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 |
|---|---|---|---|---|---|---|
Alicia starts a sequence with $m=3$. What is the fifth term of her sequence following the algorithm: Step 1: Alicia writes down the number $m$ as the first term. Step 2: If $m$ is even, Alicia sets $n= rac{1}{2} m$. If $m$ is odd, Alicia sets $n=m+1$. Step 3: Alicia writes down the number $m+n+1$ as the next term. Step... | 43 | We follow Alicia's algorithm carefully: Step 1: Alicia writes down $m=3$ as the first term. Step 2: Since $m=3$ is odd, Alicia sets $n=m+1=4$. Step 3: Alicia writes down $m+n+1=8$ as the second term. Step 4: Alicia sets $m=8$. Step 2: Since $m=8$ is even, Alicia sets $n= rac{1}{2} m=4$. Step 3: Alicia writes down $m+n+... | 1 | 2,218.0625 | 2,218.0625 | -1 |
The area of an equilateral triangle inscribed in a circle is 81 cm². Find the radius of the circle. | 6 \sqrt[4]{3} | 0.875 | 6,327.6875 | 6,061.357143 | 8,192 | |
One US dollar is the equivalent of 7 Namibian dollars. One US dollar is also the equivalent of 6 Chinese yuan. If a painting costs 105 Namibian dollars, how much would it cost in Chinese yuan? | 90 | 1 | 1,674.625 | 1,674.625 | -1 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. Given vectors $\overrightarrow{m}=(\sin B+\sin C, \sin A-\sin B)$ and $\overrightarrow{n}=(\sin B-\sin C, \sin(B+C))$, and $\overrightarrow{m} \perp \overrightarrow{n}$.
(1) Find the magnitude of angle $C$... | \frac{4\sqrt{3}-3}{10} | 0 | 5,652.375 | -1 | 5,652.375 | |
Given the function $$f(x)=\sin^{2}x+ \sqrt {3}\sin x\cos x+2\cos^{2}x，x∈R$$.
(I) Find the smallest positive period and the interval where the function is monotonically increasing;
(II) Find the maximum value of the function on the interval $$[- \frac {π}{3}， \frac {π}{12}]$$. | \frac { \sqrt {3}+3}{2} | 0 | 6,921 | -1 | 6,921 | |
What is the largest four-digit negative integer congruent to $2 \pmod{25}$? | -1023 | 0.1875 | 5,680.5625 | 4,663.333333 | 5,915.307692 | |
Let $\mathcal{R}$ be the region consisting of the set of points in the coordinate plane that satisfy both $|8 - x| + y \le 10$ and $3y - x \ge 15$. When $\mathcal{R}$ is revolved around the line whose equation is $3y - x = 15$, the volume of the resulting solid is $\frac {m\pi}{n\sqrt {p}}$, where $m$, $n$, and $p$ are... | 365 | [asy]size(280); import graph; real min = 2, max = 12; pen dark = linewidth(1); real P(real x) { return x/3 + 5; } real Q(real x) { return 10 - abs(x - 8); } path p = (2,P(2))--(8,P(8))--(12,P(12)), q = (2,Q(2))--(12,Q(12)); pair A = (8,10), B = (4.5,6.5), C= (9.75,8.25), F=foot(A,B,C), G=2*F-A; fill(A--B--C--cycle,rgb(... | 0.1875 | 7,628.5 | 6,743.666667 | 7,832.692308 |
In a square \(ABCD\), let \(P\) be a point on the side \(BC\) such that \(BP = 3PC\) and \(Q\) be the midpoint of \(CD\). If the area of the triangle \(PCQ\) is 5, what is the area of triangle \(QDA\)? | 20 | 0.8125 | 4,166.875 | 3,500.692308 | 7,053.666667 | |
The function $f(x)$ satisfies
\[f(x + f(x)) = 4f(x)\]for all $x,$ and $f(1) = 4.$ Find $f(21).$ | 64 | 0.875 | 4,857.625 | 4,595.642857 | 6,691.5 | |
For any positive integer $n$, the value of $n!$ is the product of the first $n$ positive integers. For example, $4! = 4\cdot 3\cdot 2\cdot 1 =24$. What is the greatest common divisor of $5!$ and $7!$ ? | 120 | 1 | 2,397.125 | 2,397.125 | -1 | |
