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 |
|---|---|---|---|---|---|---|
Given that $\cos \alpha$ is a root of the equation $3x^2 - x - 2 = 0$ and $\alpha$ is an angle in the third quadrant, find the value of $\frac{\sin (-\alpha + \frac{3\pi}{2}) \cos (\frac{3\pi}{2} + \alpha) \tan^2 (\pi - \alpha)}{\cos (\frac{\pi}{2} + \alpha) \sin (\frac{\pi}{2} - \alpha)}$. | \frac{5}{4} | 1 | 3,887.4375 | 3,887.4375 | -1 | |
Let $AD,BF$ and ${CE}$ be the altitudes of $\vartriangle ABC$. A line passing through ${D}$ and parallel to ${AB}$intersects the line ${EF}$at the point ${G}$. If ${H}$ is the orthocenter of $\vartriangle ABC$, find the angle ${\angle{CGH}}$. | 90^\circ |
Consider triangle \(\triangle ABC\) with altitudes \(AD\), \(BF\), and \(CE\). The orthocenter of the triangle is denoted by \(H\). A line through \(D\) that is parallel to \(AB\) intersects line \(EF\) at point \(G\).
To find the angle \(\angle CGH\), follow these steps:
1. **Identify the orthocenter \(H\):**
Si... | 0.3125 | 7,828.0625 | 7,027.4 | 8,192 |
Seven students stand in a row for a photo, among them, students A and B must stand next to each other, and students C and D must not stand next to each other. The total number of different arrangements is. | 960 | 0.3125 | 7,854.5625 | 7,251.4 | 8,128.727273 | |
Anya wants to buy ice cream that costs 19 rubles. She has two 10-ruble coins, two 5-ruble coins, and one 2-ruble coin in her pocket. Anya randomly picks three coins from her pocket without looking. Find the probability that the selected coins will be enough to pay for the ice cream. | 0.4 | 0 | 7,509.0625 | -1 | 7,509.0625 | |
How many integers $n$ are there such that $0 \le n \le 720$ and $n^2 \equiv 1$ (mod $720$ )? | 16 | 0.75 | 5,196 | 4,197.333333 | 8,192 | |
Given right triangle $ABC$ with a right angle at vertex $C$ and $AB = 2BC$, calculate the value of $\cos A$. | \frac{\sqrt{3}}{2} | 0 | 2,461.6875 | -1 | 2,461.6875 | |
Given point $P(1, 2, 3)$, the symmetric point of $P$ about the $y$-axis is $P_1$, and the symmetric point of $P$ about the coordinate plane $xOz$ is $P_2$. Find the distance $|P_1P_2|$. | 2\sqrt{14} | 0.625 | 2,244 | 2,483.9 | 1,844.166667 | |
Given the sequence $1,2,1,2,2,1,2,2,2,1,2,2,2,2,1,2,\cdots$ where the number of 2's between each pair of 1's increases by one each time, find the sum of the first 1234 terms of the sequence. | 2419 | 0.4375 | 6,662.1875 | 5,736.714286 | 7,382 | |
From the numbers \\(1, 2, \ldots, 100\\) totaling \\(100\\) numbers, three numbers \\(x, y, z\\) are chosen in sequence. The probability that these three numbers satisfy \\(x+z=2y\\) is __________. | \dfrac{1}{198} | 0 | 8,011.375 | -1 | 8,011.375 | |
Mom asks Xiao Ming to boil water and make tea for guests. Washing the kettle takes 1 minute, boiling water takes 15 minutes, washing the teapot takes 1 minute, washing the teacups takes 1 minute, and getting the tea leaves takes 2 minutes. Xiao Ming estimates that it will take 20 minutes to complete these tasks. Accord... | 16 | 0 | 6,940.8125 | -1 | 6,940.8125 | |
In triangle $\triangle ABC$, $\angle BAC = \frac{π}{3}$, $D$ is the midpoint of $AB$, $P$ is a point on segment $CD$, and satisfies $\overrightarrow{AP} = t\overrightarrow{AC} + \frac{1}{3}\overrightarrow{AB}$. If $|\overrightarrow{BC}| = \sqrt{6}$, then the maximum value of $|\overrightarrow{AP}|$ is ______. | \sqrt{2} | 0.375 | 7,897.875 | 7,544.5 | 8,109.9 | |
Selina takes a sheet of paper and cuts it into 10 pieces. She then takes one of these pieces and cuts it into 10 smaller pieces. She then takes another piece and cuts it into 10 smaller pieces and finally cuts one of the smaller pieces into 10 tiny pieces. How many pieces of paper has the original sheet been cut into? | 37 | 0.6875 | 556.125 | 475 | 734.6 | |
Observe the following three rows of numbers and complete the subsequent questions:
①-2, 4, -8, 16, ...
