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 |
|---|---|---|---|---|---|---|
Among the following functions, identify which pairs represent the same function.
1. $f(x) = |x|, g(x) = \sqrt{x^2}$;
2. $f(x) = \sqrt{x^2}, g(x) = (\sqrt{x})^2$;
3. $f(x) = \frac{x^2 - 1}{x - 1}, g(x) = x + 1$;
4. $f(x) = \sqrt{x + 1} \cdot \sqrt{x - 1}, g(x) = \sqrt{x^2 - 1}$. | (1) | 0 | 2,522.875 | -1 | 2,522.875 | |
\[\log_{10} x + \log_{\sqrt{10}} x + \log_{\sqrt[3]{10}} x + \ldots + \log_{\sqrt[1]{10}} x = 5.5\] | \sqrt[10]{10} | 0 | 6,350.8125 | -1 | 6,350.8125 | |
[asy]
draw((0,1)--(4,1)--(4,2)--(0,2)--cycle);
draw((2,0)--(3,0)--(3,3)--(2,3)--cycle);
draw((1,1)--(1,2));
label("1",(0.5,1.5));
label("2",(1.5,1.5));
label("32",(2.5,1.5));
label("16",(3.5,1.5));
label("8",(2.5,0.5));
label("6",(2.5,2.5));
[/asy]
The image above is a net of a unit cube. Let $n$ be a positive intege... | 16 | 0 | 8,192 | -1 | 8,192 | |
Given that $x - \frac{1}{x} = 4$, what is $x^3 - \frac{1}{x^3}$? | 76 | 0.875 | 4,822.5 | 4,341.142857 | 8,192 | |
Calculate the infinite sum:
\[
\sum_{n=1}^\infty \frac{n^3 - n}{(n+3)!}
\] | \frac{1}{6} | 0 | 8,155.75 | -1 | 8,155.75 | |
Let $A B C D E$ be a convex pentagon such that $\angle A B C=\angle A C D=\angle A D E=90^{\circ}$ and $A B=B C=C D=D E=1$. Compute $A E$. | 2 | By Pythagoras, $A E^{2}=A D^{2}+1=A C^{2}+2=A B^{2}+3=4$ so $A E=2$. | 0.0625 | 7,068.3125 | 4,713 | 7,225.333333 |
If $\left(a + \frac{1}{a}\right)^2 = 3$, then $a^3 + \frac{1}{a^3}$ equals: | 0 | 1. **Starting from the given equation:**
Given that $\left(a + \frac{1}{a}\right)^2 = 3$.
2. **Simplifying the square root:**
Taking the square root on both sides, we have two possible values:
\[
a + \frac{1}{a} = \sqrt{3} \quad \text{or} \quad a + \frac{1}{a} = -\sqrt{3}
\]
However, we need to c... | 0.875 | 4,596.375 | 4,082.714286 | 8,192 |
There exist constants $a_1, a_2, a_3, a_4, a_5, a_6, a_7$ such that
\[
\cos^7 \theta = a_1 \cos \theta + a_2 \cos 2 \theta + a_3 \cos 3 \theta + a_4 \cos 4 \theta + a_5 \cos 5 \theta + a_6 \cos 6 \theta + a_7 \cos 7 \theta
\]
for all angles $\theta.$ Find $a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2 + a_6^2 + a_7^2.$ | \frac{1716}{4096} | 0 | 6,451.0625 | -1 | 6,451.0625 | |
Given that the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n=\ln (1+ \frac {1}{n})$, find the value of $e^{a_7+a_8+a_9}$. | \frac {20}{21} | 0.875 | 3,984.125 | 3,536.642857 | 7,116.5 | |
In the quadratic equation $3x^{2}-6x-7=0$, the coefficient of the quadratic term is ____ and the constant term is ____. | -7 | 0.6875 | 413.75 | 418.636364 | 403 | |
Find $n$ such that $2^6 \cdot 3^3 \cdot n = 10!$. | 350 | 0 | 3,375.6875 | -1 | 3,375.6875 | |
Calculate the limit of the function:
\[ \lim _{x \rightarrow \frac{1}{2}} \frac{\sqrt[3]{\frac{x}{4}}-\frac{1}{2}}{\sqrt{\frac{1}{2}+x}-\sqrt{2x}} \] | -\frac{2}{3} | 0.3125 | 7,715.6875 | 6,667.8 | 8,192 | |
A person has a probability of $\frac{1}{2}$ to hit the target in each shot. What is the probability of hitting the target 3 times out of 6 shots, with exactly 2 consecutive hits? (Answer with a numerical value) | \frac{3}{16} | 0.1875 | 7,711.9375 | 6,035.666667 | 8,098.769231 | |
