a}} \begin{pmatrix} 3 \\ -3 \end{pmatrix} = \begin{pmatrix} -\frac{3}{5} \\ -\frac{6}{5} \end{pmatrix},$ then find $\operatorname{proj}_{\mathbf{b}} \begin{pmatrix} 3 \\ -3 \end{pmatrix}.$ | \begin{pmatrix} \frac{18}{5} \\ -\frac{9}{5} \end{pmatrix} | 1 | 2,628.9375 | 2,628.9375 | -1 | |
A mason has bricks with dimensions $2\times5\times8$ and other bricks with dimensions $2\times3\times7$ . She also has a box with dimensions $10\times11\times14$ . The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box filling its entire volume and with no bricks sti... | 24 | 0.4375 | 7,938.8125 | 7,613.285714 | 8,192 | |
In the XY-plane, mark all the lattice points $(x, y)$ where $0 \leq y \leq 10$. For an integer polynomial of degree 20, what is the maximum number of these marked lattice points that can lie on the polynomial? | 20 | 0 | 8,192 | -1 | 8,192 | |
In a city, there are 10,000 bicycles with all possible numbers from 1 to 10,000. What is the probability that the number of the first bicycle encountered does not contain the digit 8? | 0.6561 | 0 | 7,372.6875 | -1 | 7,372.6875 | |
How many integers are there between $(11.1)^3$ and $(11.2)^3$? | 37 | 0.9375 | 4,597.5 | 4,641.733333 | 3,934 | |
Given the function $f(x) = \frac{1}{2}x^2 - 2ax + b\ln(x) + 2a^2$ achieves an extremum of $\frac{1}{2}$ at $x = 1$, find the value of $a+b$. | -1 | 0.9375 | 4,897.625 | 4,678 | 8,192 | |
For a positive integer $n>1$ , let $g(n)$ denote the largest positive proper divisor of $n$ and $f(n)=n-g(n)$ . For example, $g(10)=5, f(10)=5$ and $g(13)=1,f(13)=12$ . Let $N$ be the smallest positive integer such that $f(f(f(N)))=97$ . Find the largest integer not exceeding $\sqrt{N}$ | 19 | 0.125 | 8,097 | 7,432 | 8,192 | |
How many different three-letter sets of initials are possible using the letters $A$ through $J$, where no letter is repeated in any set? | 720 | 0.5 | 2,580.375 | 2,101.5 | 3,059.25 | |
Given the parabola $C$: $y^{2}=2px$ with the focus at $F(2,0)$, and points $P(m,0)$ and $Q(-m,n)$, a line $l$ passing through $P$ with a slope of $k$ (where $k\neq 0$) intersects the parabola $C$ at points $A$ and $B$.
(Ⅰ) For $m=k=2$, if $\vec{QA} \cdot \vec{QB} = 0$, find the value of $n$.