②1, -2, 4, -8, ...
③0, -3, 3, -9, ...
(1) Consider the pattern in row ① and write the expression for the $n^{th}$ number.
(2) Denote the $m^{th}$ number in row ② as $a$ and the $m^{th}$ number in row ③ as $b$. Write... | -1 | 0 | 7,903.8125 | -1 | 7,903.8125 | |
Find all functions \( f: \mathbb{Q} \rightarrow \{-1, 1\} \) such that for all distinct \( x, y \in \mathbb{Q} \) satisfying \( xy = 1 \) or \( x + y \in \{0, 1\} \), we have \( f(x) f(y) = -1 \).
Intermediate question: Let \( f \) be a function having the above property and such that \( f(0) = 1 \). What is \( f\left... | -1 | 0 | 8,192 | -1 | 8,192 | |
Ramanujan and Hardy played a game where they both picked a complex number. If the product of their numbers was $32-8i$, and Hardy picked $5+3i$, what number did Ramanujan pick? | 4-4i | 1 | 1,956.0625 | 1,956.0625 | -1 | |
For a special event, the five Vietnamese famous dishes including Phở, (Vietnamese noodle), Nem (spring roll), Bún Chả (grilled pork noodle), Bánh cuốn (stuffed pancake), and Xôi gà (chicken sticky rice) are the options for the main courses for the dinner of Monday, Tuesday, and Wednesday. Every dish must be used exact... | 150 | 0 | 511.25 | -1 | 511.25 | |
Given that $ABCD$ is a rectangle with $AD = 10$ and the shaded area is $100, calculate the shortest distance between the semicircles. | 2.5 \pi | 0 | 7,794.5 | -1 | 7,794.5 | |
The fraction $\frac{a^{-4}-b^{-4}}{a^{-2}-b^{-2}}$ is equal to: | a^{-2}+b^{-2} | 1. **Rewrite the numerator using the difference of squares formula**:
The difference of squares formula states that $x^2 - y^2 = (x+y)(x-y)$. Applying this to $a^{-4} - b^{-4}$, where $x = a^{-2}$ and $y = b^{-2}$, we get:
\[
a^{-4} - b^{-4} = (a^{-2} - b^{-2})(a^{-2} + b^{-2})
\]
2. **Substitute the rewr... | 0.0625 | 4,194.4375 | 8,112 | 3,933.266667 |
Pile up 2019 stones into one pile. First, person A splits this pile into two piles and writes the product of the number of stones in each pile on the blackboard. Then, person A selects one pile from the two and splits it into two more piles, again writing the product of the number of stones in each pile on the blackboa... | 2037171 | 0.625 | 5,572.6875 | 4,001.1 | 8,192 | |
In the polar coordinate system, the polar coordinate equation of the curve $\Gamma$ is $\rho= \frac {4\cos \theta}{\sin ^{2}\theta}$. Establish a rectangular coordinate system with the pole as the origin, the polar axis as the positive semi-axis of $x$, and the unit length unchanged. The lines $l_{1}$ and $l_{2}$ both ... | 4 \sqrt {2} | 0 | 8,059.3125 | -1 | 8,059.3125 | |
Find four distinct positive integers $a, b, c, d$ less than $15$ which are invertible modulo $15$. Calculate the remainder when $(abc + abd + acd + bcd)(abcd)^{-1}$ is divided by $15$. | 11 | 0.0625 | 8,128.5 | 7,176 | 8,192 | |
Expand the following expression: $(13x+15)\cdot 2x$ | 26x^2+30x | 1 | 786.4375 | 786.4375 | -1 | |
The natural numbers from 1951 to 1982 are arranged in a certain order one after another. A computer reads two consecutive numbers from left to right (i.e., the 1st and 2nd, the 2nd and 3rd, etc.) until the last two numbers. If the larger number is on the left, the computer swaps their positions. Then the computer reads... | 1982 | 0 | 8,161.25 | -1 | 8,161.25 | |
In the Cartesian coordinate system $xOy$, given the parametric equations of circle $C_{1}$ as $\left\{\begin{array}{l}x=2+2\cos\alpha\\ y=1+2\sin\alpha\end{array}\right.$ ($\alpha$ is the parameter), establish a polar coordinate system with the coordinate origin as the pole and the positive x-axis as the polar axis.<br... | \frac{\sqrt{7}}{2} | 0 | 8,192 | -1 | 8,192 | |
A box contains seven cards, each with a different integer from 1 to 7 written on it. Avani takes three cards from the box and then Niamh takes two cards, leaving two cards in the box. Avani looks at her cards and then tells Niamh "I know the sum of the numbers on your cards is even." What is the sum of the numbers on A... | 12 | 0 | 8,106.0625 | -1 | 8,106.0625 | |
