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 |
|---|---|---|---|---|---|---|
Determine the number of three-element subsets of the set \(\{1, 2, 3, 4, \ldots, 120\}\) for which the sum of the three elements is a multiple of 3. | 93640 | 0.25 | 7,356.5 | 4,850 | 8,192 | |
A square and four circles, each with a radius of 5 inches, are arranged as shown. What is the area, in square inches, of the square? [asy]
unitsize(1mm);
defaultpen(linewidth(0.7pt));
draw((0,0)--(20,0)--(20,20)--(0,20)--cycle);
draw(Circle((5,5),5));
draw(Circle((15,5),5));
draw(Circle((5,15),5));
draw(Circle((15,15)... | 400 | 0.875 | 5,194.5 | 5,044.214286 | 6,246.5 | |
Simplify the fraction by rationalizing the denominator: $$\frac{4}{\sqrt{108}+2\sqrt{12}+2\sqrt{27}}.$$ | \frac{\sqrt{3}}{12} | 0 | 2,160.5625 | -1 | 2,160.5625 | |
Given that $5^{2018}$ has $1411$ digits and starts with $3$ (the leftmost non-zero digit is $3$ ), for how many integers $1\leq n\leq2017$ does $5^n$ start with $1$ ?
*2018 CCA Math Bonanza Tiebreaker Round #3* | 607 | 0 | 8,192 | -1 | 8,192 | |
Evaluate the value of $\frac{1}{4}\cdot\frac{8}{1}\cdot\frac{1}{32}\cdot\frac{64}{1} \dotsm \frac{1}{1024}\cdot\frac{2048}{1}$. | 32 | 0.125 | 6,441.4375 | 5,159.5 | 6,624.571429 | |
In a box, there are 22 kg of cranberries. How, using a single 2-kilogram weight and a two-pan scale, can you measure out 17 kg of cranberries in two weighings? | 17 | 0 | 8,192 | -1 | 8,192 | |
Choose positive integers $b_1, b_2, \dotsc$ satisfying
\[1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb\]
and let $r$ denote the largest real number satisfying $\tfrac{b_n}{n^2} \geq r$ for all positive integers $n$. What are the possible values of $r$ across all possible choices of th... | 0 \leq r \leq \frac{1}{2} |
Let \( r \) denote the largest real number satisfying \(\frac{b_n}{n^2} \geq r\) for all positive integers \( n \), where \( b_1, b_2, \dotsc \) are positive integers satisfying
\[
1 = \frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb
\]
We aim to determine the possible values of \( r \).... | 0 | 8,140.0625 | -1 | 8,140.0625 |
Find the maximum possible value of $H \cdot M \cdot M \cdot T$ over all ordered triples $(H, M, T)$ of integers such that $H \cdot M \cdot M \cdot T=H+M+M+T$. | 8 | If any of $H, M, T$ are zero, the product is 0. We can do better (examples below), so we may now restrict attention to the case when $H, M, T \neq 0$. When $M \in\{-2,-1,1,2\}$, a little casework gives all the possible $(H, M, T)=(2,1,4),(4,1,2),(-1,-2,1),(1,-2,-1)$. If $M=-2$, i.e. $H-4+T=4 H T$, then $-15=(4 H-1)(4 T... | 0.0625 | 8,190.4375 | 8,167 | 8,192 |
Find the greatest integer value of $b$ for which the expression $\frac{9x^3+4x^2+11x+7}{x^2+bx+8}$ has a domain of all real numbers. | 5 | 1 | 1,583 | 1,583 | -1 | |
For what value of $x$ does $10^{x} \cdot 100^{2x}=1000^{5}$? | 3 | 1. **Rewrite the equation using properties of exponents**:
The given equation is $10^x \cdot 100^{2x} = 1000^5$. We know that $100 = 10^2$ and $1000 = 10^3$. Substituting these values, we get:
\[
10^x \cdot (10^2)^{2x} = (10^3)^5
\]
Simplifying the exponents on both sides, we have:
\[
10^x \cdot 1... | 1 | 1,253.3125 | 1,253.3125 | -1 |
In the set of all three-digit numbers composed of the digits 0, 1, 2, 3, 4, 5, without any repeating digits, how many such numbers have a digit-sum of 9? | 12 | 0 | 6,618.875 | -1 | 6,618.875 | |
Twenty-eight 4-inch wide square posts are evenly spaced with 4 feet between adjacent posts to enclose a rectangular field. The rectangle has 6 posts on each of the longer sides (including the corners). What is the outer perimeter, in feet, of the fence? | 112 | 0.0625 | 6,293 | 2,392 | 6,553.066667 | |
Eight teams participated in a football tournament, and each team played exactly once against each other team. If a match was drawn then both teams received 1 point; if not then the winner of the match was awarded 3 points and the loser received no points. At the end of the tournament the total number of points gained b... | 17 | 0 | 8,163.6875 | -1 | 8,163.6875 | |
In a group of 10 basketball teams which includes 2 strong teams, the teams are randomly split into two equal groups for a competition. What is the probability that the 2 strong teams do not end up in the same group? | \frac{5}{9} | 0.5625 | 5,919.0625 | 4,379.444444 | 7,898.571429 | |
Points $A$ and $B$ are $5$ units apart. How many lines in a given plane containing $A$ and $B$ are $2$ units from $A$ and $3$ units from $B$? | 3 | To solve this problem, we need to consider the geometric configuration of two circles centered at points $A$ and $B$ with radii $2$ and $3$ units, respectively. We are looking for lines that are tangent to both circles.
