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 |
|---|---|---|---|---|---|---|
Let $x$ and $y$ be distinct real numbers such that
\[
\begin{vmatrix} 2 & 3 & 7 \\ 4 & x & y \\ 4 & y & x+1 \end{vmatrix}
= 0.\]Find $x + y.$ | 20 | 0.0625 | 7,959.75 | 4,759 | 8,173.133333 | |
Rectangle \(ABCD\) has area 2016. Point \(Z\) is inside the rectangle and point \(H\) is on \(AB\) so that \(ZH\) is perpendicular to \(AB\). If \(ZH : CB = 4 : 7\), what is the area of pentagon \(ADCZB\)? | 1440 | 0.125 | 7,820.8125 | 5,222.5 | 8,192 | |
Given \( x, y, z \in [0, 1] \), find the maximum value of \( M = \sqrt{|x-y|} + \sqrt{|y-z|} + \sqrt{|z-x|} \). | \sqrt{2} + 1 | 0 | 7,253.3125 | -1 | 7,253.3125 | |
Six students with distinct heights take a group photo, the photographer arranges them into two rows with three people each. What is the probability that every student in the back row is taller than the students in the front row? | \frac{1}{20} | 0.625 | 7,026.875 | 6,327.8 | 8,192 | |
There are four positive integers that are divisors of each number in the list $$20, 40, 100, 80, 180.$$ Find the sum of these four positive integers. | 12 | 0.4375 | 6,983.5625 | 5,523.714286 | 8,119 | |
Consider a $5 \times 5$ grid of squares, where each square is either colored blue or left blank. The design on the grid is considered symmetric if it remains unchanged under a 90° rotation around the center. How many symmetric designs can be created if there must be at least one blue square but not all squares can be b... | 30 | 0 | 6,431.1875 | -1 | 6,431.1875 | |
Let $ ABC$ be an isosceles triangle with $ AB\equal{}AC$ and $ \angle A\equal{}20^\circ$. On the side $ AC$ consider point $ D$ such that $ AD\equal{}BC$. Find $ \angle BDC$. | $30^\circ$ |
Let triangle \( ABC \) be an isosceles triangle with \( AB = AC \) and \( \angle A = 20^\circ \). We are given a point \( D \) on side \( AC \) such that \( AD = BC \). Our task is to find \( \angle BDC \).
#### Step-by-Step Solution:
1. **Base Angle Calculation:**
Since \( ABC \) is an isosceles triangle with \(... | 0 | 7,778.1875 | -1 | 7,778.1875 |
Given that the vertex of angle \\(θ\\) is at the origin of coordinates, its initial side coincides with the positive half-axis of \\(x\\), and its terminal side is on the line \\(4x+3y=0\\), then \\( \dfrac{\cos \left( \left. \dfrac{π}{2}+θ \right. \right)-\sin (-π-θ)}{\cos \left( \left. \dfrac{11π}{2}-θ \right. \right... | \dfrac{8}{7} | 0.6875 | 5,322.3125 | 4,819.090909 | 6,429.4 | |
Let $D$ be the circle with the equation $x^2 + 8x + 20y + 89 = -y^2 - 6x$. Find the value of $c + d + s$ where $(c, d)$ is the center of $D$ and $s$ is its radius. | -17 + 2\sqrt{15} | 0.8125 | 5,114.375 | 4,755.384615 | 6,670 | |
The cells of a $8 \times 8$ table are initially white. Alice and Bob play a game. First Alice paints $n$ of the fields in red. Then Bob chooses $4$ rows and $4$ columns from the table and paints all fields in them in black. Alice wins if there is at least one red field left. Find the least value of $n$ such that Alice ... | 13 |
Consider a \( 8 \times 8 \) table where Alice and Bob play a game. Initially, all cells in this table are white. Alice begins by painting \( n \) of the cells red. After that, Bob selects 4 rows and 4 columns and paints all cells in these rows and columns black. Alice wins if at least one red cell remains unpainted by... | 0 | 7,940.6875 | -1 | 7,940.6875 |
There are several positive numbers written on the board. The sum of the five largest numbers is 0.29 of the sum of all the numbers, and the sum of the five smallest numbers is 0.26 of the sum of all the numbers. How many numbers in total are written on the board? | 18 | 0.6875 | 6,490.4375 | 5,878.636364 | 7,836.4 | |
Each face of a die is arranged so that the sum of the numbers on opposite faces is 7. In the arrangement shown with three dice, only seven faces are visible. What is the sum of the numbers on the faces that are not visible in the given image? | 41 | 0 | 6,812.125 | -1 | 6,812.125 | |
The two banners below:
中华少年 杯赛联谊 切磋勾股
炎黄子孙 惠州弘志 振兴中华
Each character represents a non-zero natural number less than 25. Different characters represent different numbers, and the same characters represent the same number. It is known that the average value of these 34 characters is 12. What is the maximum possible sum o... | 46 | 0 | 8,158.4375 | -1 | 8,158.4375 | |
Among the 9 natural numbers $1,2,3, \cdots, 9$, if 3 numbers are chosen, let $x$ be the number of pairs of adjacent numbers among the chosen 3 numbers (for example, if the 3 chosen numbers are $1,2,3$, there are 2 pairs of adjacent numbers: 1,2 and 2,3, so the value of $x$ is 2). What is the expected value of $x$? | 2/3 | 0 | 8,192 | -1 | 8,192 | |
For each integer $n\geq3$, let $f(n)$ be the number of $3$-element subsets of the vertices of the regular $n$-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of $n$ such that $f(n+1)=f(n)+78$. | 245 | Considering $n \pmod{6}$, we have the following formulas:
Even and a multiple of 3: $\frac{n(n-4)}{2} + \frac{n}{3}$
Even and not a multiple of 3: $\frac{n(n-2)}{2}$
Odd and a multiple of 3: $\frac{n(n-3)}{2} + \frac{n}{3}$
Odd and not a multiple of 3: $\frac{n(n-1)}{2}$
To derive these formulas, we note the follo... | 0 | 7,972.0625 | -1 | 7,972.0625 |
Given that $a,b,c,d,e$ are real numbers such that
$a+b+c+d+e=8$ ,
$a^2+b^2+c^2+d^2+e^2=16$ .