(Ⅱ) If $O$ represents the o... | \frac{32\sqrt{3}}{9} | 0 | 7,849.875 | -1 | 7,849.875 | |
In a Cartesian coordinate system, the points where both the x-coordinate and y-coordinate are integers are called lattice points. How many lattice points (x, y) satisfy the inequality \((|x|-1)^{2}+(|y|-1)^{2}<2\)? | 16 | 0.25 | 7,928.9375 | 7,139.75 | 8,192 | |
Let $[x]$ denote the greatest integer not exceeding the real number $x$. If
\[ A = \left[\frac{7}{8}\right] + \left[\frac{7^2}{8}\right] + \cdots + \left[\frac{7^{2019}}{8}\right] + \left[\frac{7^{2020}}{8}\right], \]
what is the remainder when $A$ is divided by 50? | 40 | 0.4375 | 7,194.0625 | 6,345 | 7,854.444444 | |
How many odd numbers between $100$ and $999$ have distinct digits? | 320 | 0.5 | 6,931.8125 | 5,671.625 | 8,192 | |
What is the smallest number of regular hexagons of side length 1 needed to completely cover a disc of radius 1 ? | 3 | First, we show that two hexagons do not suffice. Specifically, we claim that a hexagon covers less than half of the disc's boundary. First, a hexagon of side length 1 may be inscribed in a circle, and this covers just 6 points. Translating the hexagon vertically upward (regardless of its orientation) will cause it to n... | 0 | 8,009.6875 | -1 | 8,009.6875 |
Find the sum of distances from a point on the ellipse $7x^{2}+3y^{2}=21$ to its two foci. | 2\sqrt{7} | 1 | 3,361 | 3,361 | -1 | |
Determine the volume of the region enclosed by \[|x + y + z| + |x + y - z| + |x - y + z| + |-x + y + z| \le 6.\] | 18 | 0 | 8,192 | -1 | 8,192 | |
A square field is enclosed by a wooden fence, which is made of 10-meter-long boards placed horizontally. The height of the fence is four boards. It is known that the number of boards in the fence is equal to the area of the field, expressed in hectares. Determine the dimensions of the field. | 16000 | 0 | 6,799.875 | -1 | 6,799.875 | |
What is the area enclosed by the graph of $|2x| + |5y| = 10$? | 20 | 0.9375 | 3,540.5 | 3,230.4 | 8,192 | |
Given that the sum of the first $n$ terms of the geometric sequence $\{a\_n\}$ is $S\_n$, and it satisfies $S\_n=(\frac{1}{2})^{n}-1$, find the limit as $n$ approaches infinity of $(a\_1+a\_3+...+a\_2n-1)$ . | -\frac{2}{3} | 0.8125 | 4,752.6875 | 3,969.461538 | 8,146.666667 | |
Lily has a $300 \times 300$ grid of squares. She now removes $100 \times 100$ squares from each of the four corners and colors each of the remaining 50000 squares black and white. Given that no $2 \times 2$ square is colored in a checkerboard pattern, find the maximum possible number of (unordered) pairs of squares suc... | 49998 | First we show an upper bound. Define a grid point as a vertex of one of the squares in the figure. Construct a graph as follows. Place a vertex at each grid point and draw an edge between two adjacent points if that edge forms a black-white boundary. The condition of there being no $2 \times 2$ checkerboard is equivale... | 0 | 8,192 | -1 | 8,192 |
It takes $5$ seconds for a clock to strike $6$ o'clock beginning at $6:00$ o'clock precisely. If the strikings are uniformly spaced, how long, in seconds, does it take to strike $12$ o'clock? | 11 | 1. **Understanding the Problem:** The problem states that it takes 5 seconds for a clock to strike 6 times at 6 o'clock. We need to find out how long it takes for the clock to strike 12 times.
2. **Analyzing the 6 o'clock strikes:** When the clock strikes 6 times, there are actually 5 intervals between the strikes (th... | 1 | 1,599.5625 | 1,599.5625 | -1 |
A regular hexagon of side length $1$ is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. What is the area of the region bounded by these $6$ reflected arcs? | 3\sqrt{3}-\pi | 1. **Identify the Geometry of the Problem:**
A regular hexagon with side length $s = 1$ is inscribed in a circle. This means that each side of the hexagon subtends a $60^\circ$ angle at the center of the circle.
2. **Calculate the Radius of the Circle:**
The radius $r$ of the circle can be calculated using the f... | 0 | 8,192 | -1 | 8,192 |
John is cycling east at a speed of 8 miles per hour, while Bob is also cycling east at a speed of 12 miles per hour. If Bob starts 3 miles west of John, determine the time it will take for Bob to catch up to John. | 45 | 0.0625 | 1,118.4375 | 909 | 1,132.4 | |
The numbers 407 and 370 equal the sum of the cubes of their digits. For example, \( 4^3 = 64 \), \( 0^3 = 0 \), and \( 7^3 = 343 \). Adding 64, 0, and 343 gives you 407. Similarly, the cube of 3 (27), added to the cube of 7 (343), gives 370.
