problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
The matrix
\[\begin{pmatrix} a & \frac{15}{34} \\ c & \frac{25}{34} \end{pmatrix}\]corresponds to a projection. Enter the ordered pair $(a,c).$ | \left( \frac{9}{34}, \frac{15}{34} \right) | 0.75 | 5,279.3125 | 4,308.416667 | 8,192 | |
Calculate the integral
$$
\int_{0}^{0.1} \cos \left(100 x^{2}\right) d x
$$
with an accuracy of $\alpha=0.001$. | 0.090 | 0.0625 | 8,130.0625 | 7,201 | 8,192 | |
Calculate the areas of the regions bounded by the curves given in polar coordinates.
$$
r=\cos 2 \phi
$$ | \frac{\pi}{2} | 0.25 | 7,507.125 | 5,452.5 | 8,192 | |
The market demand for a certain product over the next four years forms a sequence $\left\{a_{n}\right\}(n=1,2,3,4)$. It is predicted that the percentage increase in annual demand from the first year to the second year is $p_{1}$, from the second year to the third year is $p_{2}$, and from the third year to the fourth y... | $\frac{1}{3}$ | 0 | 8,192 | -1 | 8,192 | |
A box contains 4 white balls and 4 black balls. I draw them out of the box, one at a time. What is the probability that all of my draws alternate colors? | \dfrac{1}{35} | 0.9375 | 5,484.8125 | 5,304.333333 | 8,192 | |
For every positive integer \( n \), define \( a_{n} \) as the last digit of the sum of the digits of the number formed by writing "2005" \( n \) times consecutively. For example, \(\mathrm{a}_{1}=7\) and \(\mathrm{a}_{2}=4\).
a) What are the positive integers \( n \) such that \( a_{n}=0 \)?
b) Calculate \( a_{1}+a_{... | 9025 | 0.9375 | 4,071.9375 | 3,797.266667 | 8,192 | |
Pete thinks of a number. He doubles it, adds 10, multiplies by 4, and ends up with 120. What was his original number? | 10 | 1 | 1,288.125 | 1,288.125 | -1 | |
Given a frustum of a cone with the areas of its two bases being $4\pi$ and $25\pi$ respectively, and the height of the frustum is 4, find the volume and the lateral surface area of the frustum. | 35\pi | 1 | 2,770.1875 | 2,770.1875 | -1 | |
Given positive integers \( a, b, c, \) and \( d \) such that \( a > b > c > d \) and \( a + b + c + d = 2004 \), as well as \( a^2 - b^2 + c^2 - d^2 = 2004 \), what is the minimum value of \( a \)? | 503 | 0.0625 | 8,152.8125 | 7,565 | 8,192 | |
A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 5 units has its center at $(12, 0)$. Find the x-coordinate of the point on the $x$-axis where a line, tangent to both circles, intersects. The line should intersect the x-axis to the right of the origin. | 4.5 | 0 | 6,368.25 | -1 | 6,368.25 | |
Let $p>3$ be a prime and let $a_1,a_2,...,a_{\frac{p-1}{2}}$ be a permutation of $1,2,...,\frac{p-1}{2}$. For which $p$ is it always possible to determine the sequence $a_1,a_2,...,a_{\frac{p-1}{2}}$ if it for all $i,j\in\{1,2,...,\frac{p-1}{2}\}$ with $i\not=j$ the residue of $a_ia_j$ modulo $p$ is known? | p \geq 7 |
Given a prime number \( p > 3 \), we have a permutation \( a_1, a_2, \ldots, a_{\frac{p-1}{2}} \) of the set \( \{1, 2, \ldots, \frac{p-1}{2}\} \). The task is to determine for which primes \( p \) it is always possible to reconstruct the sequence \( a_1, a_2, \ldots, a_{\frac{p-1}{2}} \) if we know the residue of \( ... | 0 | 8,192 | -1 | 8,192 |
Suppose that the lines \(l_1\) and \(l_2\) are parallel, and on \(l_1\) and \(l_2\) there are 10 points \(A_1, A_2, \dots, A_{10}\) and \(B_1, B_2, \dots, B_{10}\), respectively. The line segments \(A_1 B_1, A_2 B_2, \dots, A_{10} B_{10}\) can divide the strip-shaped area enclosed by \(l_1\) and \(l_2\) into at most ho... | 56 | 0.5625 | 6,653.5625 | 5,457 | 8,192 | |
Label one disk "$1$", two disks "$2$", three disks "$3$", ..., fifty disks "$50$". Put these $1+2+3+ \cdots+50=1275$ labeled disks in a box. Disks are then drawn from the box at random without replacement. The minimum number of disks that must be drawn to guarantee drawing at least ten disks with the same label is | 415 | 1. **Understanding the Problem:**
We are given disks labeled from 1 to 50, where the label corresponds to the number of disks with that label. We need to determine the minimum number of disks that must be drawn to ensure that at least ten disks with the same label are drawn.