Joey wrote a system of equations on a blackboard, where each of the equations was of the form $a+b=c$ or $a \cdot b=c$ for some variables or integers $a, b, c$. Then Sean came to the board and erased all of the plus signs and multiplication signs, so that the board reads: $$\begin{array}{ll} x & z=15 \\ x & y=12 \\ x &... | 2037 | The bottom line gives $x=-6, x=6$ or $x=18$. If $x=-6, y$ can be -2 or 18 and $z$ must be 21, so the possible values for $100 x+10 y+z$ are -599 and -399. If $x=6, y$ can be 2 or 6 and $z$ must be 9, so the possible values are 629 and 669. If $x=18, y$ must be -6 and $z$ must be -3, so the only possible value is 1737. ... | 0.125 | 6,065.0625 | 6,550.5 | 5,995.714286 |
Given that six coal freight trains are organized into two groups of three trains, with trains 'A' and 'B' in the same group, determine the total number of different possible departure sequences for the six trains. | 144 | 0 | 6,923.0625 | -1 | 6,923.0625 | |
1. When a die (with faces numbered 1 through 6) is thrown twice in succession, find the probability that the sum of the numbers facing up is at least 10.
2. On a line segment MN of length 16cm, a point P is chosen at random. A rectangle is formed with MP and NP as adjacent sides. Find the probability that the area of ... | \frac{1}{4} | 1 | 3,671.875 | 3,671.875 | -1 | |
In a student speech competition held at a school, there were a total of 7 judges. The final score for a student was the average score after removing the highest and the lowest scores. The scores received by a student were 9.6, 9.4, 9.6, 9.7, 9.7, 9.5, 9.6. The mode of this data set is _______, and the student's final s... | 9.6 | 0.25 | 917.5625 | 1,108.25 | 854 | |
In the 2016 art exam of a certain high school, there were 6 contestants, including 3 females and 3 males. Now, these six contestants are to perform their talents in sequence. If any two of the three males cannot perform consecutively, and the female contestant A cannot be the first to perform, then calculate the number... | 132 | 0.375 | 6,818.875 | 6,086.166667 | 7,258.5 | |
At a physical education lesson, 29 seventh graders attended, some of whom brought one ball each. During the lesson, sometimes one seventh grader would give their ball to another seventh grader who did not have a ball.
At the end of the lesson, $N$ seventh graders said, "I received balls less often than I gave them awa... | 14 | 0.0625 | 8,115.25 | 6,964 | 8,192 | |
A fair 6-sided die is rolled once. If I roll $n$, then I win $6-n$ dollars. What is the expected value of my win, in dollars? | 2.50 | 0.6875 | 2,174.1875 | 2,098.454545 | 2,340.8 | |
A ball is dropped from a height of 150 feet and rebounds to three-fourths of the distance it fell on each bounce. How many feet will the ball have traveled when it hits the ground the fifth time? | 765.234375 | 0.25 | 7,873.5 | 7,637.25 | 7,952.25 | |
An ellipse $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1 (a>b>0)$ has its two foci and the endpoints of its minor axis all lying on the circle $x^{2}+y^{2}=1$. A line $l$ (not perpendicular to the x-axis) passing through the right focus intersects the ellipse at points A and B. The perpendicular bisector of segment AB ... | 2 \sqrt {2} | 0 | 6,271.25 | -1 | 6,271.25 | |
In triangle $ABC$, $AB = 7$, $AC = 15$, and the length of median $AM$ is 10. Find the area of triangle $ABC$. | 42 | 1 | 4,250.125 | 4,250.125 | -1 | |
What is the smallest possible real value of $x^2 + 8x$? | -16 | 1 | 2,136.125 | 2,136.125 | -1 | |
Given the equation of an ellipse is $\dfrac {x^{2}}{a^{2}} + \dfrac {y^{2}}{b^{2}} = 1 (a > b > 0)$, a line passing through the right focus of the ellipse and perpendicular to the $x$-axis intersects the ellipse at points $P$ and $Q$. The directrix of the ellipse on the right intersects the $x$-axis at point $M$. If $\... | \dfrac { \sqrt {3}}{3} | 0 | 4,918.0625 | -1 | 4,918.0625 | |
Consider the sequence created by intermixing the following sets of numbers: the first $1000$ odd numbers, and the squares of the first $100$ integers. What is the median of the new list of $1100$ numbers?
