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 |
|---|---|---|---|---|---|---|
In the sequence $\{a_n\}$, $a_1 = 1$, $a_2 = 2$, and $a_{n+2} - a_n = 1 + (-1)^n$ $(n \in \mathbb{N}^*)$, then $S_{100} = \_\_\_\_\_\_\_\_$. | 2600 | 0.6875 | 5,599.4375 | 4,421 | 8,192 | |
(1) Two poles are 6m apart with a rope tied between them, and a lamp is hung on the rope. Find the probability that the distance between the lamp and both ends is greater than 2m;
(2) From the numbers 1, 2, 3, 4, 5, 6, if two numbers are randomly selected and added together, what is the probability that their sum is ... | \frac{2}{5} | 1 | 2,288.0625 | 2,288.0625 | -1 | |
The product of four different positive integers is 360. What is the maximum possible sum of these four integers? | 66 | 0.0625 | 8,165.4375 | 7,767 | 8,192 | |
The integer $n$ has exactly six positive divisors, and they are: $1<a<b<c<d<n$ . Let $k=a-1$ . If the $k$ -th divisor (according to above ordering) of $n$ is equal to $(1+a+b)b$ , find the highest possible value of $n$ . | 2009 | 0.25 | 7,737.5 | 6,374 | 8,192 | |
Let an ordered pair of positive integers $(m, n)$ be called *regimented* if for all nonnegative integers $k$ , the numbers $m^k$ and $n^k$ have the same number of positive integer divisors. Let $N$ be the smallest positive integer such that $\left(2016^{2016}, N\right)$ is regimented. Compute the largest pos... | 10086 | 0 | 7,964.6875 | -1 | 7,964.6875 | |
If $n$ is $1$ less than a multiple of $50$, then what is the remainder when $n^2+2n+3$ is divided by $50$? | 2 | 1 | 2,427.375 | 2,427.375 | -1 | |
Find the biggest positive integer $n$ such that $n$ is $167$ times the amount of it's positive divisors. | 2004 | 0 | 8,192 | -1 | 8,192 | |
How many $y$-intercepts does the graph of the parabola $x = 2y^2 - 3y + 7$ have? | 0 | 1 | 1,955 | 1,955 | -1 | |
A two-digit number, when three times the sum of its units and tens digits is subtracted by -2, still results in the original number. Express this two-digit number algebraically. | 28 | 0 | 1,296.8125 | -1 | 1,296.8125 | |
John is tasked with creating a special mixture in his Science class, consisting of 0.05 liters of Compound X and 0.01 liters of Compound Y. He determined that each liter of this mixture has a specific ratio of Compound Y. Now, John needs to prepare 0.90 liters of this mixture. How much Compound Y will he require? | 0.15 | 0.3125 | 437.3125 | 471.2 | 421.909091 | |
Given the obtuse angle $\alpha$ that satisfies the equation $$\frac {sin\alpha-3cos\alpha}{cos\alpha -sin\alpha }=tan2\alpha$$, find the value of $tan\alpha$. | 2 - \sqrt{7} | 0.8125 | 4,634.1875 | 3,813.153846 | 8,192 | |
A point is chosen at random on the number line between 0 and 1, and the point is colored red. Then, another point is chosen at random on the number line between 0 and 1, and this point is colored blue. What is the probability that the number of the blue point is greater than the number of the red point, but less than t... | \frac{1}{9} | 0 | 6,129.6875 | -1 | 6,129.6875 | |
Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the $13$ visible numbers have the greatest possible sum. What is that sum? | 164 | To solve this problem, we need to maximize the sum of the visible numbers on three stacked cubes, each formed from a pattern with numbers 1, 2, 4, 8, 16, and 32. Each cube has six faces, but when stacked, some faces will not be visible.
1. **Maximize the sum of visible numbers on each cube:**
- Each cube has one fa... | 0 | 7,862.8125 | -1 | 7,862.8125 |
If the ratio of $2x-y$ to $x+y$ is $\frac{2}{3}$, what is the ratio of $x$ to $y$? | \frac{5}{4} | 1. **Set up the equation from the given ratio:**
Given that the ratio of $2x-y$ to $x+y$ is $\frac{2}{3}$, we can write this as:
\[
\frac{2x-y}{x+y} = \frac{2}{3}
\]
2. **Cross-multiply to eliminate the fraction:**
Cross-multiplying the equation from step 1, we get:
\[
3(2x - y) = 2(x + y)
\]
... | 1 | 1,436.1875 | 1,436.1875 | -1 |
150 people were surveyed and asked: "Do you think teal is more green or blue?" Of them, 90 believe teal is "more green," and 50 believe it's "more blue." Additionally, 40 believe it's both "more green" and "more blue." Another 20 think teal is neither "more green" nor "more blue."