2. **Calculating Total Disks:**
The t... | 0.8125 | 4,866.4375 | 5,199.692308 | 3,422.333333 |
What is the least number of colors needed to shade the tessellation shown, such that no two tiles sharing a side are the same color?
[asy]
draw((-8,-2)--(8,-2));
draw((-8,2)--(8,2));
draw((-8,6)--(8,6));
draw((-8,10)--(8,10));
draw((-8,14)--(8,14));
path a=(-8,14)--(-6.5,12)--(-10,8)--(-6.5,4)--(-10,0)--(-8,-2);
draw(... | 2 | 0.5625 | 6,536.875 | 5,775.444444 | 7,515.857143 | |
Factor $36-9x^2$. | 9(2-x)(2+x) | 1 | 3,142.5 | 3,142.5 | -1 | |
How many solutions does the equation $\tan(2x)=\cos(\frac{x}{2})$ have on the interval $[0,2\pi]?$ | 5 | To solve the equation $\tan(2x) = \cos\left(\frac{x}{2}\right)$ on the interval $[0, 2\pi]$, we analyze the behavior and intersections of the functions $y = \tan(2x)$ and $y = \cos\left(\frac{x}{2}\right)$.
1. **Properties of $y = \tan(2x)$**:
- The function $\tan(2x)$ has a period of $\frac{\pi}{2}$.
- Vertical... | 0.125 | 8,048.0625 | 7,040.5 | 8,192 |
The distance from the point of intersection of a circle's diameter with 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.
$$
AB = 18, EO = 7, AE = 2BE, R = ?
$$ | 11 | 0.5 | 6,850 | 5,508 | 8,192 | |
How many natural numbers from 1 to 700, inclusive, contain the digit 7 at least once? | 133 | 0.0625 | 7,947 | 4,272 | 8,192 | |
Find the number of the form $7x36y5$ that is divisible by 1375. | 713625 | 0 | 7,211.375 | -1 | 7,211.375 | |
Evaluate the sum \[\frac{1}{3^1} + \frac{2}{3^2} + \frac{3}{3^3} + \cdots + \frac{k}{3^k} + \cdots \] | \frac{3}{4} | 1 | 3,098.25 | 3,098.25 | -1 | |
Let \( x \in \mathbf{R} \). Find the minimum value of the algebraic expression \( (x+1)(x+2)(x+3)(x+4) + 2019 \). | 2018 | 0.6875 | 5,815.4375 | 4,976.363636 | 7,661.4 | |
Let \(p\) be the answer to this question. If a point is chosen uniformly at random from the square bounded by \(x=0, x=1, y=0\), and \(y=1\), what is the probability that at least one of its coordinates is greater than \(p\)? | \frac{\sqrt{5}-1}{2} | The probability that a randomly chosen point has both coordinates less than \(p\) is \(p^{2}\), so the probability that at least one of its coordinates is greater than \(p\) is \(1-p^{2}\). Since \(p\) is the answer to this question, we have \(1-p^{2}=p\), and the only solution of \(p\) in the interval \([0,1]\) is \(\... | 0 | 2,950.4375 | -1 | 2,950.4375 |
Two right triangles, $ABC$ and $ACD$, are joined at side $AC$. Squares are drawn on four of the sides. The areas of three of the squares are 25, 49, and 64 square units. Determine the number of square units in the area of the fourth square. | 138 | 0.125 | 7,512.25 | 5,063 | 7,862.142857 | |
If $f(x) = 2$ for all real numbers $x$, what is the value of $f(x + 2)$? | 2 | 1 | 1,167.625 | 1,167.625 | -1 | |
The number of six-digit even numbers formed by 1, 2, 3, 4, 5, 6 without repeating any digit and with neither 1 nor 3 adjacent to 5 can be calculated. | 108 | 0.1875 | 7,310.0625 | 5,392 | 7,752.692308 | |
The U.S. produces about 8 million tons of apples each year. Initially, $30\%$ of the apples are mixed with other products. If the production increases by 1 million tons, the percentage mixed with other products increases by $5\%$ for each additional million tons. Of the remaining apples, $60\%$ is used to make apple ju... | 2.24 | 0 | 4,460.5 | -1 | 4,460.5 | |
Let point $P$ lie on the curve $y= \frac {1}{2}e^{x}$, and point $Q$ lie on the curve $y=\ln (2x)$. Find the minimum value of $|PQ|$. | \sqrt {2}(1-\ln 2) | 0 | 7,470.5625 | -1 | 7,470.5625 | |
A right cone has a base with a circumference of $16\pi$ inches and a height of 30 inches. The height of this cone is reduced while the circumference stays the same. The volume of the shorter cone is $192\pi$ cubic inches. What is the ratio of the shorter height to the original height? Express your answer as a common fr... | \frac{3}{10} | 0.9375 | 2,364.8125 | 1,976.333333 | 8,192 | |
20 shareholders are seated around a round table. What is the minimum total number of their shares if it is known that:
a) any three of them together have more than 1000 shares,