- $1, 3, 5, \ldots, 1999$
- $1^2, 2^2, \ldots, 100^2$
A) $1089$
B) $1095$
C) $1100$
D) $1102$
E) $1105$ | 1100 | 0 | 7,788.75 | -1 | 7,788.75 | |
How many square units are in the area of the pentagon shown here with sides of length 15, 20, 27, 24 and 20 units?
[asy]
pair a,b,c,d,e;
a=(0,0);
b=(24,0);
c=(24,27);
d=(5.3,34);
e=(0,20);
draw((0,0)--(24,0)--(24,27)--(5.3,34)--(0,20)--cycle);
draw((4.8,32.7)--(6.1,32.2)--(6.6,33.5));
label("24",(12,0),S);
label("27... | 714 | 0 | 8,192 | -1 | 8,192 | |
Find the smallest positive integer \( n \) that is not less than 9, such that for any \( n \) integers (which can be the same) \( a_{1}, a_{2}, \cdots, a_{n} \), there always exist 9 numbers \( a_{i_{1}}, a_{i_{2}}, \cdots, a_{i_{9}} \) (where \(1 \leq i_{1} < i_{2} < \cdots < i_{9} \leq n \)) and \( b_{i} \in \{4,7\} ... | 13 | 0 | 8,192 | -1 | 8,192 | |
The area of a rhombus with diagonals of 6cm and 8cm is in cm<sup>2</sup>, and its perimeter is in cm. | 20 | 1 | 1,275.1875 | 1,275.1875 | -1 | |
How many 4-digit positive multiples of 4 can be formed from the digits 0, 1, 2, 3, 4, 5, 6 such that each digit appears without repetition? | 176 | 0 | 7,859.5 | -1 | 7,859.5 | |
Solve
\[\arctan \frac{1}{x} + \arctan \frac{1}{x^3} = \frac{\pi}{4}.\] | \frac{1 + \sqrt{5}}{2} | 0 | 5,899.75 | -1 | 5,899.75 | |
Define the operation $a\nabla b = 2 + b^a$. What is the value of $(1\nabla 2) \nabla 3$? | 83 | 1 | 1,554 | 1,554 | -1 | |
Given \( \cos \left( \frac {\pi}{2}+\alpha \right)=3\sin \left(\alpha+ \frac {7\pi}{6}\right) \), find the value of \( \tan \left( \frac {\pi}{12}+\alpha \right) = \) ______. | 2\sqrt {3} - 4 | 0 | 7,434.0625 | -1 | 7,434.0625 | |
If $y$ varies directly as $x$, and if $y=8$ when $x=4$, the value of $y$ when $x=-8$ is: | -16 | 1. **Understanding Direct Variation**:
Given that $y$ varies directly as $x$, we can express this relationship using the equation:
\[
y = kx
\]
where $k$ is the constant of proportionality.