1. **Identify the circles**:
- Circle centered at $A$ (denoted as $C_A$) has radius $2$ units.
... | 1 | 4,450.0625 | 4,450.0625 | -1 |
Given the set \( A=\left\{\left.\frac{a_{1}}{9}+\frac{a_{2}}{9^{2}}+\frac{a_{3}}{9^{3}}+\frac{a_{4}}{9^{4}} \right\rvert\, a_{i} \in\{0,1,2, \cdots, 8\}, i=1, 2, 3, 4\} \), arrange the numbers in \( A \) in descending order and find the 1997th number. | \frac{6}{9} + \frac{2}{81} + \frac{3}{729} + \frac{1}{6561} | 0 | 8,131.75 | -1 | 8,131.75 | |
Given the real numbers \( a, b, c \) satisfy \( a + b + c = 6 \), \( ab + bc + ca = 5 \), and \( abc = 1 \), determine the value of \( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \). | 38 | 0.75 | 5,155.4375 | 4,282.416667 | 7,774.5 | |
A stack of logs has 12 logs on the bottom row, and one less in each successive row, ending with three logs at the top. How many logs are in the stack? | 75 | 1 | 1,932.75 | 1,932.75 | -1 | |
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color? | \frac{1}{5} | To solve this problem, we need to calculate the probability that Cheryl gets 2 marbles of the same color. We can use the principle of symmetry and consider the probability of any specific outcome for Cheryl, as the draws are random and independent of the order in which the participants draw.
1. **Total number of ways ... | 0.375 | 7,380.5 | 6,028 | 8,192 |
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, $c$, $\left(a+c\right)\sin A=\sin A+\sin C$, $c^{2}+c=b^{2}-1$. Find:<br/>
$(1)$ $B$;<br/>
$(2)$ Given $D$ is the midpoint of $AC$, $BD=\frac{\sqrt{3}}{2}$, find the area of $\triangle ABC$. | \frac{\sqrt{3}}{2} | 0 | 5,482.4375 | -1 | 5,482.4375 | |
Given that $a\in (\frac{\pi }{2},\pi )$ and $\sin \alpha =\frac{1}{3}$,
(1) Find the value of $\sin 2\alpha$;
(2) If $\sin (\alpha +\beta )=-\frac{3}{5}$, $\beta \in (0,\frac{\pi }{2})$, find the value of $\sin \beta$. | \frac{6\sqrt{2}+4}{15} | 0 | 7,118.125 | -1 | 7,118.125 | |
Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying the following conditions: (a) $f(1)=1$ (b) $f(a) \leq f(b)$ whenever $a$ and $b$ are positive integers with $a \leq b$. (c) $f(2a)=f(a)+1$ for all positive integers $a$. How many possible values can the 2014-tuple $(f(1), f(2), \ldots, f(2014))$ take? | 1007 | Note that $f(2014)=f(1007)+1$, so there must be exactly one index $1008 \leq i \leq 2014$ such that $f(i)=f(i-1)+1$, and for all $1008 \leq j \leq 2014, j \neq i$ we must have $f(j)=f(j-1)$. We first claim that each value of $i$ corresponds to exactly one 2014-tuple $(f(1), \ldots, f(2014))$. To prove this, note that $... | 0 | 8,192 | -1 | 8,192 |
For some positive integers $a$ and $b$, the product \[\log_a(a+1) \cdot \log_{a+1} (a+2) \dotsm \log_{b-2} (b-1) \cdot\log_{b-1} b\]contains exactly $870$ terms, and its value is $2.$ Compute $a+b.$ | 930 | 1 | 1,969.625 | 1,969.625 | -1 | |
Determine the smallest positive integer $n$ such that $5^n\equiv n^5\pmod 3$. | 4 | 0.875 | 5,031.375 | 4,579.857143 | 8,192 | |
Given that $a,b$ are constants, and $a \neq 0$, $f\left( x \right)=ax^{2}+bx$, $f\left( 2 \right)=0$.
(1) If the equation $f\left( x \right)-x=0$ has a unique real root, find the analytic expression of the function $f\left( x \right)$;
(2) When $a=1$, find the maximum and minimum values of the function $f\left( x \ri... | -1 | 0.75 | 3,551.8125 | 3,046.166667 | 5,068.75 | |
A rectangular piece of cardboard was cut along its diagonal. On one of the obtained pieces, two cuts were made parallel to the two shorter sides, at the midpoints of those sides. In the end, a rectangle with a perimeter of $129 \mathrm{~cm}$ remained. The given drawing indicates the sequence of cuts.