In $\triangle PQR$, we have $PQ = QR = 34$ and $PR = 32$. Point $M$ is the midpoint of $\overline{QR}$. Find $PM$. | 3\sqrt{89} | 0.75 | 5,733.6875 | 5,156.166667 | 7,466.25 | |
Let $a$ and $b$ be angles such that $\cos a + \cos b = \frac{1}{2}$ and $\sin a + \sin b = \frac{3}{11}.$ Find
\[\tan \left( \frac{a + b}{2} \right).\] | \frac{6}{11} | 1 | 2,872.375 | 2,872.375 | -1 | |
The left and right foci of a hyperbola are $F_{1}$ and $F_{2}$, respectively. A line passing through $F_{2}$ intersects the right branch of the hyperbola at points $A$ and $B$. If $\triangle F_{1} A B$ is an equilateral triangle, what is the eccentricity of the hyperbola? | \sqrt{3} | 0.0625 | 7,900.625 | 3,530 | 8,192 | |
How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? | 100 | 0.125 | 7,692.9375 | 4,199.5 | 8,192 | |
Let $a,$ $b,$ and $t$ be real numbers such that $a + b = t.$ Find, in terms of $t,$ the minimum value of $a^2 + b^2.$ | \frac{t^2}{2} | 1 | 2,588.8125 | 2,588.8125 | -1 | |
What is the largest base-4 number that has four digits? Express your answer in base 10. | 255 | 0.9375 | 2,050.3125 | 1,954.2 | 3,492 | |
Given $m=(\sqrt{3}\sin \omega x,\cos \omega x)$, $n=(\cos \omega x,-\cos \omega x)$ ($\omega > 0$, $x\in\mathbb{R}$), $f(x)=m\cdot n-\frac{1}{2}$ and the distance between two adjacent axes of symmetry on the graph of $f(x)$ is $\frac{\pi}{2}$.
$(1)$ Find the intervals of monotonic increase for the function $f(x)$;
$(... | \frac{3\sqrt{3}}{4} | 0 | 7,369.25 | -1 | 7,369.25 | |
Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$. | 42 | Notice that the condition is equivalent to saying
\[v_2(a^6) \geq v_2(6^a) \implies 6n \geq a\] \[v_3(a^6) \geq v_3(6^a) \implies 6m \geq a.\]
Notice that we cannot have both expressions to be equality state, as that would result in $a^6 = 6^a.$ Testing, we see the possible pairs $(n, m)$ are $(1, 0), (2, 0), (3, 0), ... | 0 | 8,011.3125 | -1 | 8,011.3125 |
Given that $F$ is the right focus of the ellipse $C:\frac{x^2}{4}+\frac{y^2}{3}=1$, $P$ is a point on the ellipse $C$, and $A(1,2\sqrt{2})$, find the maximum value of $|PA|+|PF|$. | 4 + 2\sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
Given vectors $\overrightarrow{m}=( \sqrt {3}\sin x-\cos x,1)$ and $\overrightarrow{n}=(\cos x, \frac {1}{2})$, and the function $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$,
(1) Find the interval(s) where the function $f(x)$ is monotonically increasing;
(2) If $a$, $b$, $c$ are the sides opposite to angles $A$, ... | 2 \sqrt {3} | 0 | 7,946.375 | -1 | 7,946.375 | |
Square $ABCD$ has center $O,\ AB=900,\ E$ and $F$ are on $AB$ with $AE<BF$ and $E$ between $A$ and $F, m\angle EOF =45^\circ,$ and $EF=400.$ Given that $BF=p+q\sqrt{r},$ where $p,q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r.$
| 307 | 0.0625 | 8,165.125 | 7,762 | 8,192 | |
Consider a 4-by-4 grid where each of the unit squares can be colored either purple or green. Each color choice is equally likely independent of the others. Compute the probability that the grid does not contain a 3-by-3 grid of squares all colored purple. Express your result in the form $\frac{m}{n}$, where $m$ and $n$... | 255 | 0 | 7,714.1875 | -1 | 7,714.1875 | |