Determine the maximum value of $e$ . | \[
\frac{16}{5}
\] | By Cauchy Schwarz, we can see that $(1+1+1+1)(a^2+b^2+c^2+d^2)\geq (a+b+c+d)^2$ thus $4(16-e^2)\geq (8-e)^2$ Finally, $e(5e-16) \geq 0$ which means $\frac{16}{5} \geq e \geq 0$ so the maximum value of $e$ is $\frac{16}{5}$ .
from: Image from Gon Mathcenter.net | 0 | 4,419.5 | -1 | 4,419.5 |
In the Cartesian coordinate system $xOy$, a line segment of length $\sqrt{2}+1$ has its endpoints $C$ and $D$ sliding on the $x$-axis and $y$-axis, respectively. It is given that $\overrightarrow{CP} = \sqrt{2} \overrightarrow{PD}$. Let the trajectory of point $P$ be curve $E$.
(I) Find the equation of curve $E$;
(II... | \frac{\sqrt{6}}{2} | 0 | 7,963.375 | -1 | 7,963.375 | |
A park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km. How many kilometers is she from her starting point? | $\sqrt{13}$ | 0 | 7,571.0625 | -1 | 7,571.0625 | |
Let \( T \) be the set of all positive divisors of \( 60^{100} \). \( S \) is a subset of \( T \) such that no number in \( S \) is a multiple of another number in \( S \). Find the maximum value of \( |S| \). | 10201 | 0.1875 | 7,266.875 | 4,711.333333 | 7,856.615385 | |
Find the minimum positive integer $n\ge 3$, such that there exist $n$ points $A_1,A_2,\cdots, A_n$ satisfying no three points are collinear and for any $1\le i\le n$, there exist $1\le j \le n (j\neq i)$, segment $A_jA_{j+1}$ pass through the midpoint of segment $A_iA_{i+1}$, where $A_{n+1}=A_1$ | 6 |
To find the minimum positive integer \( n \geq 3 \) such that there exist \( n \) points \( A_1, A_2, \ldots, A_n \) satisfying no three points are collinear and for any \( 1 \leq i \leq n \), there exists \( 1 \leq j \leq n \) (with \( j \neq i \)), such that the segment \( A_jA_{j+1} \) passes through the midpoint o... | 0.0625 | 8,115 | 8,174 | 8,111.066667 |
Evaluate $(128)^{\frac{1}{3}}(729)^{\frac{1}{2}}$. | 108 \cdot 2^{\frac{1}{3}} | 0 | 3,242.1875 | -1 | 3,242.1875 | |
If Person B trades all their chairs for the same number of tables as Person A, Person B needs to pay an additional 320 yuan. If Person B does not pay the extra money, they would receive 5 fewer tables. It is known that the price of 3 tables is 48 yuan less than the price of 5 chairs. How many chairs does Person B origi... | 20 | 0.1875 | 7,910.5 | 7,446.333333 | 8,017.615385 | |
How many ordered quadruples $(a, b, c, d)$ of four distinct numbers chosen from the set $\{1,2,3, \ldots, 9\}$ satisfy $b<a, b<c$, and $d<c$? | 630 | Given any 4 elements $p<q<r<s$ of $\{1,2, \ldots, 9\}$, there are 5 ways of rearranging them to satisfy the inequality: prqs, psqr, $q s p r, q r p s$, and $r s p q$. This gives a total of $\binom{9}{4} \cdot 5=630$ quadruples. | 0 | 8,192 | -1 | 8,192 |
What is the sum of all the integers between -20.5 and -1.5? | -210 | 0.0625 | 4,443.25 | 3,246 | 4,523.066667 | |
Evaluate $\sqrt[3]{1+27} + \sqrt[3]{1+\sqrt[3]{27}}$. | \sqrt[3]{28} + \sqrt[3]{4} | 0.0625 | 7,937.4375 | 5,662 | 8,089.133333 | |
When two positive integers are multiplied, the result is 24. When these two integers are added, the result is 11. What is the result when the smaller integer is subtracted from the larger integer? | 5 | The positive divisor pairs of 24 are: 1 and 24, 2 and 12, 3 and 8, 4 and 6. Of these, the pair whose sum is 11 is 3 and 8. The difference between these two integers is $8 - 3 = 5$. | 1 | 1,436.5625 | 1,436.5625 | -1 |
Find the minimum value of
\[\sqrt{x^2 + (1 + 2x)^2} + \sqrt{(x - 1)^2 + (x - 1)^2}\]over all real numbers $x.$ | \sqrt{2} \times \sqrt{5} | 0 | 8,176.9375 | -1 | 8,176.9375 | |
In how many ways can 13 bishops be placed on an $8 \times 8$ chessboard such that:
(i) a bishop is placed on the second square in the second row,
(ii) at most one bishop is placed on each square,
(iii) no bishop is placed on the same diagonal as another bishop,
(iv) every diagonal contains a bishop?