Could you find a number, not containing zero and having the same property? Of... | 153 | 0.375 | 7,367.1875 | 6,132.166667 | 8,108.2 | |
The increasing sequence $2,3,5,6,7,10,11,\ldots$ consists of all positive integers that are neither the square nor the cube of a positive integer. Find the 500th term of this sequence. | 528 | Because there aren't that many perfect squares or cubes, let's look for the smallest perfect square greater than $500$. This happens to be $23^2=529$. Notice that there are $23$ squares and $8$ cubes less than or equal to $529$, but $1$ and $2^6$ are both squares and cubes. Thus, there are $529-23-8+2=500$ numbers in o... | 0.375 | 6,860.375 | 6,004.5 | 7,373.9 |
An $10 \times 25$ rectangle is divided into two congruent polygons, and these polygons are rearranged to form a rectangle again. Determine the length of the smaller side of the resulting rectangle. | 10 | 0 | 7,631.1875 | -1 | 7,631.1875 | |
Find $\csc 225^\circ.$ | -\sqrt{2} | 1 | 1,519 | 1,519 | -1 | |
Compute $$\sum_{\substack{a+b+c=12 \\ a \geq 6, b, c \geq 0}} \frac{a!}{b!c!(a-b-c)!}$$ where the sum runs over all triples of nonnegative integers $(a, b, c)$ such that $a+b+c=12$ and $a \geq 6$. | 2731 | We tile a $1 \times 12$ board with red $1 \times 1$ pieces, blue $1 \times 2$ pieces, and green $1 \times 2$ pieces. Suppose we use $a$ total pieces, $b$ blue pieces, and $c$ green pieces. Then we must have $a+b+c=12$, and the number of ways to order the pieces is $$\binom{a}{b, c, a-b-c}$$ Thus, the desired sum is the... | 0.0625 | 8,118.125 | 7,010 | 8,192 |
If $(x,y)$ is a solution to the system
\begin{align*}
xy &= 6, \\
x^2 y + xy^2 + x + y &= 63,
\end{align*}find $x^2 + y^2.$ | 69 | 1 | 2,117.0625 | 2,117.0625 | -1 | |
How many of the numbers from the set $\{1, 2, 3, \ldots, 100\}$ have a perfect square factor other than one? | 41 | 0 | 7,607.9375 | -1 | 7,607.9375 | |
A uniform tetrahedron has its four faces numbered with 1, 2, 3, and 4. It is randomly thrown twice, and the numbers on the bottom face of the tetrahedron are $x_1$ and $x_2$, respectively. Let $t = (x_{1}-3)^{2}+(x_{2}-3)^{2}$.
(1) Calculate the probabilities of $t$ reaching its maximum and minimum values, respective... | \frac{5}{16} | 0.0625 | 7,053.1875 | 5,170 | 7,178.733333 | |
Suppose \(AB = 1\), and the slanted segments form an angle of \(45^\circ\) with \(AB\). There are \(n\) vertices above \(AB\).