How many of those 150 people believe ... | 80 | 0.125 | 2,566.1875 | 781 | 2,821.214286 | |
In quadrilateral $ABCD,\ BC=8,\ CD=12,\ AD=10,$ and $m\angle A= m\angle B = 60^\circ.$ Given that $AB = p + \sqrt{q},$ where $p$ and $q$ are positive integers, find $p+q.$ | 150 | Draw the perpendiculars from $C$ and $D$ to $AB$, labeling the intersection points as $E$ and $F$. This forms 2 $30-60-90$ right triangles, so $AE = 5$ and $BF = 4$. Also, if we draw the horizontal line extending from $C$ to a point $G$ on the line $DE$, we find another right triangle $\triangle DGC$. $DG = DE - CF = 5... | 0.25 | 6,840.25 | 4,913.75 | 7,482.416667 |
The value of $\sqrt{3^{3}+3^{3}+3^{3}}$ is what? | 9 | Since $3^{3}=3 \times 3 \times 3=3 \times 9=27$, then $\sqrt{3^{3}+3^{3}+3^{3}}=\sqrt{27+27+27}=\sqrt{81}=9$. | 1 | 276 | 276 | -1 |
Solve the inequality
$$
(2+\sqrt{3})^x + 2 < 3(\sqrt{2-\sqrt{3}})^{2x}
$$
Find the sum of all integer values of \(x\) that satisfy this inequality and belong to the interval \((-20, 53)\). | -190 | 0.4375 | 7,245.875 | 6,029.428571 | 8,192 | |
Marsha has two numbers, $a$ and $b$. When she divides $a$ by 70 she gets a remainder of 64. When she divides $b$ by 105 she gets a remainder of 99. What remainder does she get when she divides $a+b$ by 35? | 23 | 1 | 2,024.625 | 2,024.625 | -1 | |
Given that each side of a triangle is an integer and none exceeds 4, determine the number of such distinct triangles. | 13 | 0.4375 | 7,257 | 6,054.857143 | 8,192 | |
Find the value of $(52+6\sqrt{43})^{3/2}-(52-6\sqrt{43})^{3/2}$. | 828 | The $3/2$ power is quite irritating to work with so we look for a way to eliminate that. Notice that squaring the expression will accomplish that. Let $S$ be the sum of the given expression. \[S^2= ((52+6\sqrt{43})^{3/2}-(52-6\sqrt{43})^{3/2})^2\] \[S^2 = (52+6\sqrt{43})^{3} + (52-6\sqrt{43})^{3} - 2((52+6\sqrt{43})(52... | 0.6875 | 5,024.625 | 4,763.636364 | 5,598.8 |
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$, the magnitude of $\overrightarrow {a}$ is the positive root of the equation x^2+x-2=0, $|\overrightarrow {b}|= \sqrt {2}$, and $(\overrightarrow {a}- \overrightarrow {b})\cdot \overrightarrow {a}=0$, find the angle between $\overrightarrow {a}$ and $\overri... | \frac{\pi}{4} | 0.8125 | 2,193.375 | 1,975 | 3,139.666667 | |
Given a cubic polynomial \( q(x) \) that satisfies \( q(3) = 2 \), \( q(8) = 20 \), \( q(18) = 12 \), and \( q(25) = 32 \). Find the sum \( q(4) + q(5) + \ldots + q(26) \). | 391 | 0 | 8,192 | -1 | 8,192 | |
Let $g$ be a function taking the positive integers to the positive integers, such that:
(i) $g$ is increasing (i.e. $g(n + 1) > g(n)$ for all positive integers $n$),
(ii) $g(mn) = g(m) g(n)$ for all positive integers $m$ and $n,$ and
(iii) if $m \neq n$ and $m^n = n^m,$ then $g(m) = n^2$ or $g(n) = m^2.$
Find the sum... | 104976 | 0 | 8,192 | -1 | 8,192 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $\frac{\cos C}{\cos B}= \frac{2a-c}{b}$.