(For the purposes o... | 1152 | 0 | 7,684.375 | -1 | 7,684.375 | |
A regular tetrahedron has the numbers $1,2,3,4$ written on its four faces. Four such identical regular tetrahedrons are thrown simultaneously onto a table. What is the probability that the product of the numbers on the four faces in contact with the table is divisible by $4$? | $\frac{13}{16}$ | 0 | 7,005.625 | -1 | 7,005.625 | |
Determine the minimum number of digits to the right of the decimal point required to express the fraction $\frac{987654321}{2^{30} \cdot 5^5}$ as a decimal. | 30 | 0.5625 | 5,489.3125 | 3,387.222222 | 8,192 | |
We divide the height of a cone into three equal parts, and through the division points, we lay planes parallel to the base. How do the volumes of the resulting solids compare to each other? | 1:7:19 | 0.125 | 7,926.1875 | 6,065.5 | 8,192 | |
The letters \( A, J, H, S, M, E \) and the numbers \( 1, 9, 8, 9 \) are "rotated" as follows:
\begin{tabular}{rrr}
AJHSME & 1989 & \\
1. JHSMEA & 9891 & (1st rotation) \\
2. HSMEAJ & 8919 & (2nd rotation) \\
3. SMEAJH & 9198 & (3rd rotation) \\
..... & &
\end{tabular}
To make AJHSME1989 reappear, the minimum number o... | 12 | 0.6875 | 5,182.6875 | 3,981.636364 | 7,825 | |
Define a function $f$ by $f(1)=1$, $f(2)=2$, and for all integers $n \geq 3$,
\[ f(n) = f(n-1) + f(n-2) + n. \]
Determine $f(10)$. | 420 | 0 | 5,033.1875 | -1 | 5,033.1875 | |
Given that $f: x \rightarrow \sqrt{x}$ is a function from set $A$ to set $B$.
1. If $A=[0,9]$, then the range of the function $f(x)$ is ________.
2. If $B={1,2}$, then $A \cap B =$ ________. | {1} | 0 | 2,880.4375 | -1 | 2,880.4375 | |
Elective 4-4: Coordinate System and Parametric Equations
Given the parametric equations of curve \\(C\\) as \\(\begin{cases}x=2\cos \left(\theta\right) \\ y= \sqrt{3}\sin \left(\theta\right)\end{cases} \\), in the same plane Cartesian coordinate system, the points on curve \\(C\\) are transformed by \\(\begin{cases} {... | \dfrac{3\sqrt{3}}{5} | 0 | 6,610.5625 | -1 | 6,610.5625 | |
Given the sequence \( a_{1}, a_{2}, \cdots, a_{n}, \cdots \) that satisfies \( a_{1}=a_{2}=1 \) and \( a_{3}=2 \), and for any natural number \( n \), \( a_{n} a_{n+1} a_{n+2} \neq 1 \), and \( a_{n} a_{n+1} a_{n+2} a_{n+3}=a_{n}+a_{n+1}+a_{n+2}+a_{n+3} \), find the value of \( a_{1}+a_{2}+\cdots+a_{100} \). | 200 | 0.75 | 4,994.9375 | 4,487.25 | 6,518 | |
In a square diagram divided into 64 smaller equilateral triangular sections, shading follows a pattern where every alternate horizontal row of triangles is filled. If this pattern begins from the first row at the bottom (considering it as filled), what fraction of the triangle would be shaded in such a 8x8 triangular-s... | \frac{1}{2} | 0 | 6,274.125 | -1 | 6,274.125 | |
Let $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8$ be a regular octagon. Let $M_1$, $M_3$, $M_5$, and $M_7$ be the midpoints of sides $\overline{A_1 A_2}$, $\overline{A_3 A_4}$, $\overline{A_5 A_6}$, and $\overline{A_7 A_8}$, respectively. For $i = 1, 3, 5, 7$, ray $R_i$ is constructed from $M_i$ towards the interior of the octagon... | 37 | Let $A_1A_2 = 2$. Then $B_1$ and $B_3$ are the projections of $M_1$ and $M_5$ onto the line $B_1B_3$, so $2=B_1B_3=-M_1M_5\cos x$, where $x = \angle A_3M_3B_1$. Then since $M_1M_5 = 2+2\sqrt{2}, \cos x = \dfrac{-2}{2+2\sqrt{2}}= 1-\sqrt{2}$, $\cos 2x = 2\cos^2 x -1 = 5 - 4\sqrt{2} = 5-\sqrt{32}$, and $m+n=\boxed{037}$. | 0 | 8,192 | -1 | 8,192 |