Let $A, B, C$ be points in that order along a line, such that $A B=20$ and $B C=18$. Let $\omega$ be a circle of nonzero radius centered at $B$, and let $\ell_{1}$ and $\ell_{2}$ be tangents to $\omega$ through $A$ and $C$, respectively. Let $K$ be the intersection of $\ell_{1}$ and $\ell_{2}$. Let $X$ lie on segment $... | 35 | Note that $B$ is the $K$-excenter of $K X Y$, so $X B$ is the angle bisector of $\angle A K Y$. As $A B$ and $X Y$ are parallel, $\angle X A B+2 \angle A X B=180^{\circ}$, so $\angle X B A=180^{\circ}-\angle A X B-\angle X A B$. This means that $A X B$ is isosceles with $A X=A B=20$. Similarly, $Y C=B C=18$. As $K X Y$... | 0 | 8,192 | -1 | 8,192 |
What is the total number of digits used when the first 4500 positive even integers are written? | 19444 | 0 | 6,386.9375 | -1 | 6,386.9375 | |
Simplify
\[\frac{\cos x}{1 + \sin x} + \frac{1 + \sin x}{\cos x}.\] | 2 \sec x | 1 | 2,926.5625 | 2,926.5625 | -1 | |
Suppose that $f(x)$ is a linear function satisfying the equation $f(x) = 4f^{-1}(x) + 6$. Given that $f(1) = 4$, find $f(2)$. | 6 | 1 | 3,129.6875 | 3,129.6875 | -1 | |
Triangle $PQR$ has sides of length 7, 24, and 25 units, and triangle $XYZ$ has sides of length 9, 40, and 41 units. Calculate the ratio of the area of triangle $PQR$ to the area of triangle $XYZ$, and also find the ratio of their perimeters. | \frac{28}{45} | 1 | 2,034 | 2,034 | -1 | |
Given that $\sin(α + \sqrt{2}\cos(α) = \sqrt{3})$, find the value of $\tan(α)$.
A) $\frac{\sqrt{2}}{2}$
B) $\sqrt{2}$
C) $-\frac{\sqrt{2}}{2}$
D) $-\sqrt{2}$ | \frac{\sqrt{2}}{2} | 0 | 6,241 | -1 | 6,241 | |
For which values of the parameter \( a \) does the equation \( x^{3}-11x^{2}+ax-8=0 \) have three distinct real roots that form a geometric progression? | 22 | 0.8125 | 5,031.625 | 4,611.846154 | 6,850.666667 | |
The journey from Petya's home to school takes him 20 minutes. One day, on his way to school, Petya remembered that he had forgotten a pen at home. If he continues his journey at the same speed, he will arrive at school 3 minutes before the bell rings. However, if he returns home for the pen and then goes to school at t... | \frac{1}{4} | 0 | 8,124.6875 | -1 | 8,124.6875 | |
Add $4_6 + 14_6.$ Express your answer in base $6.$ | 22_6 | 1 | 2,261 | 2,261 | -1 | |
The ratio of the length to the width of a rectangle is $5$ : $2$. If the rectangle has a diagonal of length $13$ units, find the constant $k$ such that the area of the rectangle can be expressed as $kd^2$. | \frac{10}{29} | 0.875 | 2,616.3125 | 2,194.428571 | 5,569.5 | |
Let $f(x) = x|x|.$ Find $f^{-1}(4) + f^{-1}(-100).$ | -8 | 1 | 2,119.125 | 2,119.125 | -1 | |
Define the sequence $ (a_p)_{p\ge0}$ as follows: $ a_p\equal{}\displaystyle\frac{\binom p0}{2\cdot 4}\minus{}\frac{\binom p1}{3\cdot5}\plus{}\frac{\binom p2}{4\cdot6}\minus{}\ldots\plus{}(\minus{}1)^p\cdot\frac{\binom pp}{(p\plus{}2)(p\plus{}4)}$ .
Find $ \lim_{n\to\infty}(a_0\plus{}a_1\plus{}\ldots\plus{}a_n)$ . | 1/3 | 0.1875 | 7,295.375 | 6,066 | 7,579.076923 | |
Crestview's school colors are purple and gold. The students are designing a flag using three solid-colored horizontal stripes, as shown. Using one or both of the school colors, how many different flags are possible if adjacent stripes may be the same color?
