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 |
|---|---|---|---|---|---|---|
Solve for $Q$ if $\sqrt{Q^3} = 16\sqrt[8]{16}$. | 8 | 0.8125 | 3,355.375 | 2,537.307692 | 6,900.333333 | |
Two distinct positive integers $x$ and $y$ are factors of 36. If $x\cdot y$ is not a factor of 36, what is the smallest possible value of $x\cdot y$? | 8 | 0.5625 | 6,223.4375 | 4,692.333333 | 8,192 | |
In a circle with a radius of 5 units, \( CD \) and \( AB \) are mutually perpendicular diameters. A chord \( CH \) intersects \( AB \) at \( K \) and has a length of 8 units, calculate the lengths of the two segments into which \( AB \) is divided. | 8.75 | 0.0625 | 6,057.0625 | 5,073 | 6,122.666667 | |
In rectangle $ABCD$, $AB=8$ and $BC=6$. Points $F$ and $G$ are on $\overline{CD}$ such that $DF=3$ and $GC=1$. Lines $AF$ and $BG$ intersect at $E$. Find the area of $\triangle AEB$. | 24 | 0 | 6,897.625 | -1 | 6,897.625 | |
For how many ordered pairs of positive integers $(x, y)$ with $x < y$ is the harmonic mean of $x$ and $y$ equal to $12^{10}$? | 220 | 0.1875 | 6,347.25 | 4,538 | 6,764.769231 | |
How many perfect squares are between 50 and 250? | 8 | 1 | 2,104.875 | 2,104.875 | -1 | |
A person receives an annuity at the end of each year for 15 years as follows: $1000 \mathrm{K}$ annually for the first five years, $1200 \mathrm{K}$ annually for the next five years, and $1400 \mathrm{K}$ annually for the last five years. If they received $1400 \mathrm{K}$ annually for the first five years and $1200 \m... | 807.95 | 0 | 7,328.375 | -1 | 7,328.375 | |
Find the largest positive integer \(n\) for which there exist \(n\) finite sets \(X_{1}, X_{2}, \ldots, X_{n}\) with the property that for every \(1 \leq a<b<c \leq n\), the equation \(\left|X_{a} \cup X_{b} \cup X_{c}\right|=\lceil\sqrt{a b c}\rceil\) holds. | 4 | First, we construct an example for \(N=4\). Let \(X_{1}, X_{2}, X_{3}, X_{4}\) be pairwise disjoint sets such that \(X_{1}=\varnothing,\left|X_{2}\right|=1,\left|X_{3}\right|=2\), and \(\left|X_{4}\right|=2\). It is straightforward to verify the condition. We claim that there are no five sets \(X_{1}, X_{2}, \ldots, X_... | 0 | 8,166.75 | -1 | 8,166.75 |
Three different natural numbers $x$, $y$ and $z$ each have exactly three natural-number factors. How many factors does $x^{2}y^{3}z^{4}$ have? | 315 | 1 | 1,897.125 | 1,897.125 | -1 | |
Find all values of the real number $a$ so that the four complex roots of
\[z^4 - 6z^3 + 11az^2 - 3(2a^2 + 3a - 3) z + 1 = 0\]form the vertices of a parallelogram in the complex plane. Enter all the values, separated by commas. | 3 | 0 | 8,057 | -1 | 8,057 | |
How many 0's are located to the right of the decimal point and before the first non-zero digit in the terminating decimal representation of $\frac{1}{2^5\cdot5^8}$? | 7 | 0.875 | 5,595.375 | 5,400.071429 | 6,962.5 | |
The Gregorian calendar defines a common year as having 365 days and a leap year as having 366 days. The $n$-th year is a leap year if and only if:
1. $n$ is not divisible by 100 and $n$ is divisible by 4, or
2. $n$ is divisible by 100 and $n$ is divisible by 400.
For example, 1996 and 2000 are leap years, whereas 19... | 365.2425 | 0.875 | 4,210.25 | 3,942.214286 | 6,086.5 | |
Triangle $DEF$ has side lengths $DE = 15$, $EF = 36$, and $FD = 39$. Rectangle $WXYZ$ has vertex $W$ on $\overline{DE}$, vertex $X$ on $\overline{DF}$, and vertices $Y$ and $Z$ on $\overline{EF}$. In terms of the side length $WX = \omega$, the area of $WXYZ$ can be expressed as the quadratic polynomial
\[ \text{Area}(W... | 17 | 0.25 | 7,656.0625 | 6,048.25 | 8,192 | |
Monica tosses a fair 6-sided die. If the roll is a prime number, then she wins that amount of dollars (so that, for example, if she rolls 3, then she wins 3 dollars). If the roll is composite, she wins nothing. Otherwise, she loses 3 dollars. What is the expected value of her winnings on one die toss? Express your an... | \$1.17 | 1 | 3,497.1875 | 3,497.1875 | -1 | |
Lines $l_1^{}$ and $l_2^{}$ both pass through the origin and make first-quadrant angles of $\frac{\pi}{70}$ and $\frac{\pi}{54}$ radians, respectively, with the positive x-axis. For any line $l^{}_{}$, the transformation $R(l)^{}_{}$ produces another line as follows: $l^{}_{}$ is reflected in $l_1^{}$, and the resultin... | 945 | Let $l$ be a line that makes an angle of $\theta$ with the positive $x$-axis. Let $l'$ be the reflection of $l$ in $l_1$, and let $l''$ be the reflection of $l'$ in $l_2$.