For any two non-zero plane vectors $\overrightarrow{\alpha}$ and $\overrightarrow{\beta}$, define $\overrightarrow{\alpha}○\overrightarrow{\beta}=\dfrac{\overrightarrow{\alpha}⋅\overrightarrow{\beta}}{\overrightarrow{\beta}⋅\overrightarrow{\beta}}$. If plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy... | \dfrac{3}{2} | 0.4375 | 7,348.375 | 6,263.714286 | 8,192 | |
The base of the pyramid is a parallelogram with adjacent sides of 9 cm and 10 cm, and one of the diagonals measuring 11 cm. The opposite lateral edges are equal, and each of the longer edges is 10.5 cm. Calculate the volume of the pyramid. | 200 | 0 | 7,382.5 | -1 | 7,382.5 | |
What is the area of the portion of the circle defined by $x^2-12x+y^2=28$ that lies above the $x$-axis and to the right of the line $y=6-x$? | 24 \pi | 0 | 8,192 | -1 | 8,192 | |
In $\triangle ABC$, $AB = 3$, $BC = 4$, and $CA = 5$. Circle $\omega$ intersects $\overline{AB}$ at $E$ and $B$, $\overline{BC}$ at $B$ and $D$, and $\overline{AC}$ at $F$ and $G$. Given that $EF=DF$ and $\frac{DG}{EG} = \frac{3}{4}$, length $DE=\frac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive inte... | 41 | Call $DE=x$ and as a result $DF=EF=\frac{x\sqrt{2}}{2}, EG=\frac{4x}{5}, GD=\frac{3x}{5}$. Since $EFGD$ is cyclic we just need to get $DG$ and using LoS(for more detail see the $2$nd paragraph of Solution $2$) we get $AG=\frac{5}{2}$ and using a similar argument(use LoS again) and subtracting you get $FG=\frac{5}{14}$ ... | 0 | 8,192 | -1 | 8,192 |
In triangle $ABC$, point $A$ is at $(1, 1)$, point $B$ is at $(4, 2)$, and point $C$ is at $(-4, 6)$.
(1) Determine the equation of the line where the median to side $BC$ lies;
(2) Determine the length of the altitude to side $BC$ and the area of triangle $ABC$. | 10 | 1 | 3,053.5625 | 3,053.5625 | -1 | |
Let $x$ and $y$ be real numbers such that $3x + 2y \le 7$ and $2x + 4y \le 8.$ Find the largest possible value of $x + y.$ | \frac{11}{4} | 0.625 | 6,563.5 | 5,586.4 | 8,192 | |
Given points P(0, -3) and Q(5, 3) in the xy-plane; point R(x, m) is taken so that PR + RQ is a minimum where x is fixed to 3, determine the value of m. | \frac{3}{5} | 0.1875 | 7,836.25 | 6,294.666667 | 8,192 | |
Given the power function $f(x) = kx^a$ whose graph passes through the point $\left( \frac{1}{3}, 81 \right)$, find the value of $k + a$. | -3 | 0.3125 | 7,111.375 | 5,873 | 7,674.272727 | |
The value of $3x + 15$ is one third of the value of $6x + 45$. After finding $x$, subtract 5 from the result. What is the final value? | -5 | 1 | 1,675.5 | 1,675.5 | -1 | |
From point $A$ to point $B$ at 13:00, a bus and a cyclist left simultaneously. After arriving at point $B$, the bus, without stopping, returned and met the cyclist at point $C$ at 13:10. Returning to point $A$, the bus again without stopping headed towards point $B$ and caught up with the cyclist at point $D$, which is... | 40 | 0 | 8,127.5 | -1 | 8,127.5 | |
A positive integer has exactly 8 divisors. The sum of its smallest 3 divisors is 15. Additionally, for this four-digit number, one prime factor minus five times another prime factor is equal to two times the third prime factor. What is this number? | 1221 | 0.25 | 7,413.9375 | 5,079.75 | 8,192 | |
Usually, I go up the escalator in the subway. I have calculated that when I walk up the moving escalator, I ascend 20 steps, and it takes me exactly 60 seconds. My wife walks up the stairs more slowly and only ascends 16 steps; therefore, her total time to ascend the escalator is longer - it is 72 seconds.