What was the peri... | 258 | 0.75 | 6,655.6875 | 6,222.166667 | 7,956.25 | |
The landlord of an apartment building needs to purchase enough digits to label all of the apartments from 100 through 125 on the first floor and 200 through 225 on the second floor. The digits can only be purchased in a package that contains one of each digit 0 through 9. How many packages must the landlord purchase? | 52 | 0 | 7,589.3125 | -1 | 7,589.3125 | |
There are 1000 rooms in a row along a long corridor. Initially, the first room contains 1000 people, and the remaining rooms are empty. Each minute, the following happens: for each room containing more than one person, someone in that room decides it is too crowded and moves to the next room. All these movements are si... | 61 | 0 | 8,192 | -1 | 8,192 | |
For how many integer values of $n$ between 1 and 1000 inclusive does the decimal representation of $\frac{n}{1400}$ terminate? | 142 | 0.75 | 5,054.3125 | 4,122.333333 | 7,850.25 | |
Given $60\%$ of students like dancing and the rest dislike it, $80\%$ of those who like dancing say they like it and the rest say they dislike it, also $90\%$ of those who dislike dancing say they dislike it and the rest say they like it. Calculate the fraction of students who say they dislike dancing but actually like... | 25\% | 0 | 4,785.875 | -1 | 4,785.875 | |
A rectangular piece of paper with dimensions 8 cm by 6 cm is folded in half horizontally. After folding, the paper is cut vertically at 3 cm and 5 cm from one edge, forming three distinct rectangles. Calculate the ratio of the perimeter of the smallest rectangle to the perimeter of the largest rectangle. | \frac{5}{6} | 0.0625 | 5,774.625 | 6,741 | 5,710.2 | |
We call a natural number \( b \) lucky if for any natural \( a \) such that \( a^{5} \) is divisible by \( b^{2} \), the number \( a^{2} \) is divisible by \( b \).
Find the number of lucky natural numbers less than 2010. | 1961 | 0 | 8,192 | -1 | 8,192 | |
In parallelogram $ABCD$, $BE$ is the height from vertex $B$ to side $AD$, and segment $ED$ is extended from $D$ such that $ED = 8$. The base $BC$ of the parallelogram is $14$. The entire parallelogram has an area of $126$. Determine the area of the shaded region $BEDC$. | 99 | 0.4375 | 6,934 | 6,022.857143 | 7,642.666667 | |
Nikola had one three-digit number and one two-digit number. Each of these numbers was positive and made up of different digits. The difference between Nikola's numbers was 976.
What was their sum? | 996 | 0.875 | 4,382.9375 | 3,838.785714 | 8,192 | |
There is a four-digit positive integer whose thousand's place is 2. If the digit 2 is moved to the unit's place, the new number formed is 66 greater than twice the original number. Let x be the original number in the units and tens places. Express the original number as 2000 + 100x + 10y + 2, and the new number formed ... | 2508 | 0 | 8,192 | -1 | 8,192 | |
Given that circle $C$ passes through the point $(0,2)$ with a radius of $2$, if there exist two points on circle $C$ that are symmetric with respect to the line $2x-ky-k=0$, find the maximum value of $k$. | \frac{4\sqrt{5}}{5} | 0 | 7,612.5 | -1 | 7,612.5 | |
A farmer claims that with four rods, he can fence a square plot of land sufficient for one sheep. If this is true, what is the minimum number of rods needed to fence an area for ten sheep? The answer depends on the shape of your fence.
How many rods are required for 10 sheep? | 12 | 0.1875 | 7,354.875 | 6,665 | 7,514.076923 | |
Let \( m \) be the largest positive integer such that for every positive integer \( n \leqslant m \), the following inequalities hold:
\[
\frac{2n + 1}{3n + 8} < \frac{\sqrt{5} - 1}{2} < \frac{n + 7}{2n + 1}
\]
What is the value of the positive integer \( m \)? | 27 | 0.3125 | 6,954.5625 | 5,609.4 | 7,566 | |
A regular hexagon $ABCDEF$ has sides of length three. Find the area of $\bigtriangleup ACE$. Express your answer in simplest radical form. | \frac{9\sqrt{3}}{4} | 0 | 4,770.5625 | -1 | 4,770.5625 | |
If $x$ and $y$ are non-zero real numbers such that
\[|x|+y=3 \qquad \text{and} \qquad |x|y+x^3=0,\]
then the integer nearest to $x-y$ is | -3 | We are given two equations involving $x$ and $y$:
1. $|x| + y = 3$
2. $|x|y + x^3 = 0$
We need to consider two cases based on the value of $x$ (positive or negative).