The angle between $l$ and $l_1$ is $\theta - \frac{\pi}{70}$, so the angle between $l_1$ and $l'$ must also be $\theta - \frac{\pi}{70}$. Thus, $l'... | 0.9375 | 5,171.9375 | 4,970.6 | 8,192 |
In a three-dimensional Cartesian coordinate system, there is a sphere with its center at the origin and a radius of 3 units. How many lattice points lie on the surface of the sphere? | 30 | 0 | 8,192 | -1 | 8,192 | |
Two positive integers differ by $60$. The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers? | 156 | Call the two integers $b$ and $b+60$, so we have $\sqrt{b}+\sqrt{b+60}=\sqrt{c}$. Square both sides to get $2b+60+2\sqrt{b^2+60b}=c$. Thus, $b^2+60b$ must be a square, so we have $b^2+60b=n^2$, and $(b+n+30)(b-n+30)=900$. The sum of these two factors is $2b+60$, so they must both be even. To maximize $b$, we want to ma... | 0.1875 | 7,409.4375 | 6,020.333333 | 7,730 |
Now a ball is launched from a vertex of an equilateral triangle with side length 5. It strikes the opposite side after traveling a distance of $\sqrt{19}$. Find the distance from the ball's point of first contact with a wall to the nearest vertex. | 2 | Consider the diagram above, where $M$ is the midpoint of $BC$. Then $AM$ is perpendicular to $BC$ since $ABC$ is equilateral, so by the Pythagorean theorem $AM = \frac{5 \sqrt{3}}{2}$. Then, using the Pythagorean theorem again, we see that $MY = \frac{1}{2}$, so that $BY = 2$. | 0.75 | 6,238.6875 | 5,587.583333 | 8,192 |
Semicircle $\widehat{AB}$ has center $C$ and radius $1$. Point $D$ is on $\widehat{AB}$ and $\overline{CD} \perp \overline{AB}$. Extend $\overline{BD}$ and $\overline{AD}$ to $E$ and $F$, respectively, so that circular arcs $\widehat{AE}$ and $\widehat{BF}$ have $B$ and $A$ as their respective centers. Circular arc $\w... | $2\pi-\pi \sqrt{2}-1$ | 1. **Identify the components of the shape**: The shape consists of sectors $ABE$, $ABF$, and $DEF$, and the semicircle $ABD$. We need to calculate the area of each component and then combine them appropriately to find the area of the shaded region "smile" $AEFBDA$.
2. **Calculate the radius and angle of sectors $ABE$ ... | 0 | 7,962.625 | -1 | 7,962.625 |
If the sum of the digits of a natural number \( n \) is subtracted from \( n \), the result is 2016. Find the sum of all such natural numbers \( n \). | 20245 | 0.375 | 7,280.25 | 6,448.5 | 7,779.3 | |
Let the arithmetic sequences $\{a_n\}$ and $\{b_n\}$ have the sum of the first $n$ terms denoted by $S_n$ and $T_n$ respectively. If for any natural number $n$ it holds that $\dfrac{S_n}{T_n} = \dfrac{2n-3}{4n-3}$, find the value of $\dfrac{a_9}{b_5+b_7} + \dfrac{a_3}{b_8+b_4}$. | \dfrac{19}{41} | 0.75 | 5,132.375 | 4,748.083333 | 6,285.25 | |
Let $P(x)$ be a polynomial with integer coefficients that satisfies $P(17)=10$ and $P(24)=17.$ Given that $P(n)=n+3$ has two distinct integer solutions $n_1$ and $n_2,$ find the product $n_1\cdot n_2.$ | 418 | We define $Q(x)=P(x)-x+7$, noting that it has roots at $17$ and $24$. Hence $P(x)-x+7=A(x-17)(x-24)$. In particular, this means that $P(x)-x-3=A(x-17)(x-24)-10$. Therefore, $x=n_1,n_2$ satisfy $A(x-17)(x-24)=10$, where $A$, $(x-17)$, and $(x-24)$ are integers. This cannot occur if $x\le 17$ or $x\ge 24$ because the pro... | 0.5 | 6,080.0625 | 4,786.625 | 7,373.5 |
If $C=\frac{5}{9}(F-32)$, what is $F$ when $C=20$? | 68 | 1 | 1,405 | 1,405 | -1 | |
Find the number of solutions to the equation
\[\tan (7 \pi \cos \theta) = \cot (7 \pi \sin \theta)\]
where $\theta \in (0, 4 \pi).$ | 28 | 0 | 8,192 | -1 | 8,192 | |
In the plane rectangular coordinate system $xOy$, the parameter equation of the line $l$ with an inclination angle $\alpha = 60^{\circ}$ is $\left\{\begin{array}{l}{x=2+t\cos\alpha}\\{y=t\sin\alpha}\end{array}\right.$ (where $t$ is the parameter). Taking the coordinate origin $O$ as the pole, and the non-negative half-... | \frac{\sqrt{7}}{3} | 0 | 7,244.4375 | -1 | 7,244.4375 | |
Given the function $f(x)=2\ln (3x)+8x+1$, find the value of $\underset{\Delta x\to 0}{{\lim }}\,\dfrac{f(1-2\Delta x)-f(1)}{\Delta x}$. | -20 | 1 | 3,432.25 | 3,432.25 | -1 | |
A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $15$ and $25$ meters. What fraction of the yard is occupied by the flower beds?