[asy]
size(75);
draw((0,0)--(0,12));
dot((0,12));
draw((0,12)... | 8 | 0.8125 | 4,338.9375 | 4,199.769231 | 4,942 | |
Star writes down the whole numbers $1$ through $40$. Emilio copies Star's numbers, but he replaces each occurrence of the digit $3$ by the digit $2$. Calculate the difference between Star's sum of the numbers and Emilio's sum of the numbers. | 104 | 0 | 8,192 | -1 | 8,192 | |
From the digits $1$, $2$, $3$, $4$, form a four-digit number with the first digit being $1$, and having exactly two identical digits in the number. How many such four-digit numbers are there? | 36 | 0.1875 | 7,288.375 | 6,163 | 7,548.076923 | |
Given that the wavelength of each light quantum is approximately $688$ nanometers and $1$ nanometer is equal to $0.000000001$ meters, express the wavelength of each light quantum in scientific notation. | 6.88\times 10^{-7} | 0.3125 | 424.4375 | 408.8 | 431.545455 | |
Thirty people are divided into three groups (I, II, and III) with 10 people in each group. How many different compositions of groups can there be? | \frac{30!}{(10!)^3} | 0.875 | 2,448.8125 | 2,457.571429 | 2,387.5 | |
If the first digit of a four-digit number, which is a perfect square, is decreased by 3, and the last digit is increased by 3, it also results in a perfect square. Find this number. | 4761 | 0.3125 | 7,500.3125 | 5,978.6 | 8,192 | |
ABCDEF is a six-digit number. All of its digits are different and arranged in ascending order from left to right. This number is a perfect square.
Determine what this number is. | 134689 | 0 | 8,192 | -1 | 8,192 | |
Seven students are standing in a row for a graduation photo. Among them, student A must stand in the middle, and students B and C must stand together. How many different arrangements are there? | 192 | 0.1875 | 6,762.25 | 6,393.333333 | 6,847.384615 | |
When $1 + 7 + 7^2 + \cdots + 7^{2004}$ is divided by $1000$, a remainder of $N$ is obtained. Determine the value of $N$.
| 801 | 0.3125 | 7,512.0625 | 6,016.2 | 8,192 | |
A school band found they could arrange themselves in rows of 6, 7, or 8 with no one left over. What is the minimum number of students in the band? | 168 | 1 | 2,647.125 | 2,647.125 | -1 | |
A satellite is launched vertically from the Earth's pole with the first cosmic velocity. What is the maximum distance the satellite will reach from the Earth's surface? (The gravitational acceleration at the Earth's surface is $g = 10 \, \mathrm{m/s^2}$, and the Earth's radius is $R = 6400 \, \mathrm{km}$). | 6400 | 0 | 8,192 | -1 | 8,192 | |
Calculate the sum:
\[
\left(\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{2016}\right)+\left(\frac{2}{3}+\frac{2}{4}+\cdots+\frac{2}{2016}\right)+\left(\frac{3}{4}+\frac{3}{5}+\cdots+\frac{3}{2016}\right)+\cdots+\left(\frac{2014}{2015}+\frac{2014}{2016}\right)+\frac{2015}{2016}.
\] | 1015560 | 0.4375 | 6,465.3125 | 4,245.285714 | 8,192 | |
A standard deck of 52 cards is randomly arranged. What is the probability that the top three cards are $\spadesuit$, $\heartsuit$, and $\spadesuit$ in that sequence? | \dfrac{78}{5100} | 0 | 5,563.125 | -1 | 5,563.125 | |
Let $a,$ $b,$ and $c$ be nonnegative numbers such that $a^2 + b^2 + c^2 = 1.$ Find the maximum value of
\[3ab \sqrt{2} + 6bc.\] | 4.5 | 0 | 7,052.0625 | -1 | 7,052.0625 | |
There are 4 children and 2 coaches in the Chess Interest Group of Hongxing Primary School. The ages of the 4 children differ by 2 years sequentially, and the ages of the 2 coaches differ by 2 years. The sum of the squares of the ages of the 6 people is 2796 years. What is the sum of their ages in years? | 106 | 0.8125 | 5,460.375 | 4,830 | 8,192 | |
In a game show, Bob is faced with 7 doors, 2 of which hide prizes. After he chooses a door, the host opens three other doors, of which one is hiding a prize. Bob chooses to switch to another door. What is the probability that his new door is hiding a prize? | \frac{5}{21} | If Bob initially chooses a door with a prize, then he will not find a prize by switching. With probability $5 / 7$ his original door does not hide the prize. After the host opens the three doors, the remaining three doors have equal probability of hiding the prize. Therefore, the probability that Bob finds the prize is... | 0.0625 | 7,420.6875 | 7,635 | 7,406.4 |
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 integers, find $m+n$.