How many st... | 40 | 0.5 | 6,791.125 | 6,228.375 | 7,353.875 | |
Find the number of pairs of union/intersection operations $\left(\square_{1}, \square_{2}\right) \in\{\cup, \cap\}^{2}$ satisfying the condition: for any sets $S, T$, function $f: S \rightarrow T$, and subsets $X, Y, Z$ of $S$, we have equality of sets $f(X) \square_{1}\left(f(Y) \square_{2} f(Z)\right)=f\left(X \squar... | 11 | If and only if $\square_{1}=\square_{2}=\cup$. See http://math.stackexchange.com/questions/359693/overview-of- | 0 | 7,677.0625 | -1 | 7,677.0625 |
2022 knights and liars are lined up in a row, with the ones at the far left and right being liars. Everyone except the ones at the extremes made the statement: "There are 42 times more liars to my right than to my left." Provide an example of a sequence where there is exactly one knight. | 48 | 0.25 | 7,388.875 | 6,038 | 7,839.166667 | |
A ball travels on a parabolic path in which the height (in feet) is given by the expression $-16t^2+80t+21$, where $t$ is the time after launch. What is the maximum height of the ball, in feet? | 121 | 1 | 3,088.25 | 3,088.25 | -1 | |
Given numbers \( x_{1}, \ldots, x_{n} \in (0,1) \), find the maximum value of the expression
$$
A = \frac{\sqrt[4]{1-x_{1}} + \ldots + \sqrt[4]{1-x_{n}}}{\frac{1}{\sqrt[4]{x_{1}}} + \ldots + \frac{1}{\sqrt[4]{x_{n}}}}
$$ | \frac{\sqrt{2}}{2} | 0 | 8,147.25 | -1 | 8,147.25 | |
Let $ABC$ be a triangle with sides 3, 4, and 5, and $DEFG$ be a 6-by-7 rectangle. A segment is drawn to divide triangle $ABC$ into a triangle $U_1$ and a trapezoid $V_1$ and another segment is drawn to divide rectangle $DEFG$ into a triangle $U_2$ and a trapezoid $V_2$ such that $U_1$ is similar to $U_2$ and $V_1$ is s... | 35 | We let $AB=3, AC=4, DE=6, DG=7$ for the purpose of labeling. Clearly, the dividing segment in $DEFG$ must go through one of its vertices, without loss of generality $D$. The other endpoint ($D'$) of the segment can either lie on $\overline{EF}$ or $\overline{FG}$. $V_2$ is a trapezoid with a right angle then, from whic... | 0 | 8,192 | -1 | 8,192 |
Given that both $\alpha$ and $\beta$ are acute angles, and $\sin \alpha = \frac{3}{5}$, $\tan (\alpha - \beta) = -\frac{1}{3}$.
(1) Find the value of $\sin (\alpha - \beta)$;
(2) Find the value of $\cos \beta$. | \frac{9\sqrt{10}}{50} | 0 | 4,131.5 | -1 | 4,131.5 | |
Two circles \( C_{1} \) and \( C_{2} \) have their centers at the point \( (3, 4) \) and touch a third circle, \( C_{3} \). The center of \( C_{3} \) is at the point \( (0, 0) \) and its radius is 2. What is the sum of the radii of the two circles \( C_{1} \) and \( C_{2} \)? | 10 | 0.375 | 5,235.25 | 4,792.5 | 5,500.9 | |
Given three points $A$, $B$, and $C$ in a plane such that $|\overrightarrow{AB}| = 3$, $|\overrightarrow{BC}| = 5$, and $|\overrightarrow{CA}| = 6$, find the value of $\overrightarrow{AB} \cdot \overrightarrow{BC} + \overrightarrow{BC} \cdot \overrightarrow{CA} + \overrightarrow{CA} \cdot \overrightarrow{AB}$. | -35 | 0.4375 | 7,210.25 | 5,948 | 8,192 | |
Arrange 5 people to be on duty from Monday to Friday, with each person on duty for one day and one person arranged for each day. The conditions are: A and B are not on duty on adjacent days, while B and C are on duty on adjacent days. The number of different arrangements is $\boxed{\text{answer}}$. | 36 | 0.0625 | 8,157.4375 | 7,639 | 8,192 | |
Find the area of the $MNRK$ trapezoid with the lateral side $RK = 3$ if the distances from the vertices $M$ and $N$ to the line $RK$ are $5$ and $7$ , respectively. | 18 | 0.125 | 8,054.6875 | 7,093.5 | 8,192 | |