[asy]
unitsize(2mm); defaultpen(... | \frac{1}{5} | 1. **Identify the dimensions of the triangles:**
The problem states that the yard contains two congruent isosceles right triangles and the parallel sides of the trapezoidal remainder are $15$ meters and $25$ meters. The difference in length between the parallel sides of the trapezoid is $25 - 15 = 10$ meters. Since ... | 0.8125 | 4,796 | 4,050.692308 | 8,025.666667 |
The real numbers $c, b, a$ form an arithmetic sequence with $a \geq b \geq c \geq 0$. The quadratic $ax^2+bx+c$ has exactly one root. What is this root? | -2-\sqrt{3} |
Given that $a, b, c$ form an arithmetic sequence and $a \geq b \geq c \geq 0$, we can express $b$ and $c$ in terms of $a$ and a common difference $d$:
\[ b = a - d, \quad c = a - 2d. \]
The quadratic equation is $ax^2 + bx + c = 0$. Since it has exactly one root, the discriminant must be zero:
\[ b^2 - 4ac = 0. \]
S... | 0 | 6,207.375 | -1 | 6,207.375 |
Along a road, there are 25 poles in a single row. Sometimes a siskin lands on one of the poles, and immediately a siskin flies off from one of the neighboring poles (if there was at least one siskin sitting on the neighboring poles at that moment). Also, each pole can hold no more than one siskin.
Initially, there are... | 24 | 0 | 7,913.125 | -1 | 7,913.125 | |
A club has increased its membership to 12 members and needs to elect a president, vice president, secretary, and treasurer. Additionally, they want to appoint two different advisory board members. Each member can hold only one position. In how many ways can these positions be filled? | 665,280 | 0 | 4,077.75 | -1 | 4,077.75 | |
In how many ways can you arrange the digits of 11250 to get a five-digit number that is a multiple of 2? | 24 | 0.125 | 6,621.625 | 4,416.5 | 6,936.642857 | |
Given real numbers $x$ and $y$ satisfying $\sqrt{x-3}+y^{2}-4y+4=0$, find the value of the algebraic expression $\frac{{x}^{2}-{y}^{2}}{xy}•\frac{1}{{x}^{2}-2xy+{y}^{2}}÷\frac{x}{{x}^{2}y-x{y}^{2}}-1$. | \frac{2}{3} | 1 | 3,452.75 | 3,452.75 | -1 | |
Real numbers $x,$ $y,$ and $z$ satisfy the following equality:
\[4(x + y + z) = x^2 + y^2 + z^2.\]Let $M$ be the maximum value of $xy + xz + yz,$ and let $m$ be the minimum value of $xy + xz + yz.$ Find $M + 10m.$ | 28 | 0.75 | 5,774.125 | 4,968.166667 | 8,192 | |
A diagonal of a polygon is a segment joining two nonconsecutive vertices of the polygon. How many diagonals does a regular octagon have? | 20 | 0.9375 | 2,064.875 | 1,656.4 | 8,192 | |
In rectangle $PQRS,$ $PQ=12$ and $PR=13.$ What is the area of rectangle $PQRS?$
[asy]
draw((0,0)--(12,0)--(12,5)--(0,5)--cycle,black+linewidth(1));
draw((0,5)--(12,0),black+linewidth(1));
label("$P$",(0,5),NW);
label("$Q$",(12,5),NE);
label("$R$",(12,0),SE);
label("$S$",(0,0),SW);
label("12",(0,5)--(12,5),N);
label("1... | 60 | 1 | 927.6875 | 927.6875 | -1 | |
A teacher received 10, 6, 8, 5, and 6 letters from Monday to Friday, respectively. The variance $s^2$ of this set of data is ______. | 3.2 | 0.3125 | 4,259.375 | 4,149.4 | 4,309.363636 | |
Let $A$ be the product of the divisors of $42$. How many distinct prime factors does $A$ have? | 3 | 1 | 2,438.6875 | 2,438.6875 | -1 | |
The equation of the line that passes through the points $(-3,5)$ and $(0,-4)$ can be expressed in the form $y=mx+b$. What is the value of $m+b$? | -7 | 1 | 1,361.75 | 1,361.75 | -1 | |
What is the positive difference of the solutions of $\dfrac{r^2-3r-17}{r+4}=2r+7$? | 12 | 1 | 1,740.4375 | 1,740.4375 | -1 | |
Consider the following sequence $$\left(a_{n}\right)_{n=1}^{\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1, \ldots)$$ Find all pairs $(\alpha, \beta)$ of positive real numbers such that $\lim _{n \rightarrow \infty} \frac{\sum_{k=1}^{n} a_{k}}{n^{\alpha}}=\beta$. | (\alpha, \beta)=\left(\frac{3}{2}, \frac{\sqrt{2}}{3}\right) | Let $N_{n}=\binom{n+1}{2}$ (then $a_{N_{n}}$ is the first appearance of number $n$ in the sequence) and consider limit of the subsequence $$b_{N_{n}}:=\frac{\sum_{k=1}^{N_{n}} a_{k}}{N_{n}^{\alpha}}=\frac{\sum_{k=1}^{n} 1+\cdots+k}{\binom{n+1}{2}^{\alpha}}=\frac{\sum_{k=1}^{n}\binom{k+1}{2}}{\binom{n+1}{2}^{\alpha}}=\f... | 0 | 8,192 | -1 | 8,192 |