What is the length of the broken line? | \sqrt{2} | 0.4375 | 7,164.125 | 5,842.571429 | 8,192 | |
A trapezoid has one base equal to twice its height, $x$, and the other base is three times as long as the height. Write the expression for the area of the trapezoid as a common fraction in terms of the height $x$. | \dfrac{5x^2}{2} | 0.3125 | 1,454.9375 | 1,441.4 | 1,461.090909 | |
In the Westeros Empire, there were 1000 cities and 2017 roads (each road connects some two cities). From each city, it was possible to travel to any other city. One day, an evil wizard cursed $N$ roads, making them impassable. As a result, 7 kingdoms formed, such that within each kingdom, it is possible to travel from ... | 1024 | 1 | 3,368.5625 | 3,368.5625 | -1 | |
If an internal point of an $n$-sided prism is connected to all its vertices, $n$ quadrangular pyramids with a common vertex at this point will be formed, with their bases being the lateral faces of the prism. Find the ratio of the sum of the volumes of these pyramids to the volume of the given prism. | \frac{2}{3} | 0 | 8,161.5625 | -1 | 8,161.5625 | |
Given two distinct numbers \(a\) and \(b\) such that \(\frac{a}{b} + a = \frac{b}{a} + b\), find \(\frac{1}{a} + \frac{1}{b}\). | -1 | 1 | 3,013.3125 | 3,013.3125 | -1 | |
In triangle $ABC$, where $AB = \sqrt{34}$ and $AC = 5$, the angle $B$ is $90^\circ$. Calculate $\tan A$. | \frac{3}{5} | 0.0625 | 7,767.125 | 7,200 | 7,804.933333 | |
Of the natural numbers greater than 1000 that are composed of the digits $0, 1, 2$ (where each digit can be used any number of times or not at all), in ascending order, what is the position of 2010? | 30 | 0 | 7,829.5 | -1 | 7,829.5 | |
Out of three hundred eleventh-grade students, 77% received excellent and good grades on the first exam, 71% on the second exam, and 61% on the third exam. What is the minimum number of participants who received excellent and good grades on all three exams? | 27 | 0.5625 | 6,342.6875 | 4,908.555556 | 8,186.571429 | |
The length of a rectangle is increased by $25\%$, but the width of the rectangle is decreased to keep the area of the rectangle unchanged. By what percent was the rectangle's width decreased? | 20 | 1 | 1,327.9375 | 1,327.9375 | -1 | |
Compute the limit of the function:
\[
\lim _{x \rightarrow \frac{\pi}{4}}(\tan x)^{1 / \cos \left(\frac{3 \pi}{4}-x\right)}
\] | e^2 | 0.875 | 5,465.875 | 5,076.428571 | 8,192 | |
Let $Q$ be the product of the first $150$ positive odd integers. Find the largest integer $k'$ such that $Q$ is divisible by $3^{k'}$. | 76 | 0.4375 | 7,032.1875 | 5,605.714286 | 8,141.666667 | |
Let $ABC$ be a triangle with incenter $I$ and $AB = 1400$ , $AC = 1800$ , $BC = 2014$ . The circle centered at $I$ passing through $A$ intersects line $BC$ at two points $X$ and $Y$ . Compute the length $XY$ .
*Proposed by Evan Chen* | 1186 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Points $X$, $Y$, $Z$, and $W$ lie on a straight line in that order. If $XY=3$ units, $YZ=4$ units, and $XW=20$ units, what is the ratio of $XZ$ to $YW$? Additionally, find the ratio of $YZ$ to $XW$. | \frac{1}{5} | 1 | 1,832.375 | 1,832.375 | -1 | |
Two players take turns placing Xs and Os in the cells of a $9 \times 9$ square (the first player places Xs, and their opponent places Os). At the end of the game, the number of rows and columns where there are more Xs than Os are counted as points for the first player. The number of rows and columns where there are mor... | 10 | 0.0625 | 8,063.3125 | 7,603 | 8,094 | |
In isosceles $\triangle ABC$, $\angle BAC=120^{\circ}$, $AB=AC=2$, $\overrightarrow{BC}=2 \overrightarrow{BD}$, $\overrightarrow{AC}=3 \overrightarrow{AE}$. Calculate the value of $\overrightarrow{AD}\cdot \overrightarrow{BE}$. | -\dfrac {2}{3} | 1 | 4,985.75 | 4,985.75 | -1 | |
The entries in a $3 \times 3$ array include all the digits from $1$ through $9$, arranged so that the entries in every row and column are in increasing order. How many such arrays are there? | 42 | To solve this problem, we need to consider the constraints given by the increasing order in rows and columns. We will analyze the possible placements of the numbers, focusing on the central number and leveraging symmetry.