Let $P$ be a moving point on curve $C_1$, and $Q$ be a moving point on curve $C_2$. The minimum value of $|PQ|$ is called the distance between curves $C_1$ and $C_2$, denoted as $d(C_1,C_2)$. If $C_1: x^{2}+y^{2}=2$, $C_2: (x-3)^{2}+(y-3)^{2}=2$, then $d(C_1,C_2)=$ \_\_\_\_\_\_ ; if $C_3: e^{x}-2y=0$, $C_4: \ln x+\ln 2... | \sqrt{2} | 0 | 7,897.1875 | -1 | 7,897.1875 | |
Given that the sequence $\{a\_n\}$ satisfies $\frac{1}{a_{n+1}} - \frac{1}{a_n} = d (n \in \mathbb{N}^*, d$ is a constant$)$, it is called a harmonic sequence. It is known that the sequence $\{\frac{1}{x\_n}\}$ is a harmonic sequence and $x\_1 + x\_2 + ... + x_{20} = 200$. Find the value of $x\_5 + x_{16}$. | 20 | 0.125 | 7,995.5625 | 6,620.5 | 8,192 | |
Triangle $ABC$ has $AB=25$ , $AC=29$ , and $BC=36$ . Additionally, $\Omega$ and $\omega$ are the circumcircle and incircle of $\triangle ABC$ . Point $D$ is situated on $\Omega$ such that $AD$ is a diameter of $\Omega$ , and line $AD$ intersects $\omega$ in two distinct points $X$ and $Y$ . C... | 252 | 0.0625 | 8,065.3125 | 7,991 | 8,070.266667 | |
Let \( x_{1}, x_{2}, \ldots, x_{60} \) be natural numbers greater than 1 (not necessarily distinct). In a \( 60 \times 60 \) table, numbers are placed as follows: in the intersection of the \( i \)-th row and the \( k \)-th column, the number written is \( \log _{x_{k}} \frac{x_{i}}{8} \). Find the smallest possible va... | -7200 | 0.125 | 7,677.875 | 6,615.5 | 7,829.642857 | |
Evaluate $\frac{5}{a+b}$ where $a=7$ and $b=3$.
A) $\frac{1}{2}$
B) $1$
C) $10$
D) $-8$
E) Meaningless | \frac{1}{2} | 0 | 2,723.25 | -1 | 2,723.25 | |
Let
\[x^6 - x^3 - x^2 - x - 1 = q_1(x) q_2(x) \dotsm q_m(x),\]
where each non-constant polynomial $q_i(x)$ is monic with integer coefficients, and cannot be factored further over the integers. Compute $q_1(2) + q_2(2) + \dots + q_m(2).$ | 14 | 0.375 | 6,772.875 | 5,338.666667 | 7,633.4 | |
Circles with centers $P, Q$ and $R$, having radii $1, 2$ and $3$, respectively, lie on the same side of line $l$ and are tangent to $l$ at $P', Q'$ and $R'$, respectively, with $Q'$ between $P'$ and $R'$. The circle with center $Q$ is externally tangent to each of the other two circles. What is the area of triangle $PQ... | \sqrt{6}-\sqrt{2} | To solve this problem, we need to determine the positions of the centers $P$, $Q$, and $R$ of the circles relative to each other and then calculate the area of triangle $PQR$.
1. **Positioning the Circles:**
- Since the circles are tangent to line $l$ and each other, we can determine the distances between the cente... | 0.3125 | 7,554.0625 | 6,150.6 | 8,192 |
Given a geometric sequence $\{a_n\}$ with the sum of its first n terms denoted as $S_n$, if $S_5$, $S_4$, and $S_6$ form an arithmetic sequence, determine the common ratio $q$ of the sequence $\{a_n\}$. | -2 | 0.375 | 7,429.8125 | 7,081 | 7,639.1 | |
Point \( D \) lies on side \( BC \) of triangle \( ABC \), and point \( O \) is located on segment \( AD \) with \( AO : OD = 9 : 4 \). A line passing through vertex \( B \) and point \( O \) intersects side \( AC \) at point \( E \) with \( BO : OE = 5 : 6 \). Determine the ratio in which point \( E \) divides side \(... | 21 : 44 | 0.125 | 7,698 | 7,050.5 | 7,790.5 | |
What is the maximum number of numbers we can choose from the first 1983 positive integers such that the product of any two chosen numbers is not among the chosen numbers? | 1939 | 0.25 | 7,911.5 | 7,070 | 8,192 | |
Suppose you have a Viennese pretzel lying on the table. What is the maximum number of parts you can cut it into with one straight swing of the knife? In which direction should this cut be made? | 10 | 0 | 6,236.375 | -1 | 6,236.375 | |
The slopes of lines $l_1$ and $l_2$ are the two roots of the equation $6x^2+x-1=0$, respectively. The angle between lines $l_1$ and $l_2$ is __________. | \frac{\pi}{4} | 0.125 | 2,778.8125 | 2,383.5 | 2,835.285714 | |