$a,b,c$ - are sides of triangle $T$ . It is known, that if we increase any one side by $1$ , we get new
a) triangle
b)acute triangle
Find minimal possible area of triangle $T$ in case of a) and in case b) | \frac{\sqrt{3}}{4} | 0 | 8,134.3125 | -1 | 8,134.3125 | |
A circle centered at $O$ is circumscribed about $\triangle ABC$ as follows: [asy]
pair pA, pB, pC, pO;
pO = (0, 0);
pA = pO + dir(-20);
pB = pO + dir(90);
pC = pO + dir(190);
draw(pA--pB--pC--pA);
draw(pO--pA);
draw(pO--pB);
draw(pO--pC);
label("$O$", pO, S);
label("$110^\circ$", pO, NE);
label("$100^\circ$", pO, NW);
... | 50^\circ | 0.3125 | 2,657.625 | 2,598.4 | 2,684.545455 | |
A regular hexagon `LMNOPQ` has sides of length 4. Find the area of triangle `LNP`. Express your answer in simplest radical form. | 8\sqrt{3} | 0.1875 | 3,981.375 | 3,435.333333 | 4,107.384615 | |
Find $t$ such that $(t,5)$ lies on the line through $(0,3)$ and $(-8,0)$. | \frac{16}{3} | 1 | 2,992.5625 | 2,992.5625 | -1 | |
To calculate $41^2$, David mentally figures the value $40^2$ and adds 81. David subtracts a number from $40^2$ to calculate $39^2$. What number does he subtract? | 79 | 1 | 1,881.3125 | 1,881.3125 | -1 | |
Given a function $f(x)$ $(x \in \mathbb{R})$ that satisfies the equation $f(-x) = 8 - f(4 + x)$, and another function $g(x) = \frac{4x + 3}{x - 2}$. If the graph of $f(x)$ has 168 intersection points with the graph of $g(x)$, denoted as $P_i(x_i, y_i)$ $(i = 1,2, \dots, 168)$, calculate the value of $(x_{1} + y_{1}) + ... | 1008 | 0.5 | 5,578.9375 | 3,841.625 | 7,316.25 | |
What is the greatest integer less than 150 for which the greatest common divisor of that integer and 18 is 6? | 144 | 0 | 5,059.5 | -1 | 5,059.5 | |
The largest divisor of a natural number \( N \), smaller than \( N \), was added to \( N \), producing a power of ten. Find all such \( N \). | 75 | 0 | 8,192 | -1 | 8,192 | |
For each positive integer $n$ let $a_n$ be the least positive integer multiple of $23$ such that $a_n \equiv 1 \pmod{2^n}.$ Find the number of positive integers $n$ less than or equal to $1000$ that satisfy $a_n = a_{n+1}.$ | 363 | Observe that if $a_{n-1} - 1$ is divisible by $2^n$, $a_n = a_{n-1}$. If not, $a_n = a_{n-1} + 23 \cdot 2^{n-1}$.
This encourages us to let $b_n = \frac{a_n - 1}{2^n}$. Rewriting the above equations, we have \[b_n = \begin{cases} \frac{b_{n-1}}{2} & \text{if } 2 \text{ } \vert \text{ } b_{n-1} \\ \frac{b_{n-1}+23}{2} ... | 0 | 8,192 | -1 | 8,192 |
Determine all such pairs pf positive integers $(a, b)$ such that $a + b + (gcd (a, b))^ 2 = lcm (a, b) = 2 \cdot lcm(a -1, b)$, where $lcm (a, b)$ denotes the smallest common multiple, and $gcd (a, b)$ denotes the greatest common divisor of numbers $a, b$. | (2, 3) \text{ and } (6, 15) |
Let us determine all pairs of positive integers \( (a, b) \) such that:
\[
a + b + (\gcd(a, b))^2 = \mathrm{lcm}(a, b) = 2 \cdot \mathrm{lcm}(a-1, b)
\]
where \(\mathrm{lcm}(a, b)\) is the least common multiple and \(\gcd(a, b)\) is the greatest common divisor of \(a\) and \(b\).