#### Step 1: Analyze the constraints
Each number from 1 to 9 must be used exactly once in the $3 ... | 0.5 | 4,872.5625 | 4,610.875 | 5,134.25 |
If $a<b<c<d<e$ are consecutive positive integers such that $b+c+d$ is a perfect square and $a+b+c+d+e$ is a perfect cube, what is the smallest possible value of $c$? | 675 | Let the numbers be $a,a+1,a+2,a+3,a+4.$ When then know $3a+6$ is a perfect cube and $5a+10$ is perfect cube. Since $5a+10$ is divisible by $5$ we know that $5a+10 = (5k)^3$ since otherwise we get a contradiction. This means $a = 25k^3 - 2$ in which plugging into the other expression we know $3(25k^3 - 2) + 6 = 75k^3$ i... | 0.625 | 6,742.75 | 5,873.2 | 8,192 |
In right triangle $ABC$ with $\angle A = 90^\circ$, $AC = 3$, $AB = 4$, and $BC = 5$, point $D$ is on side $BC$. If the perimeters of $\triangle ACD$ and $\triangle ABD$ are equal, then what is the area of $\triangle ABD$? | $\frac{12}{5}$ | 0 | 3,407.125 | -1 | 3,407.125 | |
There are 3 teachers who have all assigned homework. Determine the number of possible situations where 4 students are doing homework at the same time. | 3^{4} | 0 | 5,092.625 | -1 | 5,092.625 | |
Given that in a class of 36 students, more than half purchased notebooks from a store where each notebook had the same price in cents greater than the number of notebooks bought by each student, and the total cost for the notebooks was 990 cents, calculate the price of each notebook in cents. | 15 | 0 | 4,865.3125 | -1 | 4,865.3125 | |
Given a sufficiently large positive integer \( n \), which can be divided by all the integers from 1 to 250 except for two consecutive integers \( k \) and \( k+1 \), find \( k \). | 127 | 0 | 8,192 | -1 | 8,192 | |
The distance from the point where a diameter of a circle intersects a chord of length 18 cm to the center of the circle is 7 cm. This point divides the chord in the ratio 2:1. Find the radius.
Given:
\[ AB = 18 \, \text{cm}, \, EO = 7 \, \text{cm}, \, AE = 2 \, BE \]
Find the radius \( R \). | 11 | 0.3125 | 7,264.375 | 5,223.6 | 8,192 | |
The times between $7$ and $8$ o'clock, correct to the nearest minute, when the hands of a clock will form an angle of $84^{\circ}$ are: | 7: 23 and 7: 53 | 1. **Initial Position at 7:00**:
- The hour hand at 7:00 is at $\frac{7}{12} \times 360^\circ = 210^\circ$ from the 12 o'clock position.
- The minute hand at 7:00 is at $0^\circ$ (at the 12 o'clock position).
2. **Relative Position of Minute Hand to Hour Hand at 7:00**:
- The minute hand is $360^\circ - 210^... | 0 | 7,752.3125 | -1 | 7,752.3125 |
If $f(x)=\frac{x+4}{x^2+ax+b}$, and $f(x)$ has two vertical asymptotes at $x=1$ and $x=-2$, find the sum of $a$ and $b$. | -1 | 1 | 1,301.125 | 1,301.125 | -1 | |
If $f(x)$ is a monic quartic polynomial such that $f(-2)=-4$, $f(1)=-1$, $f(3)=-9$, and $f(5)=-25$, find $f(0)$. | -30 | 0.875 | 3,985.125 | 3,384.142857 | 8,192 | |
Let $f(x)=e^{x}$, and $f(x)=g(x)-h(x)$, where $g(x)$ is an even function, and $h(x)$ is an odd function. If there exists a real number $m$ such that the inequality $mg(x)+h(x)\geqslant 0$ holds for $x\in [-1,1]$, determine the minimum value of $m$. | \dfrac{e^{2}-1}{e^{2}+1} | 0 | 7,611.8125 | -1 | 7,611.8125 | |