(1) Find $B$;
(2) If $\tan \left(A+ \frac{π}{4}\right) =7$, find the value of $\cos C$. | \frac{-4+3\sqrt{3}}{10} | 0 | 5,379.1875 | -1 | 5,379.1875 | |
If "$x^2-2x-3>0$" is a necessary but not sufficient condition for "$x<a$", then the maximum value of $a$ is. | -1 | 0.9375 | 5,432.5 | 5,248.533333 | 8,192 | |
In triangle \(ABC\), point \(K\) on side \(AB\) and point \(M\) on side \(AC\) are positioned such that \(AK:KB = 3:2\) and \(AM:MC = 4:5\). Determine the ratio in which the line through point \(K\) parallel to side \(BC\) divides segment \(BM\). | 18/7 | 0.0625 | 6,721.8125 | 5,484 | 6,804.333333 | |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\frac{\cos B}{b} + \frac{\cos C}{c} = \frac{2\sqrt{3}\sin A}{3\sin C}$.
(1) Find the value of $b$;
(2) If $B = \frac{\pi}{3}$, find the maximum area of triangle $ABC$. | \frac{3\sqrt{3}}{16} | 0 | 7,200.5 | -1 | 7,200.5 | |
One TV was sold for a 12% profit and the other for a 12% loss at a selling price of 3080 yuan each. Determine the net profit or loss from these transactions. | -90 | 0.4375 | 1,033.75 | 787.142857 | 1,225.555556 | |
At 17:00, the speed of a racing car was 30 km/h. Every subsequent 5 minutes, the speed increased by 6 km/h. Determine the distance traveled by the car from 17:00 to 20:00 on the same day. | 425.5 | 0 | 5,976.8125 | -1 | 5,976.8125 | |
Below is the graph of $y = a \sin (bx + c)$ for some constants $a$, $b$, and $c$. Assume $a>0$ and $b>0$. Find the smallest possible value of $c$ if it is given that the graph reaches its minimum at $x = 0$. | \frac{3\pi}{2} | 0 | 8,078.4375 | -1 | 8,078.4375 | |
Ara and Shea were once the same height. Since then Shea has grown 20% while Ara has grown half as many inches as Shea. Shea is now 60 inches tall. How tall, in inches, is Ara now? | 55 | 1. **Determine Shea's original height**:
Given that Shea has grown by 20% and is now 60 inches tall, we can set up the equation for her original height $x$:
\[
1.2x = 60
\]
Solving for $x$, we divide both sides by 1.2:
\[
x = \frac{60}{1.2} = 50
\]
Thus, Shea's original height was 50 inches.... | 1 | 1,337.5625 | 1,337.5625 | -1 |
Let $A,B,C,D$ denote four points in space such that at most one of the distances $AB,AC,AD,BC,BD,CD$ is greater than $1$ . Determine the maximum value of the sum of the six distances. | \[ 5 + \sqrt{3} \] | Suppose that $AB$ is the length that is more than $1$ . Let spheres with radius $1$ around $A$ and $B$ be $S_A$ and $S_B$ . $C$ and $D$ must be in the intersection of these spheres, and they must be on the circle created by the intersection to maximize the distance. We have $AC + BC + AD + BD = 4$ .
In fact, $CD$ must ... | 0 | 8,192 | -1 | 8,192 |
In $\triangle ABC$, the side lengths are: $AB = 17, BC = 20$ and $CA = 21$. $M$ is the midpoint of side $AB$. The incircle of $\triangle ABC$ touches $BC$ at point $D$. Calculate the length of segment $MD$.
A) $7.5$
B) $8.5$
C) $\sqrt{8.75}$
D) $9.5$ | \sqrt{8.75} | 0 | 8,192 | -1 | 8,192 | |
What are all values of $p$ such that for every $q>0$, we have $$\frac{3(pq^2+p^2q+3q^2+3pq)}{p+q}>2p^2q?$$ Express your answer in interval notation in decimal form. | [0,3) | 0.375 | 7,022.625 | 5,661.166667 | 7,839.5 | |
Find the largest integer $n$ such that $n$ is divisible by all positive integers less than $\sqrt[3]{n}$. | 420 |
Given the problem, we are tasked to find the largest integer \( n \) such that \( n \) is divisible by all positive integers less than \( \sqrt[3]{n} \).