A list of five positive integers has a median of 3 and a mean of 11. What is the maximum possible value of the list's largest element? | 47 | 0.8125 | 5,383.4375 | 4,735.307692 | 8,192 | |
The number of degrees in the measures of the interior angles of a convex pentagon are five consecutive integers. What is the number of degrees in the largest angle? | 110^\circ | 1 | 1,631.5625 | 1,631.5625 | -1 | |
The whole numbers from 1 to \( 2k \) are split into two equal-sized groups in such a way that any two numbers from the same group share no more than two distinct prime factors. What is the largest possible value of \( k \)? | 44 | 0 | 8,052.625 | -1 | 8,052.625 | |
How many ways are there to arrange three indistinguishable rooks on a $6 \times 6$ board such that no two rooks are attacking each other? | 2400 | There are $6 \times 6=36$ possible places to place the first rook. Since it cannot be in the same row or column as the first, the second rook has $5 \times 5=25$ possible places, and similarly, the third rook has $4 \times 4=16$ possible places. However, the rooks are indistinguishable, so there are 3! $=6$ ways to reo... | 0.9375 | 4,341 | 4,084.266667 | 8,192 |
A rod with a length of four meters has weights attached as follows: $20 \mathrm{~kg}$ at one end, and at distances of one, two, and three meters from that end, weights of $30, 40, 50$ $\mathrm{kg}$ respectively. Additionally, a weight of $60 \mathrm{~kg}$ is attached at the other end. Where should the rod be supported ... | 2.5 | 0.1875 | 7,471.125 | 5,864 | 7,842 | |
In triangle $ABC$, $D$ is on segment $BC$ such that $BD:DC = 3:2$. Point $E$ bisects $\angle BAC$. Given $BD = 45$ units and $DC = 30$ units, find the length of $BE$. | 45 | 0.125 | 7,853.6875 | 6,173 | 8,093.785714 | |
Consider the function
\[ f(x) = \max \{-8x - 29, 3x + 2, 7x - 4\} \] defined for all real $x$. Let $q(x)$ be a quadratic polynomial tangent to the graph of $f$ at three distinct points with $x$-coordinates $a_1$, $a_2$, $a_3$. Find $a_1 + a_2 + a_3$. | -\frac{163}{22} | 0 | 8,086.4375 | -1 | 8,086.4375 | |
In how many distinct ways can you color each of the vertices of a tetrahedron either red, blue, or green such that no face has all three vertices the same color? (Two colorings are considered the same if one coloring can be rotated in three dimensions to obtain the other.) | 6 | If only two colors are used, there is only one possible arrangement up to rotation, so this gives 3 possibilities. If all three colors are used, then one is used twice. There are 3 ways to choose the color that is used twice. Say this color is red. Then the red vertices are on a common edge, and the green and blue vert... | 0 | 7,989.4375 | -1 | 7,989.4375 |
Let \( OP \) be the diameter of the circle \( \Omega \), and \( \omega \) be a circle with center at point \( P \) and a radius smaller than that of \( \Omega \). The circles \( \Omega \) and \( \omega \) intersect at points \( C \) and \( D \). The chord \( OB \) of the circle \( \Omega \) intersects the second circle... | \sqrt{5} | 0.0625 | 8,156.25 | 7,620 | 8,192 | |
A mineralogist is hosting a competition to guess the age of an ancient mineral sample. The age is provided by the digits 2, 2, 3, 3, 5, and 9, with the condition that the age must start with an odd number. | 120 | 0.375 | 1,038.9375 | 1,671 | 659.7 | |