b) any three consecutive shareholders together have more than 1000 shares? | 6674 | 0.0625 | 8,113.875 | 6,942 | 8,192 | |
Find the smallest positive real number \( r \) with the following property: For every choice of 2023 unit vectors \( \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_{2023} \in \mathbb{R}^2 \), a point \( \mathbf{p} \) can be found in the plane such that for each subset \( S \) of \(\{1, 2, \ldots, 2023\}\), the sum
\[ \... | 2023/2 | 0.1875 | 8,035.25 | 7,356 | 8,192 | |
Given that a bank has 320 employees with an average annual profit of 200,000 yuan per employee, and each remaining employee's profit increases by 20,000 yuan for every laid-off employee, while the bank must pay 60,000 yuan per year for each laid-off employee, and the number of employees required to operate normally is ... | 80 | 0.5 | 7,762 | 7,332 | 8,192 | |
Translate the graph of the function $f(x)=\sin(2x+\varphi)$ ($|\varphi| < \frac{\pi}{2}$) to the left by $\frac{\pi}{6}$ units. If the resulting graph is symmetric about the origin, determine the minimum value of the function $f(x)$ on the interval $\left[0, \frac{\pi}{2}\right]$. | -\frac{\sqrt{3}}{2} | 0 | 5,477.375 | -1 | 5,477.375 | |
There are 2008 distinct points on a circle. If you connect two of these points to form a line and then connect another two points (distinct from the first two) to form another line, what is the probability that the two lines intersect inside the circle? | 1/3 | Given four of these points, there are 3 ways in which to connect two of them and then connect the other two, and of these possibilities exactly one will intersect inside the circle. Thus $1 / 3$ of all the ways to connect two lines and then connect two others have an intersection point inside the circle. | 0.3125 | 7,636.625 | 6,810 | 8,012.363636 |
In the figure below, the smaller circle has a radius of two feet and the larger circle has a radius of four feet. What is the total area of the four shaded regions? Express your answer as a decimal to the nearest tenth.
[asy]
fill((0,0)--(12,0)--(12,-4)--(4,-4)--(4,-2)--(0,-2)--cycle,gray(0.7));
draw((0,0)--(12,0),lin... | 8.6 | 0.0625 | 7,216.25 | 4,055 | 7,427 | |
Given a quadrilateral $ABCD$ with $AB = BC =3$ cm, $CD = 4$ cm, $DA = 8$ cm and $\angle DAB + \angle ABC = 180^o$ . Calculate the area of the quadrilateral.
| 13.2 | 0 | 7,630.9375 | -1 | 7,630.9375 | |
An unfair coin lands on heads with probability $\frac34$ and tails with probability $\frac14$. A heads flip gains $\$3$, but a tails flip loses $\$8$. What is the expected worth of a coin flip? Express your answer as a decimal rounded to the nearest hundredth. | \$0.25 | 1 | 1,776.8125 | 1,776.8125 | -1 | |
Define
\[A = \frac{1}{1^2} + \frac{1}{5^2} - \frac{1}{7^2} - \frac{1}{11^2} + \frac{1}{13^2} + \frac{1}{17^2} - \dotsb,\]which omits all terms of the form $\frac{1}{n^2}$ where $n$ is an odd multiple of 3, and
\[B = \frac{1}{3^2} - \frac{1}{9^2} + \frac{1}{15^2} - \frac{1}{21^2} + \frac{1}{27^2} - \frac{1}{33^2} + \dot... | 10 | 0 | 7,199.0625 | -1 | 7,199.0625 | |
On an island, there are knights, liars, and followers; each one knows who is who among them. All 2018 island inhabitants were lined up and each was asked to answer "Yes" or "No" to the question: "Are there more knights than liars on the island?" The inhabitants answered one by one in such a way that the others could he... | 1009 | 0 | 8,192 | -1 | 8,192 | |
$M$ is the midpoint of the side $AB$ in an equilateral triangle $\triangle ABC.$ The point $D$ on the side $BC$ is such that $BD : DC = 3 : 1.$ On the line passing through $C$ and parallel to $MD$ there is a point $T$ inside the triangle $\triangle ABC$ such that $\angle CTA = 150.$ Find the $\angle MT D.$
[i](K. Ivano... | 120^\circ |
Given an equilateral triangle \(\triangle ABC\), where \(M\) is the midpoint of side \(AB\). We have a point \(D\) on side \(BC\) such that the segment \(BD : DC = 3 : 1\). We need to find \(\angle MTD\) given that there is a point \(T\) on a line passing through \(C\) and parallel to \(MD\) inside \(\triangle ABC\) s... | 0.3125 | 8,078.25 | 7,828 | 8,192 |
The graph below shows a portion of the curve defined by the quartic polynomial $P(x)=x^4+ax^3+bx^2+cx+d$.