### Step-by-Step Solution:
1. **Understanding the condition:**
We know \( n \) must be divisible by every integer \( k \) where \( k < \sqrt[3]{n} \). Therefor... | 0.3125 | 7,788.9375 | 6,960.6 | 8,165.454545 |
Vasya wrote a note on a piece of paper, folded it in four, and labeled the top with "MAME". Then he unfolded the note, added something else, folded it along the creases in a random manner (not necessarily the same as before), and left it on the table with a random side up. Find the probability that the inscription "MAM... | 1/8 | 0 | 6,198.875 | -1 | 6,198.875 | |
In Princess Sissi's garden, there is an empty water reservoir. When water is injected into the reservoir, the drainage pipe will draw water out to irrigate the flowers. Princess Sissi found that if 3 water pipes are turned on, the reservoir will be filled in 30 minutes; if 5 water pipes are turned on, the reservoir wil... | 15 | 0.3125 | 7,404.0625 | 5,713.2 | 8,172.636364 | |
In a household, when someone is at home, the probability of the phone being answered at the 1st ring is 0.1, at the 2nd ring is 0.3, at the 3rd ring is 0.4, and at the 4th ring is 0.1. What is the probability that the phone is not answered within the first 4 rings? | 0.1 | 0.375 | 5,836.625 | 3,602.166667 | 7,177.3 | |
There are 17 people standing in a circle: each of them is either truthful (always tells the truth) or a liar (always lies). All of them said that both of their neighbors are liars. What is the maximum number of liars that can be in this circle? | 11 | 0 | 8,192 | -1 | 8,192 | |
In a convex pentagon, all diagonals are drawn. For each pair of diagonals that intersect inside the pentagon, the smaller of the angles between them is found. What values can the sum of these five angles take? | 180 | 0.0625 | 7,941.6875 | 4,187 | 8,192 | |
Let \( M \) be the centroid of \( \triangle ABC \), and \( AM = 3 \), \( BM = 4 \), \( CM = 5 \). Find the area of \( \triangle ABC \). | 18 | 0.1875 | 7,078.8125 | 4,483.666667 | 7,677.692308 | |
The cube of $a$ and the fourth root of $b$ vary inversely. If $a=3$ when $b=256$, then find $b$ when $ab=81$. | 16 | 0 | 7,981.75 | -1 | 7,981.75 | |
Triangle $ABC$ is inscribed in circle $\omega$. Points $P$ and $Q$ are on side $\overline{AB}$ with $AP<AQ$. Rays $CP$ and $CQ$ meet $\omega$ again at $S$ and $T$ (other than $C$), respectively. If $AP=4,PQ=3,QB=6,BT=5,$ and $AS=7$, then $ST=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $... | 43 | By Ptolemy's Theorem applied to quadrilateral $ASTB$, we find \[AS\cdot BT+AB\cdot ST=AT\cdot BS.\] Projecting through $C$ we have \[\frac{AQ \cdot PB}{PQ \cdot AB} = (A,Q; P,B)\stackrel{C}{=}(A,T; S,B)=\frac{AT \cdot BS}{ST \cdot AB}.\] Therefore \[AT \cdot BS = \frac {AQ \cdot PB}{PQ} \times ST \implies\] \[\left(\fr... | 0 | 8,192 | -1 | 8,192 |
If $f(x) = 5x-4$, what is $f(f(f(2)))$? | 126 | 1 | 2,302.3125 | 2,302.3125 | -1 | |
Yura has a calculator that allows multiplying a number by 3, adding 3 to a number, or (if the number is divisible by 3) dividing by 3. How can you obtain the number 11 from the number 1 using this calculator? | 11 | 0.1875 | 8,013.0625 | 7,237.666667 | 8,192 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. It is given that $\frac{-b + \sqrt{2}c}{\cos B} = \frac{a}{\cos A}$,
$(I)$ Find the size of angle $A$;
$(II)$ If $a=2$, find the maximum value of the area $S$. | \sqrt{2} + 1 | 0 | 7,457.625 | -1 | 7,457.625 | |
Find all angles $\theta,$ $0 \le \theta \le 2 \pi,$ with the following property: For all real numbers $x,$ $0 \le x \le 1,$
\[x^2 \cos \theta - x(1 - x) + (1 - x)^2 \sin \theta > 0.\] | \left( \frac{\pi}{12}, \frac{5 \pi}{12} \right) | 0.1875 | 7,625.0625 | 5,713.333333 | 8,066.230769 | |
There are 8 young people, among whom 5 are capable of doing English translation work, and 4 are capable of doing computer software design work (including one person who is capable of doing both tasks). Now, 5 young people are to be selected to undertake a task, with 3 people doing English translation work and 2 people ... | 42 | 0.1875 | 7,774.75 | 7,533.666667 | 7,830.384615 | |
Let $m > n$ be positive integers such that $3(3mn - 2)^2 - 2(3m -3n)^2 = 2019$ . Find $3m + n$ .