#### Case 1: $x$ is positive
If $x > 0$, then $|x| = x$. Substituting this into the equations, we get:
- $x + y = 3$
- $xy + x^3 = 0$
From the first e... | 0.9375 | 3,668.875 | 3,367.333333 | 8,192 |
Five packages are delivered to five houses, one to each house. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to the correct houses? Express your answer as a common fraction. | \frac{1}{12} | 0.9375 | 4,122.375 | 3,851.066667 | 8,192 | |
In trapezoid $ABCD$ with $\overline{BC}\parallel\overline{AD}$, let $BC = 700$ and $AD = 1400$. Let $\angle A = 45^\circ$, $\angle D = 45^\circ$, and $P$ and $Q$ be the midpoints of $\overline{BC}$ and $\overline{AD}$, respectively. Find the length $PQ$. | 350 | 0.875 | 5,577.5625 | 5,496.785714 | 6,143 | |
Sequence $A$ is a geometric sequence. Sequence $B$ is an arithmetic sequence. Each sequence stops as soon as one of its terms is greater than $300.$ What is the least positive difference between a number selected from sequence $A$ and a number selected from sequence $B?$
$\bullet$ Sequence $A:$ $2,$ $4,$ $8,$ $16,$ $3... | 4 | 0.375 | 7,653.5 | 6,756 | 8,192 | |
What is the largest positive integer that is not the sum of a positive integral multiple of $36$ and a positive composite integer that is not a multiple of $4$? | 147 | 0 | 8,078.625 | -1 | 8,078.625 | |
Find the smallest real number \( a \) such that for any non-negative real numbers \( x \), \( y \), and \( z \) that sum to 1, the following inequality holds:
$$
a(x^{2} + y^{2} + z^{2}) + xyz \geq \frac{a}{3} + \frac{1}{27}.
$$ | \frac{2}{9} | 0 | 8,192 | -1 | 8,192 | |
Given the function $f(x)=a\ln x-x^{2}+1$.
(I) If the tangent line of the curve $y=f(x)$ at $x=1$ is $4x-y+b=0$, find the values of the real numbers $a$ and $b$;
(II) Discuss the monotonicity of the function $f(x)$. | -4 | 1 | 3,505.6875 | 3,505.6875 | -1 | |
Determine the number of minutes before Jack arrives at the park that Jill arrives at the park, given that they are 2 miles apart, Jill cycles at a constant speed of 12 miles per hour, and Jack jogs at a constant speed of 5 miles per hour. | 14 | 0.8125 | 4,386 | 4,256.230769 | 4,948.333333 | |
Jacob uses the following procedure to write down a sequence of numbers. First he chooses the first term to be 6. To generate each succeeding term, he flips a fair coin. If it comes up heads, he doubles the previous term and subtracts 1. If it comes up tails, he takes half of the previous term and subtracts 1. What is t... | \frac{5}{8} | 1. **Initial Term**: Jacob starts with the first term $a_1 = 6$.
2. **Defining the Rules**:
- If the coin flip is heads (H), the next term is $2a - 1$.
- If the coin flip is tails (T), the next term is $\frac{a}{2} - 1$.
3. **Constructing the Sequence**:
- **Second Term** ($a_2$):
- H: $2 \times 6 - 1 =... | 0.3125 | 7,053.375 | 5,037.4 | 7,969.727273 |
Consider a central regular hexagon surrounded by six regular hexagons, each of side length $\sqrt{2}$. Three of these surrounding hexagons are selected at random, and their centers are connected to form a triangle. Calculate the area of this triangle. | 2\sqrt{3} | 0 | 7,787.4375 | -1 | 7,787.4375 | |
For some constants $a$ and $b,$ let \[f(x) = \left\{
\begin{array}{cl}
ax + b & \text{if } x < 2, \\
8 - 3x & \text{if } x \ge 2.
\end{array}
\right.\]The function $f$ has the property that $f(f(x)) = x$ for all $x.$ What is $a + b?$ | \frac{7}{3} | 0.625 | 5,826.0625 | 4,406.5 | 8,192 | |
Cirlce $\Omega$ is inscribed in triangle $ABC$ with $\angle BAC=40$ . Point $D$ is inside the angle $BAC$ and is the intersection of exterior bisectors of angles $B$ and $C$ with the common side $BC$ . Tangent form $D$ touches $\Omega$ in $E$ . FInd $\angle BEC$ . | 110 | 0 | 8,153.5625 | -1 | 8,153.5625 | |
How many terms are in the arithmetic sequence $13$, $16$, $19$, $\dotsc$, $70$, $73$? | 21 | 1. **Identify the first term and common difference**: The given sequence is $13, 16, 19, \dots, 70, 73$. The first term ($a$) is $13$ and the common difference ($d$) can be calculated as $16 - 13 = 3$.