Simplify first, then evaluate: $\left(2m+n\right)\left(2m-n\right)-\left(2m-n\right)^{2}+2n\left(m+n\right)$, where $m=2$, $n=-1^{2023}$. | -12 | 1 | 2,334.4375 | 2,334.4375 | -1 | |
In \(\triangle ABC\), \(AB = 13\), \(BC = 14\), and \(CA = 15\). \(P\) is a point inside \(\triangle ABC\) such that \(\angle PAB = \angle PBC = \angle PCA\). Find \(\tan \angle PAB\). | \frac{168}{295} | 0.8125 | 6,040.4375 | 5,543.923077 | 8,192 | |
Zhang Hua has to go through four traffic posts A, B, C, and D on his way to school. The probability of encountering a red light at posts A and B is $\frac{1}{2}$ each, and at posts C and D, it is $\frac{1}{3}$ each. Assuming that the events of encountering red lights at the four traffic posts are independent, let X rep... | \frac{5}{3} | 0.4375 | 6,519.4375 | 4,369 | 8,192 | |
The real numbers \( x, y, z \) satisfy the equations \( x + y + z = 2 \) and \( xy + yz + zx = 1 \). Find the maximum possible value of \( x - y \). | \frac{2 \sqrt{3}}{3} | 0 | 6,807.4375 | -1 | 6,807.4375 | |
In the diagram, three identical circles touch each other, and each circle has a circumference of 24. Calculate the perimeter of the shaded region within the triangle formed by the centers of the circles. | 12 | 0.8125 | 3,910.5 | 3,467.230769 | 5,831.333333 | |
How many positive four-digit integers of the form $\_\_45$ are divisible by 45? | 10 | 0.9375 | 4,547.5 | 4,304.533333 | 8,192 | |
The diagram shows three triangles which are formed by the five line segments \(A C D F, B C G, G D E, A B\), and \(E F\) so that \(A C = B C = C D = G D = D F = E F\). Also, \(\angle C A B = \angle E F D\). What is the size, in degrees, of \(\angle C A B\)? | 60 | 0.25 | 7,536.375 | 6,185.75 | 7,986.583333 | |
If the distance from the foci of the hyperbola $C$ to its asymptotes is equal to the length of $C$'s real semi-axis, then the eccentricity of $C$ is \_\_\_\_\_\_. | \sqrt{2} | 1 | 3,035.375 | 3,035.375 | -1 | |
For how many integer values of \( n \) between 1 and 999 inclusive does the decimal representation of \( \frac{n}{1000} \) terminate? | 999 | 0.75 | 5,874.6875 | 5,102.25 | 8,192 | |
For any $n\in\mathbb N$ , denote by $a_n$ the sum $2+22+222+\cdots+22\ldots2$ , where the last summand consists of $n$ digits of $2$ . Determine the greatest $n$ for which $a_n$ contains exactly $222$ digits of $2$ . | 222 | 0 | 8,192 | -1 | 8,192 | |
Given an isosceles triangle $PQR$ with $PQ=QR$ and the angle at the vertex $108^\circ$. Point $O$ is located inside the triangle $PQR$ such that $\angle ORP=30^\circ$ and $\angle OPR=24^\circ$. Find the measure of the angle $\angle QOR$. | 126 | 0 | 8,170.4375 | -1 | 8,170.4375 | |
Olga Ivanovna, the class teacher of Grade 5B, is organizing a "Mathematical Ballet." She wants to arrange boys and girls so that at a distance of 5 meters from each girl there are exactly 2 boys. What is the maximum number of girls that can participate in the ballet, given that 5 boys are participating? | 20 | 0.3125 | 7,465.5625 | 6,497.4 | 7,905.636364 | |
Given vectors $\mathbf{a}$ and $\mathbf{b}$ such that $\|\mathbf{a}\| = 6,$ $\|\mathbf{b}\| = 8,$ and $\|\mathbf{a} + \mathbf{b}\| = 11.$ Find $\cos \theta,$ where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}.$ | \frac{7}{32} | 0.8125 | 3,728.375 | 2,880.769231 | 7,401.333333 | |
Add $518_{12} + 276_{12}$. Express your answer in base 12, using $A$ for $10$ and $B$ for $11$ if necessary. | 792_{12} | 1 | 3,019.3125 | 3,019.3125 | -1 | |
Consider a string of $n$ $7$'s, $7777\cdots77,$ into which $+$ signs are inserted to produce an arithmetic expression. For example, $7+77+777+7+7=875$ could be obtained from eight $7$'s in this way. For how many values of $n$ is it possible to insert $+$ signs so that the resulting expression has value $7000$?