| 46 | 0.8125 | 5,413.3125 | 4,893.846154 | 7,664.333333 | |
Given $sin(\alpha-\frac{\pi}{6})=\frac{3}{5}$, calculate $cos(\frac{2\pi}{3}-\alpha)$. | \frac{3}{5} | 0.9375 | 4,390.5625 | 4,137.133333 | 8,192 | |
Given that the perimeter of a sector is 10cm, and its area is 4cm<sup>2</sup>, find the radian measure of the central angle $\alpha$. | \frac{1}{2} | 0.375 | 5,902.3125 | 4,852.166667 | 6,532.4 | |
The coefficient of $x^2$ in the expansion of $(1+2x)^5$ is __________. | 40 | 0.9375 | 2,203.625 | 1,804.4 | 8,192 | |
How many candies were in the bag before the first day if a group of friends eat candies over five days as follows: On the first day, they eat \( \frac{1}{2} \) of the candies, on the second day \( \frac{2}{3} \) of the remaining, on the third day \( \frac{3}{4} \) of the remaining, on the fourth day \( \frac{4}{5} \) o... | 720 | We work backwards through the given information. At the end, there is 1 candy remaining. Since \( \frac{5}{6} \) of the candies are removed on the fifth day, this 1 candy represents \( \frac{1}{6} \) of the candies left at the end of the fourth day. Thus, there were \( 6 \times 1 = 6 \) candies left at the end of the f... | 0.875 | 4,398.25 | 3,856.285714 | 8,192 |
Given a finite sequence \(P = \left(p_{1}, p_{2}, \cdots, p_{n}\right)\), the Caesar sum (named after a mathematician Caesar) is defined as \(\frac{s_{1}+s_{2}+\cdots+s_{n}}{n}\), where \(s_{k} = p_{1} + p_{2} + \cdots + p_{k}\) for \(1 \leq k \leq n\). If a sequence of 99 terms \(\left(p_{1}, p_{2}, \cdots, p_{99}\rig... | 991 | 0.8125 | 5,201.6875 | 4,886.923077 | 6,565.666667 | |
A three-meter gas pipe has rusted in two places. Determine the probability that all three resulting parts can be used as connections to gas stoves, given that regulations require a stove to be no closer than 75 cm from the main gas pipeline. | \frac{1}{4} | 0.0625 | 7,558.375 | 7,612 | 7,554.8 | |
You have 50 dimes and 20 quarters. What percent of the value of your money is in quarters? | 50\% | 1 | 1,647.0625 | 1,647.0625 | -1 | |
Let $k$ and $m$ be real numbers, and suppose that the roots of the equation \[x^3 - 7x^2 + kx - m = 0\]are three distinct positive integers. Compute $k + m.$ | 22 | 1 | 2,341.6875 | 2,341.6875 | -1 | |
How many natural numbers between 150 and 300 are divisible by 9? | 17 | 0.9375 | 2,871.0625 | 3,003 | 892 | |
Suppose that $d$ is an odd integer and $e$ is an even integer. How many of the following expressions are equal to an odd integer? $d+d, (e+e) imes d, d imes d, d imes(e+d)$ | 2 | Since $d$ is an odd integer, then $d+d$ is even and $d imes d$ is odd. Since $e$ is an even integer, then $e+e$ is even, which means that $(e+e) imes d$ is even. Also, $e+d$ is odd, which means that $d imes(e+d)$ is odd. Thus, 2 of the 4 expressions are equal to an odd integer. | 1 | 2,653.625 | 2,653.625 | -1 |
Calculate the limit of the function:
\[
\lim_{x \rightarrow 0} \frac{\arcsin(2x)}{2^{-3x} - 1} \cdot \ln 2
\] | -\frac{2}{3} | 0.875 | 4,899.1875 | 4,702.428571 | 6,276.5 | |
Automobile license plates for a state consist of three letters followed by a dash and three single digits. How many different license plate combinations are possible if exactly two letters are each repeated once (yielding a total of four letters where two are the same), and the digits include exactly one repetition? | 877,500 | 0 | 6,345.1875 | -1 | 6,345.1875 | |
A plane intersects a right circular cylinder of radius $2$ forming an ellipse. If the major axis of the ellipse is $30\%$ longer than the minor axis, what is the length of the major axis? | 5.2 | 0.25 | 4,903.9375 | 3,905.75 | 5,236.666667 | |
Filling the gas tank of a small car cost, in updated values, $\mathrm{R} \$ 29.90$ in 1972 and $\mathrm{R} \$ 149.70$ in 1992. Which of the following values best approximates the percentage increase in the price of gasoline during this 20-year period?