2. **Finding the Constant of Proportionality**:
We know that $y = 8$ when $x = 4$. Substituting these values in... | 1 | 1,209.75 | 1,209.75 | -1 |
The prime factorization of 2007 is $3^{2}\times223$. How many ordered pairs of positive integers $(x,y)$ satisfy the equation $xy=2007$? | 6 | 1 | 2,106.5625 | 2,106.5625 | -1 | |
Let $a \star b = \frac{\sqrt{a^2+b}}{\sqrt{a^2 - b}}$. If $y \star 15 = 5$, find $y$. | \frac{\sqrt{65}}{2} | 0 | 4,503.375 | -1 | 4,503.375 | |
In the center of a circular field, there is a geologists' house. Eight straight roads radiate from it, dividing the field into 8 equal sectors. Two geologists set off on a journey from their house, each traveling at a speed of 4 km/h along a road chosen at random. Determine the probability that the distance between the... | 0.375 | 0 | 7,030.875 | -1 | 7,030.875 | |
The numbers $a_{1}, a_{2}, \ldots, a_{100}$ are a permutation of the numbers $1,2, \ldots, 100$. Let $S_{1}=a_{1}$, $S_{2}=a_{1}+a_{2}, \ldots, S_{100}=a_{1}+a_{2}+\ldots+a_{100}$. What maximum number of perfect squares can be among the numbers $S_{1}, S_{2}, \ldots, S_{100}$? | 60 | We add initial term \(S_{0}=0\) to the sequence \(S_{1}, S_{2}, \ldots, S_{100}\) and consider all the terms \(S_{n_{0}}<S_{n_{1}}<\ldots\) that are perfect squares: \(S_{n_{k}}=m_{k}^{2}\) (in particular, \(n_{0}=m_{0}=0\)). Since \(S_{100}=5050<72^{2}\), all the numbers \(m_{k}\) do not exceed 71. If \(m_{k+1}=m_{k}+... | 0 | 8,192 | -1 | 8,192 |
A wooden cube $n$ units on a side is painted red on all six faces and then cut into $n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $n$? | 4 | 1. **Calculate the number of small red faces:**
The original cube has $6$ faces, and each face is $n \times n$, so there are $n^2$ small faces per face of the cube. Since all faces are painted red, the total number of small red faces is $6n^2$.
2. **Calculate the total number of small faces:**
Each of the $n^3$ ... | 1 | 2,473.9375 | 2,473.9375 | -1 |
How many of the integers \(19, 21, 23, 25, 27\) can be expressed as the sum of two prime numbers? | 3 | We note that all of the given possible sums are odd, and also that every prime number is odd with the exception of 2 (which is even). When two odd integers are added, their sum is even. When two even integers are added, their sum is even. When one even integer and one odd integer are added, their sum is odd. Therefore,... | 1 | 3,435.75 | 3,435.75 | -1 |
Jenna collects stamps. She puts the same number of stamps on each page and then inserts each page into one of her two stamp books. One of her stamp books has a total of 840 stamps. The other has 1008. What is the largest number of stamps that Jenna could be putting on each page? | 168 | 1 | 1,878.0625 | 1,878.0625 | -1 | |
Starting from which number $n$ of independent trials does the inequality $p\left(\left|\frac{m}{n}-p\right|<0.1\right)>0.97$ hold, if in a single trial $p=0.8$? | 534 | 0 | 7,729.4375 | -1 | 7,729.4375 | |
A rectangular box has a total surface area of 166 square inches and the sum of the lengths of all its edges is 64 inches. Find the sum of the lengths in inches of all of its interior diagonals. | 12\sqrt{10} | 0.875 | 2,207.75 | 2,260.714286 | 1,837 | |
Selected Exercise $(4-5)$: Inequality Lecture
Given the function $f(x)=|2x-a|+|x-1|$, where $a\in R$
(1) Find the range of values for the real number $a$ if the inequality $f(x)\leqslant 2-|x-1|$ has a solution;
(2) When $a < 2$, the minimum value of the function $f(x)$ is $3$, find the value of the real number $a$. | -4 | 0.375 | 7,264.8125 | 5,719.5 | 8,192 | |
Let \(a\), \(b\), \(c\), and \(d\) be nonnegative numbers whose sum is 150. Find the largest possible value of
\[
ab + bc + cd.