2. **Formulate the general term**: The general term of an arithmetic sequence can be expressed as $a_n = a + (n-1)d$.... | 1 | 1,626.5 | 1,626.5 | -1 |
In the polar coordinate system, given the curve $C: \rho = 2\cos \theta$, the line $l: \left\{ \begin{array}{l} x = \sqrt{3}t \\ y = -1 + t \end{array} \right.$ (where $t$ is a parameter), and the line $l$ intersects the curve $C$ at points $A$ and $B$.
$(1)$ Find the rectangular coordinate equation of curve $C$ and ... | 3 + \sqrt{3} | 0.3125 | 8,006.6875 | 7,599 | 8,192 | |
A large rectangular garden contains two flower beds in the shape of congruent isosceles right triangles and a trapezoidal playground. The parallel sides of the trapezoid measure $30$ meters and $46$ meters. Determine the fraction of the garden occupied by the flower beds.
A) $\frac{1}{10}$
B) $\frac{1}{11}$
C) $\frac{4... | \frac{4}{23} | 0 | 5,586.3125 | -1 | 5,586.3125 | |
Triangle $ABC$ has $\angle BAC=90^\circ$ . A semicircle with diameter $XY$ is inscribed inside $\triangle ABC$ such that it is tangent to a point $D$ on side $BC$ , with $X$ on $AB$ and $Y$ on $AC$ . Let $O$ be the midpoint of $XY$ . Given that $AB=3$ , $AC=4$ , and $AX=\tfrac{9}{4}$ , compute th... | 39/32 | 0.5625 | 6,121.625 | 4,511.333333 | 8,192 | |
On a rectangular sheet of paper, a picture is drawn in the shape of a "cross" formed by two rectangles $ABCD$ and $EFGH$, where the sides are parallel to the edges of the sheet. It is known that $AB=9$, $BC=5$, $EF=3$, and $FG=10$. Find the area of the quadrilateral $AFCH$. | 52.5 | 0 | 7,483.875 | -1 | 7,483.875 | |
A rational number written in base eight is $\underline{ab} . \underline{cd}$, where all digits are nonzero. The same number in base twelve is $\underline{bb} . \underline{ba}$. Find the base-ten number $\underline{abc}$. | 321 | The parts before and after the decimal points must be equal. Therefore $8a + b = 12b + b$ and $c/8 + d/64 = b/12 + a/144$. Simplifying the first equation gives $a = (3/2)b$. Plugging this into the second equation gives $3b/32 = c/8 + d/64$. Multiplying both sides by 64 gives $6b = 8c + d$. $a$ and $b$ are both digits b... | 0.0625 | 7,990.1875 | 4,963 | 8,192 |
A rectangle is called cool if the number of square units in its area is equal to twice the number of units in its perimeter. A cool rectangle also must have integer side lengths. What is the sum of all the different possible areas of cool rectangles? | 236 | 1 | 2,748.8125 | 2,748.8125 | -1 | |
The function $f(n)$ is defined on the positive integers and takes non-negative integer values. $f(2)=0,f(3)>0,f(9999)=3333$ and for all $m,n:$ \[ f(m+n)-f(m)-f(n)=0 \text{ or } 1. \] Determine $f(1982)$. | 660 |
We are given that the function \( f(n) \) is defined on positive integers and it takes non-negative integer values. It satisfies:
\[ f(2) = 0, \]
\[ f(3) > 0, \]
\[ f(9999) = 3333, \]
and for all \( m, n \):
\[ f(m+n) - f(m) - f(n) = 0 \text{ or } 1. \]
We need to determine \( f(1982) \).
### Analysis of the Func... | 0.6875 | 6,367.3125 | 5,537.909091 | 8,192 |
In triangle $\triangle ABC$, $a$, $b$, $c$ are the opposite sides of the internal angles $A$, $B$, $C$, respectively, and $\sin ^{2}A+\sin A\sin C+\sin ^{2}C+\cos ^{2}B=1$.
$(1)$ Find the measure of angle $B$;
$(2)$ If $a=5$, $b=7$, find $\sin C$. | \frac{3\sqrt{3}}{14} | 0 | 4,352.375 | -1 | 4,352.375 | |
Simplify first, then evaluate: $(1+\frac{4}{a-1})÷\frac{a^2+6a+9}{a^2-a}$, where $a=2$. | \frac{2}{5} | 1 | 2,082.875 | 2,082.875 | -1 | |
Solve for $x$: $x = \dfrac{35}{6-\frac{2}{5}}$. | \frac{25}{4} | 1 | 2,084.75 | 2,084.75 | -1 | |
A collection of circles in the upper half-plane, all tangent to the $x$-axis, is constructed in layers as follows. Layer $L_0$ consists of two circles of radii $70^2$ and $73^2$ that are externally tangent. For $k \ge 1$, the circles in $\bigcup_{j=0}^{k-1}L_j$ are ordered according to their points of tangency with the... | \frac{143}{14} | 1. **Identify the radii of initial circles**: Let the two circles from $L_0$ be of radius $r_1 = 70^2$ and $r_2 = 73^2$, with $r_1 < r_2$. Let the circle of radius $r_1$ be circle $A$ and the circle of radius $r_2$ be circle $B$.