(a) $20 \%$
(b) $125 \%$
(c) $300 \%$
(d) $400 \%$
(e) $500 ... | 400\% | 1 | 445.5625 | 445.5625 | -1 | |
Find the sum of the digits in the number $\underbrace{44 \ldots 4}_{2012 \text{ times}} \cdot \underbrace{99 \ldots 9}_{2012 \text{ times}}$. | 18108 | 0.8125 | 6,237.375 | 5,786.307692 | 8,192 | |
Doug constructs a square window using $8$ equal-size panes of glass. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window? | 26 | 1. **Identify the dimensions of each pane**: Given that the ratio of the height to the width of each pane is $5:2$, let the height of each pane be $5x$ inches and the width be $2x$ inches.
2. **Calculate the total dimensions of the window**: The window is constructed with $8$ panes arranged in $2$ rows and $4$ columns... | 0.4375 | 5,610.125 | 4,007.142857 | 6,856.888889 |
Find a positive integer \( n \) less than 2006 such that \( 2006n \) is a multiple of \( 2006 + n \). | 1475 | 0 | 8,192 | -1 | 8,192 | |
Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2010,0),(2010,2011),$ and $(0,2011)$. What is the probability that $x > 3y$? Express your answer as a common fraction. | \frac{335}{2011} | 0.75 | 6,495.8125 | 5,930.416667 | 8,192 | |
What is the value of \(2^{1+2+3}-(2^1+2^2+2^3)?\) | 50 | 1. **Simplify the Exponent:**
The expression given is $2^{1+2+3} - (2^1 + 2^2 + 2^3)$. First, simplify the exponent in the term $2^{1+2+3}$:
\[
1+2+3 = 6
\]
Thus, the expression becomes:
\[
2^6 - (2^1 + 2^2 + 2^3)
\]
2. **Evaluate Powers of 2:**
Calculate each power of 2:
\[
2^6 = ... | 1 | 1,465.25 | 1,465.25 | -1 |
31 people attended a class club afternoon event, and after the program, they danced. Ági danced with 7 boys, Anikó with 8, Zsuzsa with 9, and so on, with each subsequent girl dancing with one more boy than the previously mentioned one. Finally, Márta danced with all but 3 boys. How many boys were at the club afternoon? | 20 | 0.375 | 6,267 | 3,266.833333 | 8,067.1 | |
The graphs of five functions are labelled from **(1) through (5)**. Provided below are descriptions of three:
1. The domain of function (2) is from $$\{-6,-5,-4,-3,-2,-1,0,1,2,3\}.$$ It is graphed as a set of discrete points.
2. Function (4) is defined by the equation $$y = x^3$$ and is graphed from $$x = -3$$ to... | 20 | 0 | 5,856.125 | -1 | 5,856.125 | |
Solve for $x:\ \log_2 x+\log_4 x= 6.$ | 16 | 1 | 2,171.5 | 2,171.5 | -1 | |
In rectangle $PQRS$, $PQ = 8$ and $QR = 4$. Points $T$ and $U$ are on $\overline{RS}$ such that $RT = 2$ and $SU = 3$. Lines $PT$ and $QU$ intersect at $V$. Find the area of $\triangle PVQ$. [asy]
pair P,Q,R,S,V,T,U;
P=(0,0);
Q=(8,0);
R=(8,4);
S=(0,4);
T=(2,4);
U=(6,4);
V=(3.2,6);
draw(P--Q--R--S--cycle,linewidth(0.7))... | 32 | 0.125 | 4,084.125 | 4,116.5 | 4,079.5 | |
In a new game, Jane and her brother each spin a spinner once. The spinner has six congruent sectors labeled from 1 to 6. If the non-negative difference of their numbers is less than 4, Jane wins. Otherwise, her brother wins. What is the probability that Jane wins? Express your answer as a common fraction. | \frac{5}{6} | 0.5625 | 6,595.3125 | 5,493.444444 | 8,012 | |
Joe has a collection of $23$ coins, consisting of $5$-cent coins, $10$-cent coins, and $25$-cent coins. He has $3$ more $10$-cent coins than $5$-cent coins, and the total value of his collection is $320$ cents. How many more $25$-cent coins does Joe have than $5$-cent coins? | 2 | Let $n$, $d$, and $q$ represent the number of $5$-cent coins, $10$-cent coins, and $25$-cent coins in Joe's collection, respectively. We are given the following equations based on the problem statement:
\[
n + d + q = 23 \quad \text{(1)}
\]
\[
5n + 10d + 25q = 320 \quad \text{(2)}
\]
\[
d = n + 3 \quad \text{(3)}
\]
F... | 1 | 1,885.0625 | 1,885.0625 | -1 |
Let $n>1$ be an odd integer. On an $n \times n$ chessboard the center square and four corners are deleted. We wish to group the remaining $n^{2}-5$ squares into $\frac{1}{2}(n^{2}-5)$ pairs, such that the two squares in each pair intersect at exactly one point (i.e. they are diagonally adjacent, sharing a single corner... | 3,5 | Constructions for $n=3$ and $n=5$ are easy. For $n>5$, color the odd rows black and the even rows white. If the squares can be paired in the way desired, each pair we choose must have one black cell and one white cell, so the numbers of black cells and white cells are the same. The number of black cells is $\frac{n+1}{... | 0 | 7,572.875 | -1 | 7,572.875 |