A Senate committee has 5 Democrats and 5 Republicans. Assuming all politicians are distinguishable, in how many ways can they sit around a circular table without restrictions? (Two seatings are considered the same if one is a rotation of the other.) | 362,\!880 | 0 | 3,662.8125 | -1 | 3,662.8125 | |
Three circles, each of radius $3$, are drawn with centers at $(14, 92)$, $(17, 76)$, and $(19, 84)$. A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the ab... | 24 | Notice that any line that passes through the bottom circle's center cuts it in half, so all we really care about are the top two circles. Suppose $\ell$ is the desired line. Draw lines $\ell_1$ and $\ell_2$ both parallel to $\ell$ such that $\ell_1$ passes through $(14,92)$ and $\ell_2$ passes through $(19,84)$. Clearl... | 0.25 | 7,138.6875 | 4,861 | 7,897.916667 |
A sequence of real numbers $a_{0}, a_{1}, \ldots$ is said to be good if the following three conditions hold. (i) The value of $a_{0}$ is a positive integer. (ii) For each non-negative integer $i$ we have $a_{i+1}=2 a_{i}+1$ or $a_{i+1}=\frac{a_{i}}{a_{i}+2}$. (iii) There exists a positive integer $k$ such that $a_{k}=2... | 60 | Note that $$ a_{i+1}+1=2\left(a_{i}+1\right) \text { or } a_{i+1}+1=\frac{a_{i}+a_{i}+2}{a_{i}+2}=\frac{2\left(a_{i}+1\right)}{a_{i}+2} $$ Hence $$ \frac{1}{a_{i+1}+1}=\frac{1}{2} \cdot \frac{1}{a_{i}+1} \text { or } \frac{1}{a_{i+1}+1}=\frac{a_{i}+2}{2\left(a_{i}+1\right)}=\frac{1}{2} \cdot \frac{1}{a_{i}+1}+\frac{1}{... | 0 | 8,192 | -1 | 8,192 |
Pascal has a triangle. In the $n$th row, there are $n+1$ numbers $a_{n, 0}, a_{n, 1}, a_{n, 2}, \ldots, a_{n, n}$ where $a_{n, 0}=a_{n, n}=1$. For all $1 \leq k \leq n-1, a_{n, k}=a_{n-1, k}-a_{n-1, k-1}$. What is the sum of all numbers in the 2018th row? | 2 | In general, the sum of the numbers on the $n$th row will be $$\sum_{k=0}^{n} a_{n, k}=a_{n, 0}+\sum_{k=1}^{n-1}\left(a_{n-1, k}-a_{n-1, k-1}\right)+a_{n, n}=a_{n, 0}+\left(a_{n-1, n-1}-a_{n-1,0}\right)+a_{n, n}=2$$ | 0.8125 | 4,166.9375 | 3,238.076923 | 8,192 |
Given real numbers \( x, y \in (1,+\infty) \) such that \( xy - 2x - y + 1 = 0 \), find the minimum value of \( \frac{3}{2} x^{2} + y^{2} \). | 15 | 0.625 | 6,368.5625 | 5,274.5 | 8,192 | |
Let $\mathcal{C}$ be the hyperbola $y^{2}-x^{2}=1$. Given a point $P_{0}$ on the $x$-axis, we construct a sequence of points $\left(P_{n}\right)$ on the $x$-axis in the following manner: let $\ell_{n}$ be the line with slope 1 passing through $P_{n}$, then $P_{n+1}$ is the orthogonal projection of the point of intersec... | 254 | Let $P_{n}=\left(x_{n}, 0\right)$. Then the $\ell_{n}$ meet $\mathcal{C}$ at $\left(x_{n+1}, x_{n+1}-x_{n}\right)$. Since this point lies on the hyperbola, we have $\left(x_{n+1}-x_{n}\right)^{2}-x_{n+1}^{2}=1$. Rearranging this equation gives $$x_{n+1}=\frac{x_{n}^{2}-1}{2x_{n}}$$ Choose a $\theta_{0} \in(0, \pi)$ wit... | 0.1875 | 8,043 | 7,881.333333 | 8,080.307692 |