2. **Constructing the circle in $L_1$**: A new circle $C$ in $L_1$ is constructed externa... | 0 | 8,192 | -1 | 8,192 |
A cryptographer devises the following method for encoding positive integers. First, the integer is expressed in base $5$.
Second, a 1-to-1 correspondence is established between the digits that appear in the expressions in base $5$ and the elements of the set
$\{V, W, X, Y, Z\}$. Using this correspondence, the cryptog... | 108 | 1. **Identify the pattern and the base-5 system**: The problem states that three consecutive integers are coded as $VYZ, VYX, VVW$. We know that in base-5, each digit represents powers of 5, starting from the rightmost digit (units place) to the leftmost digit.
2. **Analyze the change from $VYX$ to $VVW$**: Since $VYX... | 0.1875 | 6,792.375 | 5,615.333333 | 7,064 |
In triangle $\triangle ABC$, $AC=2AB=4$ and $\cos A=\frac{1}{8}$. Calculate the length of side $BC$. | 3\sqrt{2} | 0.75 | 3,076.5625 | 3,013.333333 | 3,266.25 | |
Given that $x$ is a multiple of $15336$, what is the greatest common divisor of $f(x)=(3x+4)(7x+1)(13x+6)(2x+9)$ and $x$? | 216 | 0.5 | 6,714.625 | 5,237.25 | 8,192 | |
A perfect power is an integer $n$ that can be represented as $a^{k}$ for some positive integers $a \geq 1$ and $k \geq 2$. Find the sum of all prime numbers $0<p<50$ such that $p$ is 1 less than a perfect power. | 41 | First, it is known that $a^{k}-1=(a-1)\left(a^{k-1}+a^{k-2}+\ldots\right)$. This means either $a-1$ or $a^{k-1}+a^{k-2}+\ldots+1$ must be 1 in order for $a^{k}-1$ to be prime. But this only occurs when $a$ is 2 . Thus, the only possible primes are of the form $2^{k}-1$ for some integer $k>1$. One can check that the pri... | 0.9375 | 5,464.25 | 5,282.4 | 8,192 |
What fraction of the pizza is left for Wally if Jovin takes $\frac{1}{3}$ of the pizza, Anna takes $\frac{1}{6}$ of the pizza, and Olivia takes $\frac{1}{4}$ of the pizza? | \frac{1}{4} | Since Jovin, Anna and Olivia take $\frac{1}{3}, \frac{1}{6}$ and $\frac{1}{4}$ of the pizza, respectively, then the fraction of the pizza with which Wally is left is
$$ 1-\frac{1}{3}-\frac{1}{6}-\frac{1}{4}=\frac{12}{12}-\frac{4}{12}-\frac{2}{12}-\frac{3}{12}=\frac{3}{12}=\frac{1}{4} $$ | 1 | 1,862.9375 | 1,862.9375 | -1 |
Using the digits 0, 1, 2, 3, 4, and 5, form six-digit numbers without repeating any digit.
(1) How many such six-digit odd numbers are there?
(2) How many such six-digit numbers are there where the digit 5 is not in the unit place?
(3) How many such six-digit numbers are there where the digits 1 and 2 are not adj... | 408 | 0 | 7,850.1875 | -1 | 7,850.1875 | |
A sphere is inscribed in a cone, and the surface area of the sphere is equal to the area of the base of the cone. Find the cosine of the angle at the vertex in the axial section of the cone. | \frac{7}{25} | 0.75 | 5,338.5 | 4,821.083333 | 6,890.75 | |
Let $a,$ $b,$ and $c$ be the roots of $x^3 - 7x^2 + 5x + 2 = 0.$ Find
\[\frac{a}{bc + 1} + \frac{b}{ac + 1} + \frac{c}{ab + 1}.\] | \frac{15}{2} | 0.4375 | 6,369.125 | 5,929.142857 | 6,711.333333 | |
From the set \( M = \{1, 2, \cdots, 2008\} \) of the first 2008 positive integers, a \( k \)-element subset \( A \) is chosen such that the sum of any two numbers in \( A \) cannot be divisible by the difference of those two numbers. What is the maximum value of \( k \)? | 670 | 0.0625 | 8,167.9375 | 7,807 | 8,192 | |
The smaller square has an area of 16 and the grey triangle has an area of 1. What is the area of the larger square?