Let \( x_{1} \) and \( x_{2} \) be the largest roots of the polynomials \( f(x) = 1 - x - 4x^{2} + x^{4} \) and \( g(x) = 16 - 8x - 16x^{2} + x^{4} \), respectively. Find \( \frac{x_{1}}{x_{2}} \). | 1/2 | 0.1875 | 8,033.8125 | 7,348.333333 | 8,192 | |
The sequence 12, 15, 18, 21, 51, 81, $\ldots$ consists of all positive multiples of 3 that contain at least one digit that is a 1. What is the $50^{\mathrm{th}}$ term of the sequence? | 318 | 0 | 7,993.875 | -1 | 7,993.875 | |
The sum of four numbers is one-half. What is the mean of the four numbers? Express your answer as a common fraction. | \frac{1}{8} | 1 | 942.4375 | 942.4375 | -1 | |
We have a triangle $\triangle ABC$ and a point $K$ on $BC$ such that $AK$ is an altitude of $\triangle ABC$. If $AC = 10,$ $BK = 7$, and $BC = 13,$ then what is the area of $\triangle ABC$? | 52 | 1 | 2,462.25 | 2,462.25 | -1 | |
For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as
$$a_{n+1} =
\begin{cases}
\sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\
a_n + 3 & \text{otherwise.}
\end{cases}
$$
Determine all values of $a_0$ such that there exists a number $A$ such that $a_n = A$ for infinitel... | 3 \mid a_0 |
We are given a sequence defined by \( a_0, a_1, a_2, \ldots \) where the recurrence relation for \( n \geq 0 \) is:
\[
a_{n+1} =
\begin{cases}
\sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer}, \\
a_n + 3 & \text{otherwise}.
\end{cases}
\]
The goal is to determine all starting values \( a_0 \) such that the se... | 0 | 8,192 | -1 | 8,192 |
The side surface of a cylinder unfolds into a rectangle with side lengths of $6\pi$ and $4\pi$. The surface area of the cylinder is ______. | 24\pi^2 + 8\pi | 0 | 8,141.1875 | -1 | 8,141.1875 | |
What is the smallest positive integer that has exactly eight distinct positive factors? | 24 | 1 | 2,517.9375 | 2,517.9375 | -1 | |
If \( 8 + 6 = n + 8 \), what is the value of \( n \)? | 6 | Since \( 8+6=n+8 \), then subtracting 8 from both sides, we obtain \( 6=n \) and so \( n \) equals 6. | 1 | 1,443.125 | 1,443.125 | -1 |
The matrix
\[\begin{pmatrix} a & b \\ -\frac{4}{5} & \frac{3}{5} \end{pmatrix}\]corresponds to a reflection. Enter the ordered pair $(a,b).$ | \left( -\frac{3}{5}, -\frac{4}{5} \right) | 1 | 4,413.75 | 4,413.75 | -1 | |
On a table there are $100$ red and $k$ white buckets for which all of them are initially empty. In each move, a red and a white bucket is selected and an equal amount of water is added to both of them. After some number of moves, there is no empty bucket and for every pair of buckets that are selected together at l... | 100 | 0.25 | 7,146.75 | 4,011 | 8,192 | |
Let vertices $A, B, C$, and $D$ form a regular tetrahedron with each edge of length 1 unit. Define point $P$ on edge $AB$ such that $P = tA + (1-t)B$ for some $t$ in the range $0 \leq t \leq 1$ and point $Q$ on edge $CD$ such that $Q = sC + (1-s)D$ for some $s$ in the range $0 \leq s \leq 1$. Determine the minimum poss... | \frac{\sqrt{2}}{2} | 0 | 6,935.375 | -1 | 6,935.375 | |
Convert the following expressions between different bases: $110011_{(2)} = \_{(10)} = \_{(5)}$ | 51_{(10)} = 201_{(5)} | 0 | 2,018.9375 | -1 | 2,018.9375 | |
In a pocket, there are eight cards of the same size, among which three are marked with the number $1$, three are marked with the number $2$, and two are marked with the number $3$. The first time, a card is randomly drawn from the pocket and then put back. After that, a second card is drawn randomly. Let the sum of the... | \dfrac {15}{4} | 0.875 | 3,662.9375 | 3,191.214286 | 6,965 | |
Ctibor marked a square land plot on a map with a scale of 1:50000 and calculated that its side corresponds to $1 \mathrm{~km}$ in reality. He then resized the map on a copier such that the marked square had an area $1.44 \mathrm{~cm}^{2}$ smaller than the original.