| 164 | 0.125 | 8,047.375 | 7,035 | 8,192 | |
In triangle $ABC$, $\angle C = 90^\circ$, $AC = 4$, and $AB = \sqrt{41}$. What is $\tan B$? | \frac{4}{5} | 1 | 1,885.25 | 1,885.25 | -1 | |
For an even integer positive integer $n$ Kevin has a tape of length $4 n$ with marks at $-2 n,-2 n+1, \ldots, 2 n-1,2 n$. He then randomly picks $n$ points in the set $-n,-n+1,-n+2, \ldots, n-1, n$, and places a stone on each of these points. We call a stone 'stuck' if it is on $2 n$ or $-2 n$, or either all the points... | \frac{1}{n-1} | After we have selected the positions of the initial $n$ stones, we number their positions: $a_{1}<a_{2}<\ldots<a_{n}$. The conditions on how we move the stones imply that the expected value of $\left(a_{i}-a_{j}\right)$ after $t$ minutes is still equal to $a_{i}-a_{j}$. In addition, if $b_{i}$ is the final position of ... | 0 | 8,110.1875 | -1 | 8,110.1875 |
Evaluate the expression $\sqrt{7+4\sqrt{3}} - \sqrt{7-4\sqrt{3}}$. | 2\sqrt{3} | 1 | 2,139.625 | 2,139.625 | -1 | |
The dilation, centered at $2 + 3i,$ with scale factor 3, takes $-1 - i$ to which complex number? | -7 - 9i | 0.9375 | 3,042 | 2,698.666667 | 8,192 | |
A subset of a student group is called an [i]ideal company[/i] if
1) in this subset, all girls are liked by all young men,
2) no one can be added to this subset without violating condition $1$.
In a certain group, $9$ female students and $15$ students study. Warden of the group made a list of all kinds of ideal companie... | 512 |
To solve this problem, we need to understand the concept of an "ideal company" as defined by the question. An ideal company is a subset of the student group where all female students in the subset are liked by all male students within the same subset, and no additional student can be added to this group without breaki... | 0 | 8,075.5625 | -1 | 8,075.5625 |
Expanding the expression \((1+\sqrt{11})^{212}\) using the binomial theorem, we obtain terms of the form \(C_{212}^{k}(\sqrt{11})^{k}\). Find the value of \(k\) for which this term takes on the greatest value. | 163 | 0.75 | 6,503.8125 | 6,175.75 | 7,488 | |
A market survey shows that the price $f(t)$ and sales volume $g(t)$ of a particular product in Oriental Department Store over the past month (calculated based on 30 days) approximately satisfy the functions $f(t)=100(1+ \frac {1}{t})$ and $g(t)= \begin{cases} 100+t & 1\leqslant t < 25,t\in N \\ 150-t & 25\leqslant t\le... | 12100 | 0.1875 | 7,212.9375 | 7,469.666667 | 7,153.692308 | |
How many integer pairs $(x,y)$ are there such that \[0\leq x < 165, \quad 0\leq y < 165 \text{ and } y^2\equiv x^3+x \pmod {165}?\] | 99 | 0.5 | 5,881.5625 | 5,359 | 6,404.125 | |
How many positive three-digit integers with each digit greater than 4 are divisible by 6? | 16 | 0.3125 | 7,581.75 | 6,704.4 | 7,980.545455 | |
If $f(x)=\frac{x^4+x^2}{x+1}$, then $f(i)$, where $i=\sqrt{-1}$, is equal to | 0 | To find $f(i)$ for the function $f(x) = \frac{x^4 + x^2}{x + 1}$, we substitute $i$ for $x$:
1. **Calculate the numerator**:
\[
i^4 + i^2 = 1 - 1 = 0
\]
Here, $i^4 = (i^2)^2 = (-1)^2 = 1$ and $i^2 = -1$.