\] | 5625 | 0.75 | 7,598.0625 | 7,400.083333 | 8,192 | |
Given Ben's test scores $95, 85, 75, 65,$ and $90$, and his goal to increase his average by at least $5$ points and score higher than his lowest score of $65$ with his next test, calculate the minimum test score he would need to achieve both goals. | 112 | 0.875 | 3,778.5 | 3,148 | 8,192 | |
Let $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$ be two distinct polynomials with real coefficients such that the $x$-coordinate of the vertex of $f$ is a root of $g,$ and the $x$-coordinate of the vertex of $g$ is a root of $f,$ and both $f$ and $g$ have the same minimum value. If the graphs of the two polynomials... | -400 | 0.625 | 6,526.9375 | 5,861.4 | 7,636.166667 | |
What is the least positive integer with exactly $12$ positive factors? | 150 | 0 | 4,144.1875 | -1 | 4,144.1875 | |
A line is expressed in the form
\[\begin{pmatrix} 1 \\ 3 \end{pmatrix} \cdot \left( \begin{pmatrix} x \\ y \end{pmatrix} - \begin{pmatrix} -2 \\ 8 \end{pmatrix} \right) = 0.\]The equation of the line can be expressed in the form $y = mx + b.$ Enter the ordered pair $(m,b).$ | \left( -\frac{1}{3}, \frac{22}{3} \right) | 0.9375 | 2,651 | 2,281.6 | 8,192 | |
Determine the number of distinct terms in the simplified expansion of $[(x+5y)^3(x-5y)^3]^{3}$. | 10 | 0.8125 | 3,823.9375 | 4,109 | 2,588.666667 | |
If there are 3 identical copies of the Analects and 6 different modern literary masterpieces in the classroom, and 3 books are selected from these 9 books, determine the number of different ways to make the selection. | 42 | 0.8125 | 5,063.75 | 4,341.846154 | 8,192 | |
What is the remainder when $2001 \cdot 2002 \cdot 2003 \cdot 2004 \cdot 2005$ is divided by 19? | 11 | 0.8125 | 5,770.375 | 5,211.538462 | 8,192 | |
The organizers of a ping-pong tournament have only one table. They call two participants to play, who have not yet played against each other. If after the game the losing participant suffers their second defeat, they are eliminated from the tournament (since there are no ties in tennis). After 29 games, it turned out t... | 16 | 0.0625 | 6,862.3125 | 4,479 | 7,021.2 | |
If $x<0$, then $|x-\sqrt{(x-1)^2}|$ equals | 1-2x | 1. **Understanding the expression**: We start by simplifying the expression inside the absolute value:
\[
\left|x - \sqrt{(x-1)^2}\right|
\]
We know that the square root of a square gives the absolute value, i.e., $\sqrt{a^2} = |a|$. Applying this to the expression, we get:
\[
\sqrt{(x-1)^2} = |x-1|
... | 0.8125 | 2,867.5 | 2,941.230769 | 2,548 |
Let $a,$ $b,$ $c,$ $z$ be complex numbers such that $|a| = |b| = |c| = 1$ and $\arg(c) = \arg(a) + \arg(b)$. Suppose that
\[ a z^2 + b z + c = 0. \]
Find the largest possible value of $|z|$. | \frac{1 + \sqrt{5}}{2} | 0 | 7,740.25 | -1 | 7,740.25 | |
In the diagram, \(\angle AFC = 90^\circ\), \(D\) is on \(AC\), \(\angle EDC = 90^\circ\), \(CF = 21\), \(AF = 20\), and \(ED = 6\). Determine the total area of quadrilateral \(AFCE\). | 297 | 0.1875 | 7,643.5625 | 6,545 | 7,897.076923 | |
Find all positive integer pairs $(a,n)$ such that $\frac{(a+1)^n-a^n}{n}$ is an integer. | (a, n) = (a, 1) |
We need to find all positive integer pairs \((a, n)\) such that \(\frac{(a+1)^n - a^n}{n}\) is an integer.
First, observe that for \(\frac{(a+1)^n - a^n}{n}\) to be an integer, \((a+1)^n - a^n\) must be divisible by \(n\).
Consider the smallest prime divisor \(p\) of \(n\). We have:
\[
(a+1)^n \equiv a^n \pmod{p}.
... | 0 | 7,477.0625 | -1 | 7,477.0625 |
Simplify $\sqrt{288}$. | 12\sqrt{2} | 1 | 2,290.1875 | 2,290.1875 | -1 | |
The four complex roots of
\[2z^4 + 8iz^3 + (-9 + 9i)z^2 + (-18 - 2i)z + (3 - 12i) = 0,\]when plotted in the complex plane, form a rhombus. Find the area of the rhombus. | \sqrt{10} | 0 | 8,192 | -1 | 8,192 | |
The area of polygon $ABCDEF$ is 52 with $AB=8$, $BC=9$ and $FA=5$. What is $DE+EF$? [asy]
pair a=(0,9), b=(8,9), c=(8,0), d=(4,0), e=(4,4), f=(0,4);
draw(a--b--c--d--e--f--cycle);
draw(shift(0,-.25)*a--shift(.25,-.25)*a--shift(.25,0)*a);
draw(shift(-.25,0)*b--shift(-.25,-.25)*b--shift(0,-.25)*b);
draw(shift(-.25,0)*c--... | 9 | 0 | 8,034.125 | -1 | 8,034.125 | |
The numbers \( a, b, c, d \) belong to the interval \([-4 ; 4]\). Find the maximum value of the expression \( a + 2b + c + 2d - ab - bc - cd - da \). | 72 | 0.0625 | 8,122.5625 | 7,081 | 8,192 | |
The base of the quadrilateral prism $A B C D A_{1} B_{1} C_{1} D_{1}$ is a rhombus $A B C D$, where $B D=3$ and $\angle A D C=60^{\circ}$. A sphere passes through the vertices $D, C, B, B_{1}, A_{1}, D_{1}$.