[asy]
unitsize(0.8 cm);
int i;
real func (real x) {
return(0.5*(x^4/4 - 2*x^3/3 - 3/2*x^2) + 2.7);
}
draw(graph(func,-4.5,4.5));
draw((-4.5,0)--(4.5,0));
draw((0,-5.5)--(0,5.5));
for (i = -4; i <= 4; ++i) {
... | \text{C} | 0 | 8,192 | -1 | 8,192 | |
A sequence of distinct circles $\omega_1, \omega_2, \cdots$ is inscribed in the parabola $y=x^2$ so that $\omega_n$ and $\omega_{n+1}$ are tangent for all $n$ . If $\omega_1$ has diameter $1$ and touches the parabola at $(0,0)$ , find the diameter of $\omega_{1998}$ . | 3995 | 0.0625 | 8,054.6875 | 5,995 | 8,192 | |
Given the function $f(x)=x+\sin \pi x-3$, calculate the value of $f\left( \dfrac {1}{2015}\right)+f\left( \dfrac {2}{2015}\right)+f\left( \dfrac {3}{2015}\right)+\ldots+f\left( \dfrac {4029}{2015}\right)$. | -8058 | 0.625 | 6,713.875 | 5,827 | 8,192 | |
In $\triangle ABC$, the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$ respectively. If $\angle B = \angle C$ and $7a^2 + b^2 + c^2 = 4\sqrt{3}$, find the maximum value of the area of $\triangle ABC$. | \frac{\sqrt{5}}{5} | 0 | 7,360.5 | -1 | 7,360.5 | |
In circle $O$ with radius 10 units, chords $AC$ and $BD$ intersect at right angles at point $P$. If $BD$ is a diameter of the circle, and the length of $PC$ is 3 units, calculate the product $AP \cdot PB$. | 51 | 0 | 8,066.6875 | -1 | 8,066.6875 | |
For any integer $k\geq 1$, let $p(k)$ be the smallest prime which does not divide $k.$ Define the integer function $X(k)$ to be the product of all primes less than $p(k)$ if $p(k)>2$, and $X(k)=1$ if $p(k)=2.$ Let $\{x_n\}$ be the sequence defined by $x_0=1$, and $x_{n+1}X(x_n)=x_np(x_n)$ for $n\geq 0.$ Find the smalle... | 149 | We go through the terms and look for a pattern. We find that
$x_0 = 1$ $x_8 = 7$
$x_1 = 2$ $x_9 = 14$
$x_2 = 3$ $x_{10} = 21$
$x_3 = 6$ $x_{11} = 42$
$x_4 = 5$ $x_{12} = 35$
$x_5 = 10$ $x_{13} = 70$
$x_6 = 15$ $x_{14} = 105$
$x_7 = 30$ $x_{15} = 210$
Commit to the bash. Eventually, you will receive that $x_{149} = 209... | 0 | 8,192 | -1 | 8,192 |
Find the difference between $1234_5$ and $432_5$ in base $5$. | 302_5 | 0.5 | 6,308.1875 | 4,424.375 | 8,192 | |
Twelve candidates for mayor are participating in a televised debate. At some point, one of them says, "So far, we've lied once." A second then says, "Now it's twice." A third exclaims, "Three times now," and so on, up to the twelfth who claims that before him, they lied twelve times. The presenter then stops the discus... | 11 | 0.0625 | 8,077.125 | 6,354 | 8,192 | |
Find the product of all constants $t$ such that the quadratic $x^2 + tx - 12$ can be factored in the form $(x+a)(x+b)$, where $a$ and $b$ are integers. | 1936 | 0 | 4,654.6875 | -1 | 4,654.6875 | |
The sum of three numbers is $20$. The first is four times the sum of the other two. The second is seven times the third. What is the product of all three? | 28 |
Let the three numbers be $x$, $y$, and $z$. According to the problem, we have the following conditions:
1. $x + y + z = 20$
2. $x = 4(y + z)$
3. $y = 7z$
We can use these equations to find the values of $x$, $y$, and $z$.