In the plane rectangular coordinate system $xOy$, the equation of the hyperbola $C$ is $x^{2}-y^{2}=1$. Find all real numbers $a$ greater than 1 that satisfy the following requirement: Through the point $(a, 0)$, draw any two mutually perpendicular lines $l_{1}$ and $l_{2}$. If $l_{1}$ intersects the hyperbola $C$ at p... | \sqrt{2} | 0.3125 | 7,814.25 | 6,983.2 | 8,192 | |
Let \( f \) be a function taking positive integers to positive integers, such that:
(i) \( f \) is increasing (\( f(n + 1) > f(n) \) for all positive integers \( n \))
(ii) \( f(mn) = f(m) f(n) \) for all positive integers \( m \) and \( n \)
(iii) if \( m \neq n \) and \( m^n = n^m \), then \( f(m) = n \) or \( f(n) ... | 3600 | 0.0625 | 8,178.75 | 7,980 | 8,192 | |
Let the complex numbers \(z\) and \(w\) satisfy \(|z| = 3\) and \((z + \bar{w})(\bar{z} - w) = 7 + 4i\), where \(i\) is the imaginary unit and \(\bar{z}\), \(\bar{w}\) denote the conjugates of \(z\) and \(w\) respectively. Find the modulus of \((z + 2\bar{w})(\bar{z} - 2w)\). | \sqrt{65} | 0.6875 | 5,791.375 | 5,078.090909 | 7,360.6 | |
Misha rolls a standard, fair six-sided die until she rolls $1-2-3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 647 | Let $P_n$ be the probability of getting consecutive $1,2,3$ rolls in $n$ rolls and not rolling $1,2,3$ prior to the nth roll.
Let $x = P_3+P_5+...=1-(P_4+P_6+..)$. Following Solution 2, one can see that \[P_{n+1}=P_{n}-\frac{P_{n-2}}{6^3}\] for all positive integers $n \ge 5$. Summing for $n=5,7,...$ gives \[(1-x)-\fr... | 0 | 8,170 | -1 | 8,170 |
How many times does the digit 0 appear in the integer equal to \( 20^{10} \)? | 11 | 0.1875 | 7,498.6875 | 6,504 | 7,728.230769 | |
A construction company built a cottage village consisting of three identical streets. The houses in the village are identical according to the plan and are up for sale at the same price. The business plan of the company, which includes expenses for all stages of construction, contains an estimate of the total cost of ... | 42 | 0 | 3,970.625 | -1 | 3,970.625 | |
Given the series $2+1+\frac {1}{2}+\frac {1}{4}+\cdots$ and the following five statements:
(1) the sum increases without limit
(2) the sum decreases without limit
(3) the difference between any term of the sequence and zero can be made less than any positive quantity no matter how small
(4) the difference between the ... | 4 and 5 | 1. **Identify the type of series**: The given series is $2 + 1 + \frac{1}{2} + \frac{1}{4} + \cdots$. This is a geometric series where the first term $a = 2$ and the common ratio $r = \frac{1}{2}$.
2. **Formula for the sum of an infinite geometric series**: The sum $S$ of an infinite geometric series can be calculated... | 0 | 3,402.375 | -1 | 3,402.375 |
When Bendegúz boarded the 78-seat train car with his valid seat reservation, he was shocked to find that all seats were already taken. What had happened was that Dömötör boarded without a seat reservation. The other 77 passengers, including Elek, had purchased a seat reservation, but did not necessarily sit in their as... | 1/2 | 0.3125 | 7,803.3125 | 6,948.2 | 8,192 | |
When $x=3$ and $y=4$, what is the value of the expression $xy-x$? | 9 | When $x=3$ and $y=4$, we get $xy-x=3 \times 4-3=12-3=9$. Alternatively, $xy-x=x(y-1)=3 \times 3=9$. | 1 | 368.3125 | 368.3125 | -1 |
Suppose that $a$ is a multiple of $3$ and $b$ is a multiple of $6$. Which of the following statements must be true?
A. $b$ is a multiple of $3$.
B. $a-b$ is a multiple of $3$.
C. $a-b$ is a multiple of $6$.
D. $a-b$ is a multiple of $2$.
List the choices in your answer separated by commas. For example, if you thin... | \text{A, B} | 0 | 777.6875 | -1 | 777.6875 | |
In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitting on pad 1. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches p... | \frac{63}{146} | Define \( P(N) \) as the probability that the frog survives starting from pad \( N \). We need to find \( P(1) \), the probability that the frog escapes starting from pad 1.