a) Find the area of the circle obtained in the cross-section of the sphere by the plane passing through the poi... | 3\sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
How many triples \((a, b, c)\) of positive integers satisfy the conditions \( 6ab = c^2 \) and \( a < b < c \leq 35 \)? | 8 | There are 8 such triplets: \((2,3,6), (3,8,12), (4,6,12), (6,9,18), (6,16,24), (8,12,24), (6,25,30), (10,15,30)\). | 0.625 | 7,289 | 6,832.1 | 8,050.5 |
In the coordinate plane, a rectangle is drawn with vertices at $(34,0),(41,0),(34,9),(41,9)$. Find the smallest value of the parameter $a$ for which the line $y=ax$ divides this rectangle into two parts such that the area of one part is twice the area of the other. If the answer is not an integer, write it as a decimal... | 0.08 | 0 | 8,070.4375 | -1 | 8,070.4375 | |
Compute the sum:
\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2)))))))) \] | 1022 | 0.0625 | 5,078.5 | 3,778 | 5,165.2 | |
The integer 636405 may be written as the product of three 2-digit positive integers. What is the sum of these three integers? | 259 | We begin by factoring the given integer into prime factors. Since 636405 ends in a 5, it is divisible by 5, so $636405=5 \times 127281$. Since the sum of the digits of 127281 is a multiple of 3, then it is a multiple of 3, so $636405=5 \times 3 \times 42427$. The new quotient (42427) is divisible by 7, which gives $636... | 0.625 | 6,531.625 | 5,535.4 | 8,192 |
An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4$. One vertex of the triangle is $(0,1)$, one altitude is contained in the y-axis, and the square of the length of each side is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
| 937 | 0.875 | 4,840.3125 | 4,361.5 | 8,192 | |
A section of a book fell out. The first page of this section is numbered 143, and the number of the last page consists of the same digits but in a different order. How many pages fell out of the book? | 172 | 0.375 | 6,003.8125 | 5,221.333333 | 6,473.3 | |
Let $S$ be the set of points $(x, y)$ in the Cartesian plane that satisfy
\[\Big|\big| |x|-2\big|-1\Big|+\Big|\big| |y|-2\big|-1\Big|=1.\]What is the total length of all the lines that make up $S$? | 64\sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
Suppose 9 people are arranged in a line randomly. What is the probability that person A is in the middle, and persons B and C are adjacent? | \frac{1}{42} | 0 | 7,173.125 | -1 | 7,173.125 | |
In a similar setup, square $PQRS$ is constructed along diameter $PQ$ of a semicircle. The semicircle and square $PQRS$ are coplanar. Line segment $PQ$ has a length of 8 centimeters. If point $N$ is the midpoint of arc $PQ$, what is the length of segment $NS$? | 4\sqrt{10} | 0.125 | 4,889.25 | 4,157.5 | 4,993.785714 | |
On the complex plane, consider the parallelogram formed by the points 0, $z,$ $\frac{1}{z},$ and $z + \frac{1}{z}$ where the area of the parallelogram is $\frac{24}{25}.$ If the real part of $z$ is positive, determine the smallest possible value of $\left| z + \frac{1}{z} \right|.$ Compute the square of this value. | \frac{36}{25} | 0.1875 | 7,771.4375 | 7,445.666667 | 7,846.615385 | |
Given circle $O$: $x^{2}+y^{2}=10$, a line $l$ passing through point $P(-3,-4)$ intersects with circle $O$ at points $A$ and $B$. If the area of triangle $AOB$ is $5$, find the slope of line $l$. | \frac{11}{2} | 0.125 | 7,915.5625 | 5,980.5 | 8,192 | |
There are three pairs of real numbers \left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), and \left(x_{3}, y_{3}\right) that satisfy both $x^{3}-3 x y^{2}=2005$ and $y^{3}-3 x^{2} y=2004$. Compute \left(1-\frac{x_{1}}{y_{1}}\right)\left(1-\frac{x_{2}}{y_{2}}\right)\left(1-\frac{x_{3}}{y_{3}}\right). | 1/1002 | By the given, 2004 \left(x^{3}-3 x y^{2}\right)-2005\left(y^{3}-3 x^{2} y\right)=0. Dividing both sides by $y^{3}$ and setting $t=\frac{x}{y}$ yields $2004\left(t^{3}-3 t\right)-2005\left(1-3 t^{2}\right)=0$. A quick check shows that this cubic has three real roots. Since the three roots are precisely \frac{x_{1}}{y_{1... | 0.25 | 7,473.75 | 5,319 | 8,192 |
Given a right triangle with integer leg lengths $a$ and $b$ and a hypotenuse of length $b+2$, where $b<100$, determine the number of possible integer values for $b$. | 10 | 0 | 3,393.875 | -1 | 3,393.875 | |
Bangladesh National Mathematical Olympiad 2016 Higher Secondary
<u>**Problem 2:**</u>
(a) How many positive integer factors does $6000$ have?