#### Step 1: Substitute $y$ in terms of $z$ into the equations
From equation (3), we have $y = ... | 1 | 1,829.25 | 1,829.25 | -1 |
The diagram shows a semicircle with diameter $20$ and the circle with greatest diameter that fits inside the semicircle. The area of the shaded region is $N\pi$ , where $N$ is a positive integer. Find $N$ . | 25 | 0.75 | 5,592.75 | 4,726.333333 | 8,192 | |
A circle has an area of $16\pi$ square units. What are the lengths of the circle's diameter and circumference, in units? | 8\pi | 1 | 1,104.375 | 1,104.375 | -1 | |
Five cards have the numbers 101, 102, 103, 104, and 105 on their fronts. On the reverse, each card has one of five different positive integers: \(a, b, c, d,\) and \(e\) respectively. We know that \(a + 2 = b - 2 = 2c = \frac{d}{2} = e^2\).
Gina picks up the card which has the largest integer on its reverse. What numb... | 105 | 0.125 | 6,473.375 | 933.5 | 7,264.785714 | |
If $8^x = 32$, then $x$ equals: | \frac{5}{3} | 1. **Express 8 and 32 as powers of 2**:
We know that $8 = 2^3$ and $32 = 2^5$. Therefore, the equation $8^x = 32$ can be rewritten using the base 2:
\[
(2^3)^x = 2^5
\]
2. **Simplify the left-hand side**:
Using the power of a power property $(a^m)^n = a^{mn}$, we can simplify the left-hand side:
\[
... | 1 | 2,432.6875 | 2,432.6875 | -1 |
Juan takes a number, adds $2$ to it, multiplies the answer by $2$, subtracts $2$ from the result, and finally divides that number by $2$. If his answer is $7$, what was the original number? | 6 | 1 | 1,715.8125 | 1,715.8125 | -1 | |
Given the fractional equation about $x$: $\frac{x+m}{x+2}-\frac{m}{x-2}=1$ has a solution not exceeding $6$, and the inequality system about $y$: $\left\{\begin{array}{l}{m-6y>2}\\{y-4\leq 3y+4}\end{array}\right.$ has exactly four integer solutions, then the sum of the integers $m$ that satisfy the conditions is ____. | -2 | 0.1875 | 7,659.8125 | 8,187 | 7,538.153846 | |
Given $f(x)=\frac{1}{x}$, calculate the limit of $\frac{f(2+3\Delta x)-f(2)}{\Delta x}$ as $\Delta x$ approaches infinity. | -\frac{3}{4} | 0 | 3,591.9375 | -1 | 3,591.9375 | |
An artist wants to completely cover a rectangle with identically sized squares which do not overlap and do not extend beyond the edges of the rectangle. If the rectangle is \(60 \frac{1}{2} \mathrm{~cm}\) long and \(47 \frac{2}{3} \mathrm{~cm}\) wide, what is the minimum number of squares required? | 858 | 0.6875 | 6,539 | 5,787.636364 | 8,192 | |
Let $a,b$ be integers greater than $1$. Find the largest $n$ which cannot be written in the form $n = 7a + 5b$. | 47 | 0.1875 | 7,769.75 | 7,343.666667 | 7,868.076923 | |
Point \( M \) divides the side \( BC \) of parallelogram \( ABCD \) in the ratio \( BM: MC = 1: 3 \). Line \( AM \) intersects diagonal \( BD \) at point \( K \). Find the area of quadrilateral \( CMKD \) if the area of parallelogram \( ABCD \) is 1. | \frac{19}{40} | 0.375 | 7,613.1875 | 6,900.666667 | 8,040.7 | |
In trapezoid $PQRS$ with $\overline{QR}\parallel\overline{PS}$, let $QR = 1500$ and $PS = 3000$. Let $\angle P = 37^\circ$, $\angle S = 53^\circ$, and $X$ and $Y$ be the midpoints of $\overline{QR}$ and $\overline{PS}$, respectively. Find the length $XY$. | 750 | 0.5625 | 6,489.5625 | 5,858.888889 | 7,300.428571 | |
How many four-digit positive integers exist, all of whose digits are 0's, 2's, and/or 5's, and the number does not start with 0? | 54 | 1 | 1,871.125 | 1,871.125 | -1 | |
Add $91.234$ to $42.7689$ and round your answer to the nearest hundredth. | 134.00 | 0.8125 | 4,667.625 | 4,246.846154 | 6,491 | |
Given the array: $(1,1,1)$, $(2,2,4)$, $(3,4,12)$, $(4,8,32)$, $\ldots$, $(a_{n}, b_{n}, c_{n})$, find the value of $c_{7}$. | 448 | 0.8125 | 4,401.25 | 3,526.461538 | 8,192 | |
Given a rectangular grid measuring 8 by 6, there are $48$ grid points, including those on the edges. Point $P$ is placed at the center of the rectangle. Find the probability that the line $PQ$ is a line of symmetry of the rectangle, given that point $Q$ is randomly selected from the other $47$ points. | \frac{12}{47} | 0.1875 | 6,137.0625 | 3,608.333333 | 6,720.615385 | |
Cátia leaves school every day at the same time and returns home by bicycle. When she pedals at $20 \mathrm{~km/h}$, she arrives home at $4:30$ PM. If she pedals at $10 \mathrm{~km/h}$, she arrives home at $5:15$ PM. At what speed should she pedal to arrive home at $5:00$ PM? | 12 | 0.5625 | 4,843 | 3,678.222222 | 6,340.571429 | |
Consider an equilateral triangle $ABC$ with side length $3$. Right triangle $CBD$ is constructed outwardly on side $BC$ of triangle $ABC$ such that $CB = BD$ and $BCD$ is a right angle at $B$. Find $\sin^2\left(\angle CAD\right)$.