The recursive relationship given in the problem is:
\[ P(N) = \frac{N}{10} P(N-1) + \left(1 - \frac{N}{10}\right) P(N+1) \]
for \( 0 < N < 10 \).... | 0 | 8,192 | -1 | 8,192 |
Find the number of solutions in natural numbers to the equation \(\left\lfloor \frac{x}{10} \right\rfloor = \left\lfloor \frac{x}{11} \right\rfloor + 1\). | 110 | 0 | 8,192 | -1 | 8,192 | |
A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a_{}$ and $b_{}$ are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that $|a+bi|=8$ and that $b^2=m/n,$ where $m_{}$ and $n_{}$ are relative... | 259 | Plugging in $z=1$ yields $f(1) = a+bi$. This implies that $a+bi$ must fall on the line $Re(z)=a=\frac{1}{2}$, given the equidistant rule. By $|a+bi|=8$, we get $a^2 + b^2 = 64$, and plugging in $a=\frac{1}{2}$ yields $b^2=\frac{255}{4}$. The answer is thus $\boxed{259}$. | 1 | 2,268.625 | 2,268.625 | -1 |
Let $T$ denote the sum of all four-digit positive integers where each digit is distinct and ranges from 1 to 9. Compute the remainder when $T$ is divided by $1000$. | 320 | 0.875 | 4,874.6875 | 4,400.785714 | 8,192 | |
In triangle $DEF$, $\angle E = 45^\circ$, $DE = 100$, and $DF = 100 \sqrt{2}$. Find the sum of all possible values of $EF$. | \sqrt{30000 + 5000(\sqrt{6} - \sqrt{2})} | 0 | 7,645.75 | -1 | 7,645.75 | |
One angle of a triangle is twice another, and the sides opposite these angles have lengths 15 and 9. Compute the length of the third side of the triangle. | 16 | 0.875 | 5,017 | 4,563.428571 | 8,192 | |
What is the sum of every third odd integer between $200$ and $500$? | 17400 | 0.6875 | 6,043.75 | 5,358.636364 | 7,551 | |
In triangle \( \triangle ABC \), it is known that \( AC=3 \) and the three interior angles \( A \), \( B \), and \( C \) form an arithmetic sequence.
(1) If \( \cos C= \frac {\sqrt{6}}{3} \), find \( AB \);
(2) Find the maximum value of the area of \( \triangle ABC \). | \frac {9 \sqrt {3}}{4} | 0 | 7,666.9375 | -1 | 7,666.9375 | |
The greatest common divisor of two positive integers is $(x+6)$ and their least common multiple is $x(x+6)$, where $x$ is a positive integer. If one of the integers is 36, what is the smallest possible value of the other one? | 24 | 0.5 | 6,776.1875 | 5,588.375 | 7,964 | |
A journalist wants to take a photo of 5 volunteers and 2 elderly individuals they are assisting, requiring them to line up in a row. The 2 elderly individuals must stand next to each other but not at the ends. The total number of different arrangements is ____. | 960 | 0.375 | 6,821.3125 | 5,672.833333 | 7,510.4 | |
What is the radius of the smallest circle into which any system of points with a diameter of 1 can be enclosed? | \frac{\sqrt{3}}{3} | 0 | 7,310.9375 | -1 | 7,310.9375 | |
If circular arcs $AC$ and $BC$ have centers at $B$ and $A$, respectively, then there exists a circle tangent to both $\overarc {AC}$ and $\overarc{BC}$, and to $\overline{AB}$. If the length of $\overarc{BC}$ is $12$, then the circumference of the circle is | 27 |
1. **Identify the Geometry of the Problem:**
Since the centers of the arcs $AC$ and $BC$ are at $B$ and $A$ respectively, and each arc is part of a circle with radius equal to $AB$, triangle $ABC$ is equilateral. This is because all sides $AB$, $BC$, and $CA$ are radii of the respective circles and hence equal.
2.... | 0.75 | 4,302.75 | 4,200.666667 | 4,609 |
If vectors $\mathbf{a} = (1,0)$, $\mathbf{b} = (0,1)$, $\mathbf{c} = k\mathbf{a} + \mathbf{b}$ ($k \in \mathbb{R}$), $\mathbf{d} = \mathbf{a} - \mathbf{b}$, and $\mathbf{c} \parallel \mathbf{d}$, calculate the value of $k$ and the direction of vector $\mathbf{c}$. | -1 | 0.6875 | 2,281.6875 | 2,351.818182 | 2,127.4 | |
Let $a_n$ be the integer obtained by writing all the integers from $1$ to $n$ from left to right. For example, $a_3 = 123$ and $a_{11} = 1234567891011$. Compute the remainder when $a_{44}$ is divided by $45$. | 9 | 0.625 | 6,167.9375 | 4,953.5 | 8,192 | |
What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are $(\cos 40^\circ,\sin 40^\circ)$, $(\cos 60^\circ,\sin 60^\circ)$, and $(\cos t^\circ,\sin t^\circ)$ is isosceles? | 380 | Let $A = (\cos 40^\circ, \sin 40^\circ)$, $B = (\cos 60^\circ, \sin 60^\circ)$, and $C = (\cos t^\circ, \sin t^\circ)$. We need to find the values of $t$ such that $\triangle ABC$ is isosceles. We consider three cases based on which sides of the triangle are equal.