2. **Check the denominator**:
\[
i + 1 \neq 0
\]
The denominator $i + 1$ is not zero beca... | 1 | 3,156.5625 | 3,156.5625 | -1 |
In a rectangular coordinate system, a point whose coordinates are both integers is called a lattice point. How many lattice points \((x, y)\) satisfy the inequality \((|x|-1)^{2} + (|y|-1)^{2} < 2\)? | 16 | 0.4375 | 7,809.8125 | 7,318.428571 | 8,192 | |
How many five-digit natural numbers are divisible by 9, where the last digit is greater than the second last digit by 2? | 800 | 0.0625 | 7,946.625 | 6,127 | 8,067.933333 | |
In $\triangle ABC, D$ and $E$ are the midpoints of $BC$ and $CA$, respectively. $AD$ and $BE$ intersect at $G$. Given that $GEC$D is cyclic, $AB=41$, and $AC=31$, compute $BC$. | 49 | By Power of a Point, $$\frac{2}{3}AD^{2}=AD \cdot AG=AE \cdot AC=\frac{1}{2} \cdot 31^{2}$$ so $AD^{2}=\frac{3}{4} \cdot 31^{2}$. The median length formula yields $$AD^{2}=\frac{1}{4}\left(2AB^{2}+2AC^{2}-BC^{2}\right)$$ whence $$BC=\sqrt{2AB^{2}+2AC^{2}-4AD^{2}}=\sqrt{2 \cdot 41^{2}+2 \cdot 31^{2}-3 \cdot 31^{2}}=49$$ | 0 | 8,192 | -1 | 8,192 |
Start with an angle of \(60^\circ\) and bisect it, then bisect the lower \(30^\circ\) angle, then the upper \(15^\circ\) angle, and so on, always alternating between the upper and lower of the previous two angles constructed. This process approaches a limiting line that divides the original \(60^\circ\) angle into two ... | 20 | 0.4375 | 7,245.0625 | 6,027.571429 | 8,192 | |
Let $(x,y)$ be an ordered pair of real numbers that satisfies the equation $x^2+y^2=14x+48y$. What is the minimum value of $y$? | -1 | 1 | 2,140.4375 | 2,140.4375 | -1 | |
All positive integers whose digits add up to 11 are listed in increasing order: $29, 38, 47, ...$. What is the eleventh number in that list? | 137 | 0.375 | 6,889.8125 | 5,653.333333 | 7,631.7 | |
Find the distance \( B_{1} H \) from point \( B_{1} \) to the line \( D_{1} B \), given \( B_{1}(5, 8, -3) \), \( D_{1}(-3, 10, -5) \), and \( B(3, 4, 1) \). | 2\sqrt{6} | 0.875 | 4,834.6875 | 4,355.071429 | 8,192 | |
The task is given a finite increasing sequence \( a_{1}, a_{2}, \ldots, a_{n} \) (\(n \geq 3\)) of natural numbers, and for all \( k \leq n-2 \), the equality \( a_{k+2}=3 a_{k+1}-2 a_{k}-1 \) holds. The sequence must necessarily contain the term \( a_{k}=2021 \). Determine the maximum number of three-digit numbers div... | 36 | 0 | 8,192 | -1 | 8,192 | |
Suppose the curve C has the polar coordinate equation $ρ\sin^2θ - 8\cos θ = 0$. Establish a rectangular coordinate system $xoy$ with the pole as the origin and the non-negative semi-axis of the polar axis as the $x$-axis. A line $l$, with an inclination angle of $α$, passes through point $P(2, 0)$.
(1) Write the recta... | 16\sqrt{2} | 0.5625 | 6,813.1875 | 5,740.777778 | 8,192 | |
Let $AC$ and $CE$ be two diagonals of a regular hexagon $ABCDEF$. Points $M$ and $N$ divide $AC$ and $CE$ internally such that $AM:AC = CN:CE = r$. If $B$, $M$, and $N$ are collinear, find $r$. | \frac{1}{\sqrt{3}} | 0 | 6,238.9375 | -1 | 6,238.9375 | |
Given that Jo and Blair take turns counting from 1, with Jo adding 2 to the last number said and Blair subtracting 1 from the last number said, determine the 53rd number said. | 79 | 0 | 6,742.5 | -1 | 6,742.5 | |
The Intermediate Maths Challenge has 25 questions with the following scoring rules:
5 marks are awarded for each correct answer to Questions 1-15;
6 marks are awarded for each correct answer to Questions 16-25;
Each incorrect answer to Questions 16-20 loses 1 mark;
Each incorrect answer to Questions 21-25 loses 2 marks... | 16 | 0 | 7,311.5625 | -1 | 7,311.5625 | |
Given an ellipse \( C: \frac{x^{2}}{2}+y^{2}=1 \) with left and right foci \( F_{1}, F_{2} \) respectively, let \( P \) be a point on the ellipse \( C \) in the first quadrant. The extended lines \( P F_{1}, P F_{2} \) intersect the ellipse \( C \) at points \( Q_{1}, Q_{2} \) respectively. Find the maximum value of th... | \frac{2\sqrt{2}}{3} | 0 | 8,192 | -1 | 8,192 | |
A circle has center $(-10, -4)$ and has radius $13$. Another circle has center $(3, 9)$ and radius $\sqrt{65}$. The line passing through the two points of intersection of the two circles has equation $x+y=c$. What is $c$? | 3 | 1. **Identify the equations of the circles**:
The first circle has center $(-10, -4)$ and radius $13$. Its equation is:
\[
(x+10)^2 + (y+4)^2 = 169.
\]
The second circle has center $(3, 9)$ and radius $\sqrt{65}$. Its equation is:
\[
(x-3)^2 + (y-9)^2 = 65.