A) $\frac{3}{4}$
B) $\frac{1}{4}$
C) $\frac{1}{2}$
D) $\frac{\sqrt{2}}{2}$
E) $\f... | \frac{1}{2} | 0 | 6,592.6875 | -1 | 6,592.6875 | |
Find the quadratic polynomial $p(x)$ such that $p(-3) = 10,$ $p(0) = 1,$ and $p(2) = 5.$ | x^2 + 1 | 1 | 2,986.625 | 2,986.625 | -1 | |
Find the integer $n,$ $-180 < n < 180,$ such that $\tan n^\circ = \tan 345^\circ.$ | -15 | 0.5625 | 6,342.9375 | 5,883.888889 | 6,933.142857 | |
How many different numbers are obtainable from five 5s by first concatenating some of the 5s, then multiplying them together? For example, we could do $5 \cdot 55 \cdot 55,555 \cdot 55$, or 55555, but not $5 \cdot 5$ or 2525. | 7 | If we do 55555, then we're done. Note that $5,55,555$, and 5555 all have completely distinguishable prime factorizations. This means that if we are given a product of them, we can obtain the individual terms. The number of 5555's is the exponent of 101, the number of 555's is the exponent of 37, the number of 55's is t... | 0.0625 | 8,158.5625 | 7,657 | 8,192 |
Given a quadrilateral $ABCD$ where the internal angles form an arithmetic progression, the angles are such that $\angle ADB$ is double the $\angle DBA$. Triangles $ABD$ and $DCB$ are similar with $\angle DBA = \angle DCB$ and $\angle ADB = \angle CBD$. Furthermore, the angles within triangle $ABD$ form an arithmetic pr... | 90 | 0.0625 | 8,065.5625 | 6,326 | 8,181.533333 | |
A restricted path of length $n$ is a path of length $n$ such that for all $i$ between 1 and $n-2$ inclusive, if the $i$th step is upward, the $i+1$st step must be rightward. Find the number of restricted paths that start at $(0,0)$ and end at $(7,3)$. | 56 | This is equal to the number of lattice paths from $(0,0)$ to $(7,3)$ that use only rightward and diagonal (upward+rightward) steps plus the number of lattice paths from $(0,0)$ to $(7,2)$ that use only rightward and diagonal steps, which is equal to the number of paths (as defined above) from $(0,0)$ to $(4,3)$ plus th... | 0.375 | 6,972.4375 | 4,939.833333 | 8,192 |
What is the area of the smallest square that can enclose a circle with a radius of 5? | 100 | 0.625 | 4,231.6875 | 3,425.1 | 5,576 | |
Evaluate the infinite sum $\sum_{n=1}^{\infty} \frac{n}{n^{4}+4}$. | \frac{3}{8} | We have $$\begin{aligned} \sum_{n=1}^{\infty} \frac{n}{n^{4}+4} & =\sum_{n=1}^{\infty} \frac{n}{\left(n^{2}+2 n+2\right)\left(n^{2}-2 n+2\right)} \\ & =\frac{1}{4} \sum_{n=1}^{\infty}\left(\frac{1}{n^{2}-2 n+2}-\frac{1}{n^{2}+2 n+2}\right) \\ & =\frac{1}{4} \sum_{n=1}^{\infty}\left(\frac{1}{(n-1)^{2}+1}-\frac{1}{(n+1)^... | 0.5 | 7,279.375 | 6,854.875 | 7,703.875 |
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $|\overrightarrow {a}|=5$, $|\overrightarrow {a}- \overrightarrow {b}|=6$, and $|\overrightarrow {a}+ \overrightarrow {b}|=4$, find the projection of vector $\overrightarrow {b}$ on vector $\overrightarrow {a}$. | -1 | 0.8125 | 4,239.75 | 4,298.692308 | 3,984.333333 | |
A particular fortune cookie lists four two-digit positive integers as your lucky numbers. The first three are 57, 13, and 72, but you got sauce on the last one and can't read it. If the sum of the digits of all four numbers equals $\frac{1}{5}$ of the sum of all four numbers, what is the smallest possibility for the fo... | 38 | 0.875 | 6,375.625 | 6,116.142857 | 8,192 | |
Given the function $f(x)=2 \sqrt {3}\sin \frac {ωx}{2}\cos \frac {ωx}{2}-2\sin ^{2} \frac {ωx}{2}(ω > 0)$ with a minimum positive period of $3π$.