**Case 1: $AB = AC$**
For $AB = AC$, $C$ must lie on... | 0.3125 | 7,807.5625 | 7,291.2 | 8,042.272727 |
The bases of a trapezoid are 2 cm and 3 cm long. A line passing through the intersection point of the diagonals and parallel to the bases intersects the legs at points X and Y. What is the distance between points X and Y? | 2.6 | 0 | 4,933.3125 | -1 | 4,933.3125 | |
Twelve mayoral candidates each made a statement about how many times lies had been told before their turn. The first candidate said, "Before me, one lie was told." The second candidate said, "Now, two lies have been told." The third candidate said, "Now, three lies have been told," and so on, until the twelfth candidat... | 11 | 0 | 8,192 | -1 | 8,192 | |
Find the smallest real number $a$ such that for any non-negative real numbers $x, y, z$ whose sum is 1, the inequality $a\left(x^2 + y^2 + z^2\right) + xyz \geq \frac{9}{3} + \frac{1}{27}$ holds. | \frac{2}{9} | 0 | 8,133.6875 | -1 | 8,133.6875 | |
Euler's formula states that for a convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces, $V-E+F=2$. A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V$ vertices, $T$ triangular faces and $P$ pentagonal faces meet. What is the value of $100P+10T+V$? | 250 | We know that $V-E = -30 \implies V = E-30$ based off the problem condition. Furthermore, if we draw out a few pentagons as well as triangles on each of side of the pentagons, it's clear that each vertex has 4 edges connected to it, with two triangles and two pentagons for each vertex. However, each edge is used for two... | 0.5625 | 6,201.3125 | 4,817.444444 | 7,980.571429 |
Find the largest integer $n$ such that $3^{512}-1$ is divisible by $2^{n}$. | 11 | Write $$ \begin{aligned} 3^{512}-1 & =\left(3^{256}+1\right)\left(3^{256}-1\right)=\left(3^{256}+1\right)\left(3^{128}+1\right)\left(3^{128}-1\right) \\ & =\cdots=\left(3^{256}+1\right)\left(3^{128}+1\right) \cdots(3+1)(3-1) \end{aligned} $$ Now each factor $3^{2^{k}}+1, k \geq 1$, is divisible by just one factor of 2 ... | 0.25 | 7,339.25 | 4,781 | 8,192 |
Suppose that $x_1+1=x_2+2=x_3+3=\cdots=x_{2008}+2008=x_1+x_2+x_3+\cdots+x_{2008}+2009$. Find the value of $\left\lfloor|S|\right\rfloor$, where $S=\sum_{n=1}^{2008}x_n$.
| 1005 | 0.875 | 5,540.6875 | 5,172.5 | 8,118 | |
Let $r_1,$ $r_2,$ and $r_3$ be the roots of
\[x^3 - 3x^2 + 8 = 0.\]Find the monic polynomial, in $x,$ whose roots are $2r_1,$ $2r_2,$ and $2r_3.$ | x^3 - 6x^2 + 64 | 0.8125 | 4,171.1875 | 3,674.307692 | 6,324.333333 | |
Find all real numbers $x,y,z$ so that
\begin{align*}
x^2 y + y^2 z + z^2 &= 0 \\
z^3 + z^2 y + z y^3 + x^2 y &= \frac{1}{4}(x^4 + y^4).
\end{align*} | (0, 0, 0) |
To solve the system of equations for real numbers \( x, y, \) and \( z \):
\[
x^2 y + y^2 z + z^2 = 0
\]
\[
z^3 + z^2 y + z y^3 + x^2 y = \frac{1}{4}(x^4 + y^4),
\]
we proceed with the following approach:
### Step 1: Analyze the First Equation
The first equation is:
\[
x^2 y + y^2 z + z^2 = 0.
\]
One obvious solu... | 0 | 8,192 | -1 | 8,192 |
Calculate the arithmetic mean of 17, 29, 45, and 64. | 38.75 | 1 | 558.875 | 558.875 | -1 | |
The table below displays the grade distribution of the $30$ students in a mathematics class on the last two tests. For example, exactly one student received a 'D' on Test 1 and a 'C' on Test 2. What percent of the students received the same grade on both tests? | 40\% | To solve this problem, we need to determine the number of students who received the same grade on both Test 1 and Test 2, and then calculate this as a percentage of the total number of students.