\]
2. **Set up the equation for the ... | 0.9375 | 3,975 | 3,693.866667 | 8,192 |
Consider a $4 \times 4$ grid of squares, each of which are originally colored red. Every minute, Piet can jump on one of the squares, changing the color of it and any adjacent squares (two squares are adjacent if they share a side) to blue. What is the minimum number of minutes it will take Piet to change the entire gr... | 4 | Piet can change the colors of at most 5 squares per minute, so as there are 16 squares, it will take him at least four minutes to change the colors of every square. Some experimentation yields that it is indeed possible to make the entire grid blue after 4 minutes; one example is shown below: Here, jumping on the squar... | 0.0625 | 8,049.5 | 7,389 | 8,093.533333 |
A class has a group of 7 students, and now select 3 of them to swap seats with each other, while the remaining 4 students keep their seats unchanged. Calculate the number of different ways to adjust their seats. | 70 | 0.75 | 5,384.75 | 4,859.833333 | 6,959.5 | |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) and the circle $x^2 + y^2 = a^2 + b^2$ in the first quadrant, find the eccentricity of the hyperbola, where $|PF_1| = 3|PF_2|$. | \frac{\sqrt{10}}{2} | 0 | 7,246.75 | -1 | 7,246.75 | |
In a regular quadrilateral prism $ABCDA'A'B'C'D'$ with vertices on the same sphere, $AB = 1$ and $AA' = \sqrt{2}$, calculate the spherical distance between points $A$ and $C$. | \frac{\pi}{2} | 0.875 | 5,942.5625 | 5,621.214286 | 8,192 | |
Given a company needs to select 8 engineering and technical personnel from its 6 subsidiaries to form a task force, with each subsidiary contributing at least one person, calculate the total number of ways to allocate these 8 positions. | 21 | 0 | 6,485 | -1 | 6,485 | |
In the sum shown, each letter represents a different digit with $T \neq 0$ and $W \neq 0$. How many different values of $U$ are possible?
\begin{tabular}{rrrrr}
& $W$ & $X$ & $Y$ & $Z$ \\
+ & $W$ & $X$ & $Y$ & $Z$ \\
\hline & $W$ & $U$ & $Y$ & $V$
\end{tabular} | 3 | Since $WXYZ$ is a four-digit positive integer, then $WXYZ \leq 9999$. (In fact $WXYZ$ cannot be this large since all of its digits must be different.) Since $WXYZ \leq 9999$, then $TWUYV \leq 2(9999) = 19998$. Since $T \neq 0$, then $T = 1$. Next, we note that the 'carry' from any column to the next cannot be larger th... | 0.0625 | 8,122.9375 | 7,087 | 8,192 |
Let \( x \) be a real number with the property that \( x+\frac{1}{x} = 4 \). Define \( S_m = x^m + \frac{1}{x^m} \). Determine the value of \( S_6 \). | 2702 | 0.875 | 5,159.6875 | 4,726.5 | 8,192 | |
If an integer \( n \) that is greater than 8 is a solution to the equation \( x^2 - ax + b = 0 \), and the representation of \( a \) in base \( n \) is 18, then the representation of \( b \) in base \( n \) is: | 80 | 0.9375 | 2,665.6875 | 2,297.266667 | 8,192 | |
Given the following four propositions:
(1) Two lines parallel to the same plane are parallel to each other.
(2) Two lines perpendicular to the same line are parallel to each other.
(3) Through a point outside a known plane, there exists exactly one plane parallel to the given plane.
(4) Through a line outside a... | (3) | 0 | 1,728.8125 | -1 | 1,728.8125 | |
Given a parabola C: $y^2=2px$ $(p>0)$ whose focus F coincides with the right focus of the ellipse $\frac{x^2}{4}+\frac{y^2}{3}=1$, and a line l passing through point F intersects the parabola at points A and B.
(1) Find the equation of the parabola C.