(I) Find the interval where the function $f(x)$ is monotonically increasing.
(II) In $\triangle ABC$, $a$, $b$, and $c$ correspond to angles $A$, $B$, and $C$ respectively, w... | \frac {5 \sqrt {3}+12}{26} | 0 | 7,921.5625 | -1 | 7,921.5625 | |
For $\mathbf{v} = \begin{pmatrix} -10 \\ 6 \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} 15 \\ -9 \end{pmatrix}$, compute $\text{proj}_{\mathbf{w}} \mathbf{v}$. | \begin{pmatrix} -10 \\ 6 \end{pmatrix} | 1 | 2,609.6875 | 2,609.6875 | -1 | |
A fruit store sells a type of fruit. The purchase price is $30$ yuan per kilogram. According to a survey, when the selling price is $40$ yuan per kilogram, the store can sell an average of $500$ kilograms per day. For every $1$ yuan increase in the selling price, the store sells $10$ kilograms less. Let the selling pri... | 60 | 0.875 | 4,779.9375 | 4,554 | 6,361.5 | |
In right triangle $XYZ$, we have $\angle X = \angle Y$ and $XY = 8\sqrt{2}$. What is the area of $\triangle XYZ$? | 32 | 1 | 1,449.3125 | 1,449.3125 | -1 | |
Triangle $ABC$ is a right triangle with $AC = 7,$ $BC = 24,$ and right angle at $C.$ Point $M$ is the midpoint of $AB,$ and $D$ is on the same side of line $AB$ as $C$ so that $AD = BD = 15.$ Given that the area of triangle $CDM$ may be expressed as $\frac {m\sqrt {n}}{p},$ where $m,$ $n,$ and $p$ are positive integers... | 578 | 0.0625 | 8,141.875 | 7,390 | 8,192 | |
30 students from five courses created 40 problems for the olympiad, with students from the same course creating the same number of problems, and students from different courses creating different numbers of problems. How many students created exactly one problem? | 26 | 0 | 8,192 | -1 | 8,192 | |
Consider a rectangle $ABCD$ with $BC = 2 \cdot AB$ . Let $\omega$ be the circle that touches the sides $AB$ , $BC$ , and $AD$ . A tangent drawn from point $C$ to the circle $\omega$ intersects the segment $AD$ at point $K$ . Determine the ratio $\frac{AK}{KD}$ .
*Proposed by Giorgi Arabidze, Georgia* | 1/2 | 0.5 | 7,430.125 | 6,668.25 | 8,192 | |
Evaluate $\left\lfloor -\frac{5}{3}\right\rfloor.$ | -2 | 1 | 2,965.75 | 2,965.75 | -1 | |
The length of the chord cut by one of the asymptotes of the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \ (a > 0, b > 0)$ on the circle $x^2 + y^2 - 6x + 5 = 0$ is $2$. Find the eccentricity of the hyperbola. | \dfrac{\sqrt{6}}{2} | 0 | 4,551.875 | -1 | 4,551.875 | |
How many triangles can be formed using the vertices of a regular hexacontagon (a 60-sided polygon), avoiding the use of any three consecutive vertices in forming these triangles? | 34160 | 0.4375 | 6,839.5625 | 5,622.428571 | 7,786.222222 | |
In the plane rectangular coordinate system $O-xy$, if $A(\cos\alpha, \sin\alpha)$, $B(\cos\beta, \sin\beta)$, $C\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$, then one possible value of $\beta$ that satisfies $\overrightarrow{OC}=\overrightarrow{OB}-\overrightarrow{OA}$ is ______. | \frac{2\pi}{3} | 0.1875 | 7,893.9375 | 7,320.666667 | 8,026.230769 | |
What is the minimum length of the second longest side of a triangle with an area of one unit? | \sqrt{2} | 0.5 | 7,464 | 6,736 | 8,192 | |
To walk 2 km, ride 3 km by bicycle, and drive 20 km by car, Uncle Vanya needs 1 hour and 6 minutes. If he needs to walk 5 km, ride 8 km by bicycle, and drive 30 km by car, it takes him 2 hours and 24 minutes. How much time will Uncle Vanya need to walk 4 km, ride 5 km by bicycle, and drive 80 km by car? | 174 | 0.0625 | 7,182.8125 | 5,614 | 7,287.4 | |
Bag A has 3 white marbles and 4 black marbles. Bag B has 6 yellow marbles and 4 blue marbles. Bag C has 2 yellow marbles and 5 blue marbles. A marble is drawn at random from Bag A. If it is white, a marble is drawn at random from Bag B, otherwise, if it is black, a marble is drawn at random from Bag C. What is the... | \frac{103}{245} | 0.9375 | 4,722.9375 | 4,491.666667 | 8,192 | |