(b) How many positive integer factors of $6000$ are not perfect squares? | 34 | 0.8125 | 3,853.0625 | 3,294.615385 | 6,273 | |
The measure of each exterior angle of a regular polygon is \(20^\circ\). What is the sum of the measures of the interior angles and the total number of diagonals of this polygon? | 135 | 0.0625 | 2,370 | 1,217 | 2,446.866667 | |
Solve for $x$: $$\sqrt[3]{3-\frac{1}{x}}=-4$$ | x=\frac{1}{67} | 1 | 1,424.6875 | 1,424.6875 | -1 | |
Find the smallest number, written using only ones and zeros, that would be divisible by 225. | 11111111100 | 0.625 | 6,226.5625 | 5,047.3 | 8,192 | |
Given quadrilateral $ABCD$ where $AC \perp BD$ and $AC=2$, $BD=3$, find the minimum value of $\overrightarrow{AB} \cdot \overrightarrow{CD}$. | - \dfrac{13}{4} | 0.4375 | 6,556.4375 | 5,660 | 7,253.666667 | |
Find the phase shift of the graph of $y = 3 \sin \left( x - \frac{\pi}{5} \right).$ | \frac{\pi}{5} | 1 | 1,763 | 1,763 | -1 | |
Cedric has deposited $\$12,\!000$ into an account that pays $5\%$ interest compounded annually.
Daniel has deposited $\$12,\!000$ into an account that pays $7\%$ simple annual interest.
In $15$ years Cedric and Daniel compare their respective balances. To the nearest dollar, what is the positive difference between th... | \$347 | 0.875 | 4,678 | 4,700.857143 | 4,518 | |
How many four-digit positive integers are multiples of 7? | 1286 | 1 | 3,618.1875 | 3,618.1875 | -1 | |
Petya and his three classmates started a 100-meter race simultaneously, and Petya finished first. Twelve seconds after the race began, no one had finished yet, and all four participants had collectively run a total of 288 meters. When Petya finished the race, the other three participants had a combined distance of 40 m... | 80 | 0 | 7,994.6875 | -1 | 7,994.6875 | |
Given the sequences \(\{a_{n}\}\) and \(\{b_{n}\}\) with general terms \(a_{n} = 2^{n}\) and \(b_{n} = 5n - 2\), find the sum of all elements in the set \(\{a_{1}, a_{2}, \cdots, a_{2019}\} \cap \{b_{1}, b_{2}, \cdots, b_{2019}\}\). | 2184 | 0.1875 | 5,930.625 | 4,927.666667 | 6,162.076923 | |
Compute $\cos 210^\circ$. | -\frac{\sqrt{3}}{2} | 0 | 2,059.875 | -1 | 2,059.875 | |
Without using a calculator, compute $1003^2-997^2-1001^2+999^2$. | 8000 | 0.75 | 5,779.25 | 4,975 | 8,192 | |
The sequence $\{a_n\}$ is an arithmetic sequence, and the sequence $\{b_n\}$ satisfies $b_n = a_na_{n+1}a_{n+2} (n \in \mathbb{N}^*)$. Let $S_n$ be the sum of the first $n$ terms of $\{b_n\}$. If $a_{12} = \frac{3}{8}a_5 > 0$, then when $S_n$ reaches its maximum value, the value of $n$ is equal to . | 16 | 0.125 | 8,006.0625 | 7,664 | 8,054.928571 |
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