What was the scale of the resized map?
Hint: What we... | 1:62500 | 0.3125 | 5,423.0625 | 3,254.8 | 6,408.636364 | |
Let \(\triangle A B C\) be a right triangle with right angle \(C\). Let \(I\) be the incenter of \(A B C\), and let \(M\) lie on \(A C\) and \(N\) on \(B C\), respectively, such that \(M, I, N\) are collinear and \(\overline{M N}\) is parallel to \(A B\). If \(A B=36\) and the perimeter of \(C M N\) is 48, find the are... | 252 | Note that \(\angle M I A=\angle B A I=\angle C A I\), so \(M I=M A\). Similarly, \(N I=N B\). As a result, \(C M+M N+N C=C M+M I+N I+N C=C M+M A+N B+N C=A C+B C=48\). Furthermore \(A C^{2}+B C^{2}=36^{2}\). As a result, we have \(A C^{2}+2 A C \cdot B C+B C^{2}=48^{2}\), so \(2 A C \cdot B C=48^{2}-36^{2}=12 \cdot 84\)... | 0.375 | 7,131.8125 | 5,848.333333 | 7,901.9 |
Someone forms an integer by writing the integers from 1 to 82 in ascending order, i.e. 1234567891011 ...808182. Find the sum of the digits of this integer. | 667 | 0.1875 | 7,756.875 | 6,128 | 8,132.769231 | |
What is the value of $\sqrt[4]{2^3 + 2^4 + 2^5 + 2^6}$? | 2^{3/4} \cdot 15^{1/4} | 0 | 7,400.0625 | -1 | 7,400.0625 | |
Convert the decimal number 2011 to a base-7 number. | 5602_7 | 0.4375 | 953.6875 | 924.285714 | 976.555556 | |
Given an ellipse $\frac{x^2}{25}+\frac{y^2}{m^2}=1\left(m \gt 0\right)$ with one focus at $F\left(0,4\right)$, find $m$. | \sqrt{41} | 1 | 1,840 | 1,840 | -1 | |
How many numbers are in the following list: $$-4, -1, 2, 5,\ldots, 32$$ | 13 | 1 | 1,648.9375 | 1,648.9375 | -1 | |
My five friends and I play doubles badminton every weekend. Each weekend, two of us play as a team against another two, while the remaining two rest. How many different ways are there for us to choose the two teams and the resting pair? | 45 | 0.3125 | 6,287.75 | 5,952 | 6,440.363636 | |
How many prime numbers are divisible by $39$ ? | 0 | 1 | 1,486.6875 | 1,486.6875 | -1 | |
Find the smallest positive real t such that
\[ x_1 + x_3 = 2tx_2, \]
\[ x_2 + x_4 = 2tx_3, \]
\[ x_3 + x_5 = 2tx_4 \]
has a solution \( x_1, x_2, x_3, x_4, x_5 \) in non-negative reals, not all zero. | \frac{1}{\sqrt{2}} | 0 | 7,781.5625 | -1 | 7,781.5625 | |
Given the set \( A \) formed by exponential functions, there are 10 odd functions, 8 increasing functions defined on \((-\infty, \infty)\), and 12 functions whose graphs pass through the origin. Determine the minimum number of elements in set \( A \). | 14 | 0.0625 | 7,780.6875 | 8,192 | 7,753.266667 |
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