| 108 | 0 | 8,192 | -1 | 8,192 | |
The sequence $\left\{a_{n}\right\}$ consists of 9 terms, where $a_{1} = a_{9} = 1$, and for each $i \in \{1,2, \cdots, 8\}$, we have $\frac{a_{i+1}}{a_{i}} \in \left\{2,1,-\frac{1}{2}\right\}$. Find the number of such sequences. | 491 | 0.25 | 6,914.4375 | 4,577.75 | 7,693.333333 | |
The region consisting of all points in three-dimensional space within $4$ units of line segment $\overline{CD}$ has volume $544\pi$. Calculate the length of $CD$. | \frac{86}{3} | 1 | 2,625.9375 | 2,625.9375 | -1 | |
Let $m$ and $n$ be positive integers satisfying the conditions
$\quad\bullet\ \gcd(m+n,210)=1,$
$\quad\bullet\ m^m$ is a multiple of $n^n,$ and
$\quad\bullet\ m$ is not a multiple of $n.$
Find the least possible value of $m+n.$
| 407 | 0 | 7,988.0625 | -1 | 7,988.0625 | |
Solve for $r$: $$\frac{r+3}{r-2} = \frac{r-1}{r+1}.$$Express your answer as a fraction. | -\frac{1}{7} | 1 | 1,920.5 | 1,920.5 | -1 | |
Let $ABC$ be a triangle such that $AB=2$ , $CA=3$ , and $BC=4$ . A semicircle with its diameter on $BC$ is tangent to $AB$ and $AC$ . Compute the area of the semicircle. | \frac{27\pi}{40} | 0.5625 | 6,456.75 | 5,501.444444 | 7,685 | |
The sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2,\ldots$ consists of $1$’s separated by blocks of $2$’s with $n$ $2$’s in the $n^{th}$ block. The sum of the first $1234$ terms of this sequence is | 2419 | 1. **Understanding the Sequence**: The sequence given is $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, \ldots$ where each $1$ is followed by an increasing number of $2$s. Specifically, the $n$-th block of $2$s contains $n$ instances of $2$.
2. **Summing Blocks**: The sum of each block can be described as follows:
-... | 0.625 | 6,941.9375 | 6,877.9 | 7,048.666667 |
Find the value of cos $$\frac {π}{11}$$cos $$\frac {2π}{11}$$cos $$\frac {3π}{11}$$cos $$\frac {4π}{11}$$cos $$\frac {5π}{11}$$\=\_\_\_\_\_\_． | \frac {1}{32} | 0.4375 | 6,937.3125 | 5,324.142857 | 8,192 | |
Pirate Bob shares his treasure with Pirate Sam in a peculiar manner. Bob first declares, ``One for me, one for you,'' keeping one coin for himself and starting Sam's pile with one coin. Then Bob says, ``Two for me, and two for you,'' adding two more coins to his pile but updating Sam's total to two coins. This continue... | 20 | 0.4375 | 6,694 | 5,373.428571 | 7,721.111111 | |
At a conference with 35 businessmen, 18 businessmen drank coffee, 15 businessmen drank tea, and 8 businessmen drank juice. Six businessmen drank both coffee and tea, four drank both tea and juice, and three drank both coffee and juice. Two businessmen drank all three beverages. How many businessmen drank only one type ... | 21 | 0.75 | 4,092 | 4,578.333333 | 2,633 | |
What is the smallest positive integer that ends in 9 and is divisible by 7? | 49 | 0.9375 | 2,462.6875 | 2,080.733333 | 8,192 | |
The line $\sqrt{2}ax+by=1$ intersects the circle $x^{2}+y^{2}=1$ at points $A$ and $B$ (where $a$ and $b$ are real numbers), and $\triangle AOB$ is a right-angled triangle (where $O$ is the origin). The maximum distance between point $P(a,b)$ and point $(0,1)$ is ______. | \sqrt{2} + 1 | 0 | 7,502.5 | -1 | 7,502.5 | |
Circles of radii $5, 5, 8,$ and $\frac mn$ are mutually externally tangent, where $m$ and $n$ are relatively prime positive integers. Find $m + n.$ | 17 | We may also use Descartes' theorem, $k_4=k_1+k_2+k_3\pm 2\sqrt{k_1k_2+k_2k_3+k_3k_1}$ where each of $k_i$ is the curvature of a circle with radius $r_i$, and the curvature is defined as $k_i=\frac{1}{r_i}$. The larger solution for $k_4$ will give the curvature of the circle externally tangent to the other circles, whil... | 0.625 | 6,363.375 | 5,266.2 | 8,192 |
A given sequence $r_1, r_2, \dots, r_n$ of distinct real numbers can be put in ascending order by means of one or more "bubble passes". A bubble pass through a given sequence consists of comparing the second term with the first term, and exchanging them if and only if the second term is smaller, then comparing the thir... | 931 | 0 | 7,420.625 | -1 | 7,420.625 |
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