(2) If the line l intersects the y-axis at point M, and $\overrighta... | -1 | 0.5 | 6,924.75 | 5,657.5 | 8,192 | |
In a tetrahedron V-ABC with edge length 10, point O is the center of the base ABC. Segment MN has a length of 2, with one endpoint M on segment VO and the other endpoint N inside face ABC. If point T is the midpoint of segment MN, then the area of the trajectory formed by point T is __________. | 2\pi | 0.0625 | 8,075.6875 | 6,331 | 8,192 | |
Find $(4^4 \div 4^3) \cdot 2^8$. | 1024 | 1 | 2,033.4375 | 2,033.4375 | -1 | |
Given triangle \( \triangle ABC \) with \( AB = 1 \), \( AC = 2 \), and \( \cos B + \sin C = 1 \), find the length of side \( BC \). | \frac{3 + 2 \sqrt{21}}{5} | 0 | 7,000.75 | -1 | 7,000.75 | |
Given externally tangent circles with centers at points $A$ and $B$ and radii of lengths $6$ and $4$, respectively, a line externally tangent to both circles intersects ray $AB$ at point $C$. Calculate $BC$. | 20 | 0.75 | 4,770.8125 | 4,108.25 | 6,758.5 | |
What is the probability that Robbie will win if he and Francine each roll a special six-sided die three times, and after two rolls each, Robbie has a score of 8 and Francine has a score of 10? | \frac{55}{441} | Robbie has a score of 8 and Francine has a score of 10 after two rolls each. Thus, in order for Robbie to win (that is, to have a higher total score), his third roll must be at least 3 larger than that of Francine. If Robbie rolls 1, 2 or 3, his roll cannot be 3 larger than that of Francine. If Robbie rolls a 4 and win... | 0 | 5,261.25 | -1 | 5,261.25 |
The number obtained from the last two nonzero digits of $80!$ is equal to $n$. Find the value of $n$. | 12 | 0 | 8,132.375 | -1 | 8,132.375 | |
For a four-digit natural number $M$, let the digit in the thousands place be $a$, in the hundreds place be $b$, in the tens place be $c$, and in the units place be $d$. The two-digit number formed by the thousands and units digits of $M$ is $A=10a+d$, and the two-digit number formed by the tens and hundreds digits of $... | 8892 | 0.125 | 7,994.1875 | 6,609.5 | 8,192 | |
A wholesaler purchased $50$ packs of shirts of size $L$ from a clothing manufacturer. Due to the negligence of the packaging workers, some packs were mixed with shirts of size $M$. The number of $M$ shirts mixed in (pieces) and the corresponding number of packs (packs) are shown in the table below:
|M shirts (pieces)|... | \frac{3}{25} | 0.4375 | 3,666.5625 | 3,418 | 3,859.888889 | |
The altitudes of an acute isosceles triangle, where \(AB = BC\), intersect at point \(H\). Find the area of triangle \(ABC\), given \(AH = 5\) and the altitude \(AD\) is 8. | 40 | 0.5625 | 7,037.0625 | 6,183 | 8,135.142857 | |
What is the least positive integer that can be added to 412 to yield a multiple of 3? | 2 | 1 | 1,708.8125 | 1,708.8125 | -1 | |
In the diagram, $ABCD$ is a trapezoid with bases $AB$ and $CD$ such that $AB$ is parallel to $CD$ and $CD$ is three times the length of $AB$. The area of $ABCD$ is $27$. Find the area of $\triangle ABC$.
[asy]
draw((0,0)--(3,6)--(9,6)--(12,0)--cycle);
draw((3,6)--(0,0));
label("$A$",(0,0),W);
label("$B$",(3,6),NW);
la... | 6.75 | 0 | 6,511.125 | -1 | 6,511.125 | |
If $x+y=8$ and $xy=12$, what is the value of $x^3+y^3$ and $x^2 + y^2$? | 40 | 0.75 | 1,863 | 1,976.166667 | 1,523.5 | |
Given that the sum of the first n terms of a geometric sequence {a_n} (where all terms are real numbers) is S_n, if S_10=10 and S_30=70, determine the value of S_40. | 150 | 0.9375 | 5,095.5625 | 4,890.866667 | 8,166 | |
Let $p$ and $q$ satisfy $pq=9$ and $p+q=6$. What is the value of $p^2 + q^2$? | 18 | 1 | 1,479.625 | 1,479.625 | -1 | |
Given that $0 < α < \dfrac{\pi }{2}$, $-\dfrac{\pi }{2} < β < 0$, $\cos \left( \dfrac{\pi }{4}+α \right)=\dfrac{1}{3}$, and $\cos \left( \dfrac{\pi }{4}-\dfrac{\beta }{2} \right)=\dfrac{\sqrt{3}}{3}$, calculate the value of $\cos \left( α +\dfrac{\beta }{2} \right)$. | \dfrac{5\sqrt{3}}{9} | 0 | 5,363.5625 | -1 | 5,363.5625 | |