A) 17
B) 18
C) 19
D) 20
E) 21 | 18 | 0 | 5,521.1875 | -1 | 5,521.1875 | |
Calculate the limit of the function:
$$
\lim _{x \rightarrow 1}\left(\frac{e^{\sin \pi x}-1}{x-1}\right)^{x^{2}+1}
$$ | \pi^2 | 0.5 | 6,315.6875 | 5,199.75 | 7,431.625 | |
Given that Steve's empty swimming pool holds 30,000 gallons of water when full and will be filled by 5 hoses, each supplying 2.5 gallons of water per minute, calculate the time required to fill the pool. | 40 | 0.375 | 442.875 | 481.333333 | 419.8 | |
A bag contains three balls labeled 1, 2, and 3. A ball is drawn from the bag, its number is recorded, and then it is returned to the bag. This process is repeated three times. If each ball has an equal chance of being drawn, calculate the probability of the number 2 being drawn three times given that the sum of the num... | \frac{1}{7} | 1 | 3,371.5 | 3,371.5 | -1 | |
Consider all ordered pairs $(m, n)$ of positive integers satisfying $59 m - 68 n = mn$ . Find the sum of all the possible values of $n$ in these ordered pairs. | 237 | 0.75 | 4,371.875 | 4,218.333333 | 4,832.5 | |
Some bugs are sitting on squares of $10\times 10$ board. Each bug has a direction associated with it **(up, down, left, right)**. After 1 second, the bugs jump one square in **their associated**direction. When the bug reaches the edge of the board, the associated direction reverses (up becomes down, left becomes righ... | 40 | 0 | 7,982.6875 | -1 | 7,982.6875 | |
Fisica and Ritmo discovered a piece of Notalium shaped like a rectangular box, and wanted to find its volume. To do so, Fisica measured its three dimensions using a ruler with infinite precision, multiplied the results and rounded the product to the nearest cubic centimeter, getting a result of $V$ cubic centimeters. R... | 4035 | The only possible way for Ritmo to get 2017 cubic centimeters is to have his measurements rounded to $1,1,2017$ centimeters respectively. Therefore the largest value of $V$ is achieved when the dimensions are $(1.5-\epsilon)(1.5-\epsilon)(2017.5-\epsilon)=4539.375-\epsilon^{\prime}$ for some very small positive real $\... | 0 | 7,374.1875 | -1 | 7,374.1875 |
In the diagram, there are several triangles formed by connecting points in a shape. If each triangle has the same probability of being selected, what is the probability that a selected triangle includes a vertex marked with a dot? Express your answer as a common fraction.
[asy]
draw((0,0)--(2,0)--(1,2)--(0,0)--cycle,l... | \frac{1}{2} | 0.1875 | 7,403.5 | 6,328.333333 | 7,651.615385 | |
Let $P(x)=x^3+ax^2+bx+c$ be a polynomial where $a,b,c$ are integers and $c$ is odd. Let $p_{i}$ be the value of $P(x)$ at $x=i$ . Given that $p_{1}^3+p_{2}^{3}+p_{3}^{3}=3p_{1}p_{2}p_{3}$ , find the value of $p_{2}+2p_{1}-3p_{0}.$ | 18 | 0.75 | 5,475.4375 | 4,569.916667 | 8,192 | |
Determine $\sqrt[7]{218618940381251}$ without a calculator. | 102 | 0 | 8,192 | -1 | 8,192 | |
Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages? | 18 | 1. **Identify the factors of 128**: We know that the product of the ages of Kiana and her twin brothers is 128. Since 128 can be expressed as $128 = 2^7$, the ages must be factors of 128.
2. **List possible ages for the twins**: The twins are older than Kiana and have the same age. Possible ages for the twins (both be... | 0.9375 | 2,132.1875 | 2,232.266667 | 631 |
In $\triangle ABC$, the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$ respectively, and
$$
a=5, \quad b=4, \quad \cos(A-B)=\frac{31}{32}.
$$
Find the area of $\triangle ABC$. | \frac{15 \sqrt{7}}{4} | 0 | 7,305.75 | -1 | 7,305.75 | |
Given that point $A(-2,3)$ lies on the axis of parabola $C$: $y^{2}=2px$, and the line passing through point $A$ is tangent to $C$ at point $B$ in the first quadrant. Let $F$ be the focus of $C$. Then, $|BF|=$ _____ . | 10 | 0.1875 | 7,734.5 | 5,752 | 8,192 | |
The graph of $y = \frac{p(x)}{q(x)}$ where $p(x)$ is quadratic and $q(x)$ is quadratic is given conceptually (imagine a graph with necessary features). The function has vertical asymptotes at $x = -4$ and $x = 1$. The graph passes through the point $(0,0)$ and $(2,-1)$. Determine $\frac{p(-1)}{q(-1)}$ if $q(x) = (x+4)(... | -\frac{1}{2} | 0.5625 | 4,771.1875 | 2,434.222222 | 7,775.857143 | |
In parallelogram ABCD, AB=2AD=4, ∠BAD=60°, E is the midpoint of BC, calculate the dot product of vectors BD and AE. | -12 | 0.625 | 5,021.0625 | 3,842.7 | 6,985 | |
A checkerboard of $13$ rows and $17$ columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered $1,2,\ldots,17$, the second row $18,19,\ldots,34$, and so on down the board. If the board is renumbered so that the left column, top to bottom, is $1,2,\ldots,13,$, the... | 555 | 1. **Indexing and Numbering the Board:**
- Let $i$ be the index for rows, where $i = 1, 2, 3, \ldots, 13$.