Find the sum of the first 10 elements that appear both among the terms of the arithmetic progression $\{4,7,10,13, \ldots\}$ and the geometric progression $\{20,40,80,160, \ldots\}$. | 13981000 | 0.75 | 6,024.9375 | 5,740.333333 | 6,878.75 | |
Find the terminating decimal expansion of $\frac{13}{200}$. | 0.052 | 0 | 1,833.875 | -1 | 1,833.875 | |
A school has between 150 and 250 students enrolled. Each day, all the students split into eight different sections for a special workshop. If two students are absent, each section can contain an equal number of students. Find the sum of all possible values of student enrollment at the school. | 2626 | 0.5625 | 4,914.4375 | 3,082 | 7,270.428571 | |
There is a $6 \times 6$ square in which all cells are white. In one move, you are allowed to change the color of both cells in any domino (rectangle consisting of two cells) to the opposite color. What is the minimum number of moves required to obtain a square with a checkerboard pattern? Do not forget to explain why f... | 18 | 0.375 | 7,510.5625 | 6,374.833333 | 8,192 | |
Given \((1+x-x^2)^{10} = a_0 + a_1 x + a_2 x^2 + \cdots + a_{20} x^{20}\), find \( a_0 + a_1 + 2a_2 + 3a_3 + \cdots + 20a_{20} \). | -9 | 0.375 | 6,283.3125 | 6,262.666667 | 6,295.7 | |
Two circles have radius 5 and 26. The smaller circle passes through center of the larger one. What is the difference between the lengths of the longest and shortest chords of the larger circle that are tangent to the smaller circle?
*Ray Li.* | 52 - 2\sqrt{235} | 0 | 7,890.375 | -1 | 7,890.375 | |
Let $S$ be the sum of all positive integers $n$ such that $\frac{3}{5}$ of the positive divisors of $n$ are multiples of $6$ and $n$ has no prime divisors greater than $3$ . Compute $\frac{S}{36}$ . | 2345 | 0.9375 | 4,936.625 | 4,875.4 | 5,855 | |
Two diameters and one radius are drawn in a circle of radius 1, dividing the circle into 5 sectors. The largest possible area of the smallest sector can be expressed as $\frac{a}{b} \pi$, where $a, b$ are relatively prime positive integers. Compute $100a+b$. | 106 | Let the two diameters split the circle into four sectors of areas $A, B$, $A$, and $B$, where $A+B=\frac{\pi}{2}$. Without loss of generality, let $A \leq B$. If our radius cuts into a sector of area $A$, the area of the smallest sector will be of the form $\min (x, A-x)$. Note that $\min (A-x, x) \leq \frac{A}{2} \leq... | 0 | 8,192 | -1 | 8,192 |
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(0)+1=f(1)$ and for any real numbers $x$ and $y$,
$$ f(xy-x)+f(x+f(y))=yf(x)+3 $$ | f(x) = x + 1 |
We are given the functional equation and conditions to determine all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that:
\[ f(0) + 1 = f(1) \]
and for all real numbers \( x \) and \( y \),
\[ f(xy - x) + f(x + f(y)) = y f(x) + 3. \]
To solve this, we will proceed as follows:
### Step 1: Simplify Using Sp... | 0 | 7,402.9375 | -1 | 7,402.9375 |
Determine the smallest positive real number $ k$ with the following property. Let $ ABCD$ be a convex quadrilateral, and let points $ A_1$, $ B_1$, $ C_1$, and $ D_1$ lie on sides $ AB$, $ BC$, $ CD$, and $ DA$, respectively. Consider the areas of triangles $ AA_1D_1$, $ BB_1A_1$, $ CC_1B_1$ and $ DD_1C_1$; let $ S$ be... | 1 |
To determine the smallest positive real number \( k \) such that for any convex quadrilateral \( ABCD \) with points \( A_1 \), \( B_1 \), \( C_1 \), and \( D_1 \) on sides \( AB \), \( BC \), \( CD \), and \( DA \) respectively, the inequality \( kS_1 \ge S \) holds, where \( S \) is the sum of the areas of the two s... | 0 | 8,192 | -1 | 8,192 |
Five people are sitting around a round table, with identical coins placed in front of each person. Everyone flips their coin simultaneously. If the coin lands heads up, the person stands up; if it lands tails up, the person remains seated. Determine the probability that no two adjacent people stand up. | \frac{11}{32} | 0.125 | 7,976.625 | 6,469 | 8,192 |
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