Given that $\angle A$ and $\angle B$ are within the interval $\left(0, \frac{\pi}{2}\right)$, and that $\frac{\sin A}{\sin B} = \sin (A+B)$, find the maximum value of $\tan A$. | 4/3 | 0.3125 | 7,508 | 6,003.2 | 8,192 | |
Let
\[f(x) = x^3 + 6x^2 + 16x + 28.\]The graphs of $y = f(x)$ and $y = f^{-1}(x)$ intersect at exactly one point $(a,b).$ Enter the ordered pair $(a,b).$ | (-4,-4) | 1 | 3,216.25 | 3,216.25 | -1 | |
Given that the plane vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ satisfy $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and for all $t\in \mathbb{R}$, $|\overrightarrow{b}+t\overrightarrow{a}| \geq |\overrightarrow{b}-\overrightarrow{a}|$ always holds, determine the angle between $2\ov... | \frac{2\pi}{3} | 0.1875 | 6,429.0625 | 4,532.333333 | 6,866.769231 | |
Let $x$ be a complex number such that $x+x^{-1}$ is a root of the polynomial $p(t)=t^{3}+t^{2}-2 t-1$. Find all possible values of $x^{7}+x^{-7}$. | 2 | Since $x+x^{-1}$ is a root, $$\begin{aligned} 0 & =\left(x+x^{-1}\right)^{3}+\left(x+x^{-1}\right)^{2}-2\left(x+x^{-1}\right)-1 \\ & =x^{3}+x^{-3}+3 x+3 x^{-1}+x^{2}+2+x^{-2}-2 x-2 x^{-1}-1 \\ & =x^{3}+x^{-3}+x^{2}+x^{-2}+x+x^{-1}+1 \\ & =x^{-3}\left(1+x+x^{2}+\cdots+x^{6}\right) \end{aligned}$$ Since $x \neq 0$, the a... | 0.4375 | 7,311.25 | 6,178.857143 | 8,192 |
In every cell of a \(5 \times 5\) square, there is a number. The numbers in any given row (any row) and any given column (any column) form an arithmetic progression. The numbers in the corners of the square are \(1, 25, 17, 81\). What number is in the center cell? Do not forget to justify your answer.
\[
\begin{array}... | 31 | 0.0625 | 8,192 | 8,192 | 8,192 | |
A five-digit natural number $\overline{a_1a_2a_3a_4a_5}$ is considered a "concave number" if and only if $a_1 > a_2 > a_3$ and $a_3 < a_4 < a_5$, with each $a_i \in \{0,1,2,3,4,5\}$ for $i=1,2,3,4,5$. Calculate the number of possible "concave numbers". | 146 | 0 | 8,176.4375 | -1 | 8,176.4375 | |
To welcome the 2008 Olympic Games, a craft factory plans to produce the Olympic logo "China Seal" and the Olympic mascot "Fuwa". The factory mainly uses two types of materials, A and B. It is known that producing a set of the Olympic logo requires 4 boxes of material A and 3 boxes of material B, and producing a set of ... | 2400 | 0.5625 | 1,686.25 | 2,124.555556 | 1,122.714286 | |
In a WeChat group, members A, B, C, D, and E simultaneously grab 4 red envelopes. Each person can grab at most one red envelope, and all red envelopes are grabbed. Among the 4 red envelopes, there are two worth 2 yuan and two worth 3 yuan (red envelopes with the same amount are considered the same). The number of situa... | 18 | 0.1875 | 7,772.25 | 6,625.333333 | 8,036.923077 | |
A bag contains 20 candies: 4 chocolate, 6 mint, and 10 butterscotch. Candies are removed randomly from the bag and eaten. What is the minimum number of candies that must be removed to be certain that at least two candies of each flavor have been eaten? | 18 | 0.3125 | 7,293.1875 | 5,972 | 7,893.727273 | |
Find all positive integers $x,y$ satisfying the equation \[9(x^2+y^2+1) + 2(3xy+2) = 2005 .\] | \[
\boxed{(7, 11), (11, 7)}
\] | Solution 1
We can re-write the equation as:
$(3x)^2 + y^2 + 2(3x)(y) + 8y^2 + 9 + 4 = 2005$
or $(3x + y)^2 = 4(498 - 2y^2)$
The above equation tells us that $(498 - 2y^2)$ is a perfect square.
Since $498 - 2y^2 \ge 0$ . this implies that $y \le 15$
Also, taking $mod 3$ on both sides we see that $y$ cannot be a multi... | 0 | 5,403.3125 | -1 | 5,403.3125 |
Calculate: $5 \times 13 \times 31 \times 73 \times 137$ | 20152015 | 0 | 927.1875 | -1 | 927.1875 | |
In triangle $ABC$, $AB = 10$ and $AC = 17$. Let $D$ be the foot of the perpendicular from $A$ to $BC$. If $BD:CD = 2:5$, then find $AD$. | 8 | 1 | 2,166.4375 | 2,166.4375 | -1 |
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