According to census statistics, the probability of a woman of childbearing age giving birth to a boy or a girl is equal. If a second child is allowed, calculate the probability that a woman of childbearing age will have two girls. | \frac {1}{4} | 0.75 | 1,870.9375 | 1,788.666667 | 2,117.75 | |
Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be three mutually orthogonal unit vectors, such that
\[\mathbf{a} = p (\mathbf{a} \times \mathbf{b}) + q (\mathbf{b} \times \mathbf{c}) + r (\mathbf{c} \times \mathbf{a})\]for some scalars $p,$ $q,$ and $r,$ and $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 1.$ Fin... | 1 | 0.625 | 6,976.0625 | 6,246.5 | 8,192 | |
Mr. Patrick teaches math to $15$ students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was $80$. After he graded Payton's test, the test average became $81$. What was Payton's score on the test? | 95 | 1. **Calculate the total score of the first 14 students**:
Given that the average score of the first 14 students is 80, we can find the total score of these students by multiplying the average by the number of students:
\[
\text{Total score of 14 students} = 14 \times 80 = 1120.
\]
2. **Calculate the new ... | 1 | 1,366.75 | 1,366.75 | -1 |
Find the ordered pair $(x,y)$ if
\begin{align*}
x+y&=(5-x)+(5-y),\\
x-y&=(x-1)+(y-1).
\end{align*} | (4,1) | 0.625 | 692.8125 | 716.1 | 654 | |
Some expressions containing square roots can be written as the square of another expression, such as $3+2\sqrt{2}={(1+\sqrt{2})}^{2}$. Let $a+b\sqrt{2}=(m+n\sqrt{2})^{2}$ (where $a$, $b$, $m$, $n$ are all positive integers), then we have $a+b\sqrt{2}=m^{2}+2n^{2}+2mn\sqrt{2}$, so $a=m^{2}+2m^{2}$, $b=2mn$. This method ... | \frac{13}{6}+\frac{\sqrt{7}}{6} | 0 | 4,178.5625 | -1 | 4,178.5625 | |
Given the function $y=x^2+10x+21$, what is the least possible value of $y$? | -4 | 1 | 1,858.4375 | 1,858.4375 | -1 | |
Sandhya must save 35 files onto disks, each with 1.44 MB space. 5 of the files take up 0.6 MB, 18 of the files take up 0.5 MB, and the rest take up 0.3 MB. Files cannot be split across disks. Calculate the smallest number of disks needed to store all 35 files. | 12 | 0 | 8,192 | -1 | 8,192 | |
Given Jones traveled 100 miles on his first trip and 500 miles on a subsequent trip at a speed four times as fast, compare his new time to the old time. | 1.25 | 0.75 | 1,910.25 | 1,755.5 | 2,374.5 | |
Given that $\tan \alpha = 2$, where $\alpha$ is an angle in the first quadrant, find the value of $\sin 2\alpha + \cos \alpha$. | \dfrac{4 + \sqrt{5}}{5} | 0 | 2,837.875 | -1 | 2,837.875 | |
Complex numbers \( a \), \( b \), and \( c \) form an equilateral triangle with side length 24 in the complex plane. If \( |a + b + c| = 48 \), find \( |ab + ac + bc| \). | 768 | 0.1875 | 8,084.6875 | 7,619.666667 | 8,192 | |
Two sides of a right triangle have the lengths 8 and 15. What is the product of the possible lengths of the third side? Express the product as a decimal rounded to the nearest tenth. | 215.7 | 0.8125 | 4,046.8125 | 3,767.307692 | 5,258 | |
What is the sum of all integer values $n$ for which $\binom{20}{n}+\binom{20}{10}=\binom{21}{11}$? | 20 | 0.75 | 4,559.375 | 3,348.5 | 8,192 |
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