- Let $j$ be the index for columns, where $j = 1, 2, 3, \ldots, 17$.
- The numbering of the board in the first system (row-wise) for a cell in row $i$ and column $j$ is given by:
\[
f(i, j) = 17... | 0.9375 | 2,989.5625 | 2,970.533333 | 3,275 |
Points $K$, $L$, $M$, and $N$ lie in the plane of the square $ABCD$ such that $AKB$, $BLC$, $CMD$, and $DNA$ are isosceles right triangles. If the area of square $ABCD$ is 25, find the area of $KLMN$. | 25 | 0 | 8,147.75 | -1 | 8,147.75 | |
For any natural number $n$ , expressed in base $10$ , let $S(n)$ denote the sum of all digits of $n$ . Find all positive integers $n$ such that $n^3 = 8S(n)^3+6S(n)n+1$ . | 17 | 0.0625 | 8,002.5 | 5,160 | 8,192 | |
Josanna's test scores to date are $90, 80, 70, 60,$ and $85$. Her goal is to raise her test average at least $3$ points with her next test. What is the minimum test score she would need to accomplish this goal? | 95 | 1. **Calculate the current average score**:
Josanna's current test scores are $90, 80, 70, 60,$ and $85$. The average of these scores is calculated as follows:
\[
\text{Average} = \frac{90 + 80 + 70 + 60 + 85}{5} = \frac{385}{5} = 77
\]
2. **Determine the desired average score**:
Josanna wants to raise ... | 1 | 2,892.25 | 2,892.25 | -1 |
Equilateral triangles $ABC$ and $A_{1}B_{1}C_{1}$ with a side length of 12 are inscribed in a circle $S$ such that point $A$ lies on the arc $B_{1}C_{1}$, and point $B$ lies on the arc $A_{1}B_{1}$. Find $AA_{1}^{2} + BB_{1}^{2} + CC_{1}^{2}$. | 288 | 0 | 8,070.375 | -1 | 8,070.375 | |
There exist $s$ unique nonnegative integers $m_1 > m_2 > \cdots > m_s$ and $s$ integers $b_k$ ($1\le k\le s$), with each $b_k$ either $1$ or $-1$, such that \[b_13^{m_1} + b_23^{m_2} + \cdots + b_s3^{m_s} = 1007.\] Find $m_1 + m_2 + \cdots + m_s$. | 15 | 0 | 6,780.75 | -1 | 6,780.75 | |
A line contains the points $(6,8)$, $(-2, k)$ and $(-10, 4)$. What is the value of $k$? | 6 | 1 | 2,008.5625 | 2,008.5625 | -1 | |
There are 100 points on a coordinate plane. Let \( N \) be the number of triplets \((A, B, C)\) that satisfy the following conditions: the vertices are chosen from these 100 points, \( A \) and \( B \) have the same y-coordinate, and \( B \) and \( C \) have the same x-coordinate. Find the maximum value of \( N \). | 8100 | 0 | 8,192 | -1 | 8,192 | |
The cafeteria in a certain laboratory is open from noon until 2 in the afternoon every Monday for lunch. Two professors eat 15 minute lunches sometime between noon and 2. What is the probability that they are in the cafeteria simultaneously on any given Monday? | 15 | 15. | 0 | 4,974.4375 | -1 | 4,974.4375 |
Let $a,$ $b,$ $c$ be nonzero real numbers such that $a + b + c = 0,$ and $ab + ac + bc \neq 0.$ Find all possible values of
\[\frac{a^7 + b^7 + c^7}{abc (ab + ac + bc)}.\] | -7 | 0.0625 | 8,028.8125 | 8,192 | 8,017.933333 | |
Xiao Li and Xiao Hua are racing up the stairs. When Xiao Li reaches the 5th floor, Xiao Hua has reached the 3rd floor. At this rate, how many floors will Xiao Hua have reached when Xiao Li reaches the 25th floor? | 13 | 0.25 | 573.5 | 587 | 569 | |
Find the number of 10-tuples $(x_1, x_2, \dots, x_{10})$ of real numbers such that
\[(1 - x_1)^2 + (x_1 - x_2)^2 + (x_2 - x_3)^2 + \dots + (x_9 - x_{10})^2 + x_{10}^2 = \frac{1}{11}.\] | 1 | 0.0625 | 7,910.5 | 7,354 | 7,947.6 | |
In preparation for the family's upcoming vacation, Tony puts together five bags of jelly beans, one bag for each day of the trip, with an equal number of jelly beans in each bag. Tony then pours all the jelly beans out of the five bags and begins making patterns with them. One of the patterns that he makes has one je... | 45 | 0.3125 | 6,958.4375 | 5,304.8 | 7,710.090909 |
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