1. **Identify students with the same grades on both tests**: According to the problem, the diagonal entries in the table rep... | 0.0625 | 7,523.8125 | 6,580 | 7,586.733333 |
The graph of the function \( y = x^2 + ax + b \) is drawn on a board. Julia drew two lines parallel to the \( O x \) axis on the same diagram. The first line intersects the graph at points \( A \) and \( B \), and the second line intersects the graph at points \( C \) and \( D \). Find the distance between the lines, g... | 24 | 1 | 2,971.0625 | 2,971.0625 | -1 | |
Let $S$ be a set of size 3. How many collections $T$ of subsets of $S$ have the property that for any two subsets $U \in T$ and $V \in T$, both $U \cap V$ and $U \cup V$ are in $T$ ? | 74 | Let us consider the collections $T$ grouped based on the size of the set $X=\bigcup_{U \in T} U$, which we can see also must be in $T$ as long as $T$ contains at least one set. This leads us to count the number of collections on a set of size at most 3 satisfying the desired property with the additional property that t... | 0 | 8,018.9375 | -1 | 8,018.9375 |
From three red balls numbered $1$, $2$, $3$ and two white balls numbered $2$, $3$, find the probability that two balls drawn at random have different numbers and colors. | \dfrac{2}{5} | 0.4375 | 7,675.25 | 7,010.857143 | 8,192 | |
Positive real numbers \( a \) and \( b \) satisfy \( \log _{9} a = \log _{12} b = \log _{16}(3a + b) \). Find the value of \(\frac{b}{a}\). | \frac{1+\sqrt{13}}{2} | 0 | 5,121.25 | -1 | 5,121.25 | |
If I expand $25\cdot 24\cdot 23 \cdots 3 \cdot 2 \cdot 1$, how many zeros are there at the end of the number I get? | 6 | 1 | 1,825.1875 | 1,825.1875 | -1 | |
A triangular box is to be cut from an equilateral triangle of length 30 cm. Find the largest possible volume of the box (in cm³). | 500 | 0 | 4,303.5 | -1 | 4,303.5 | |
A road of 1500 meters is being repaired. In the first week, $\frac{5}{17}$ of the total work was completed, and in the second week, $\frac{4}{17}$ was completed. What fraction of the total work was completed in these two weeks? And what fraction remains to complete the entire task? | \frac{8}{17} | 0.8125 | 747.625 | 834.615385 | 370.666667 | |
Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $39x + y = 1953$. Find the number of such distinct triangles whose area is a positive integer and where $x_1 \neq x_2$. | 625 | 0.75 | 5,798 | 5,000 | 8,192 | |
At 30 palm trees on different parts of an uninhabited island, a sign is attached.
- On 15 of them it says: "Exactly under 15 signs a treasure is buried."
- On 8 of them it says: "Exactly under 8 signs a treasure is buried."
- On 4 of them it says: "Exactly under 4 signs a treasure is buried."
- On 3 of them it says: "... | 15 | 0.0625 | 7,841.8125 | 6,766 | 7,913.533333 | |
Given that $m \angle A= 60^\circ$, $BC=12$ units, $\overline{BD} \perp \overline{AC}$, $\overline{CE} \perp \overline{AB}$ and $m \angle DBC = 3m \angle ECB$, the length of segment $EC$ can be expressed in the form $a(\sqrt{b}+\sqrt{c})$ units where $b$ and $c$ have no perfect-square factors. What is the value of $a+b... | 11 | 0.125 | 8,015.6875 | 6,781.5 | 8,192 | |
Given the equation $\frac{20}{x^2 - 9} - \frac{3}{x + 3} = 2$, determine the root(s). | \frac{-3 - \sqrt{385}}{4} | 0 | 3,678.6875 | -1 | 3,678.6875 | |
In the arithmetic sequence $\{a_n\}$, $d=-2$, $a_1+a_4+a_7+\ldots+a_{31}=50$. Calculate the value of $a_2+a_6+a_{10}+\ldots+a_{42}$. | 82 | 0 | 6,752.4375 | -1 | 6,752.4375 | |
How many unordered pairs of edges of a given square pyramid determine a plane? | 22 | 0 | 7,871.8125 | -1 | 7,871.8125 | |
Ten chairs are evenly spaced around a round table and numbered clockwise from $1$ through $10$. Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse. How many seating arrangements are possible? | 480 | 1. **Fixing the first man's position**: We can fix the first man in any of the 10 seats. This is a common strategy in circular arrangements to avoid equivalent rotations being counted multiple times. Thus, we have 10 choices for the first man.
2. **Seating the other men**: After placing the first man, we must place th... | 0 | 8,192 | -1 | 8,192 |
A solid right prism $ABCDEF$ has a height of 16, as shown. Also, its bases are equilateral triangles with side length 12. Points $X$, $Y$, and $Z$ are the midpoints of edges $AC$, $BC$, and $DC$, respectively. Determine the perimeter of triangle $XYZ$. [asy]
pair A, B, C, D, E, F, X, Y, Z;
A=(0,0);
B=(12,0);
C=(6,-6);
... | 26 | 0.375 | 5,520.0625 | 4,725.666667 | 5,996.7 |
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