### Step 1: Understanding the Equat... | 0.1875 | 7,789 | 6,774 | 8,023.230769 |
A five-digit number is called a "hill" if its first three digits are in ascending order and its last three digits are in descending order. For example, 13760 and 28932 are hills, whereas 78821 and 86521 are not hills. How many hills exist that are greater than the number 77777? | 36 | 0.1875 | 8,009.5 | 7,218.666667 | 8,192 | |
Six people are arranged in a row. In how many ways can the three people A, B, and C be arranged such that they are not adjacent to each other? | 144 | 0.25 | 7,982.8125 | 7,355.25 | 8,192 | |
In $\triangle ABC$, $\overrightarrow {AD}=3 \overrightarrow {DC}$, $\overrightarrow {BP}=2 \overrightarrow {PD}$, if $\overrightarrow {AP}=λ \overrightarrow {BA}+μ \overrightarrow {BC}$, then $λ+μ=\_\_\_\_\_\_$. | - \frac {1}{3} | 0.5 | 6,691.5625 | 6,047.375 | 7,335.75 | |
The first term of a sequence is 934. Each subsequent term is equal to the sum of the digits of the previous term multiplied by 13. Find the 2019th term of the sequence. | 130 | 0.8125 | 5,194.3125 | 4,502.538462 | 8,192 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $c = a \cos B + 2b \sin^2 \frac{A}{2}$.
(1) Find angle $A$.
(2) If $b=4$ and the length of median drawn to side $AC$ is $\sqrt{7}$, find $a$. | \sqrt{13} | 0.9375 | 4,430.3125 | 4,179.533333 | 8,192 | |
Let $M = 36 \cdot 36 \cdot 77 \cdot 330$. Find the ratio of the sum of the odd divisors of $M$ to the sum of the even divisors of $M$. | 1 : 62 | 0 | 6,867.5625 | -1 | 6,867.5625 | |
Find the maximum value of the function
$$
f(x)=\sin (x+\sin x)+\sin (x-\sin x)+\left(\frac{\pi}{2}-2\right) \sin (\sin x)
$$ | \frac{\pi - 2}{\sqrt{2}} | 0 | 7,977.875 | -1 | 7,977.875 | |
The expression $\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right)\left(1+\frac{1}{4}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{6}\right)\left(1+\frac{1}{7}\right)\left(1+\frac{1}{8}\right)\left(1+\frac{1}{9}\right)$ is equal to what? | 5 | The expression is equal to $\left(\frac{3}{2}\right)\left(\frac{4}{3}\right)\left(\frac{5}{4}\right)\left(\frac{6}{5}\right)\left(\frac{7}{6}\right)\left(\frac{8}{7}\right)\left(\frac{9}{8}\right)\left(\frac{10}{9}\right)$ which equals $\frac{3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10}{2 \cdot 3 \cdot 4... | 0.9375 | 2,371 | 1,982.933333 | 8,192 |
Let $m$ and $n$ be positive integers with $m\le 2000$ and $k=3-\frac{m}{n}$. Find the smallest positive value of $k$. | $\boxed{ \frac{1}{667}} .$ |
Given the problem with positive integers \( m \) and \( n \) such that \( m \leq 2000 \), and \( k = 3 - \frac{m}{n} \). We are tasked to find the smallest positive value of \( k \).
Firstly, to ensure \( k \) is positive, we need:
\[
k = 3 - \frac{m}{n} > 0,
\]
which implies:
\[
3 > \frac{m}{n}.
\]
Rearranging give... | 0 | 7,736.6875 | -1 | 7,736.6875 |
In $\triangle ABC$, $D$ is a point on side $\overline{AC}$ such that $BD=DC$ and $\angle BCD$ measures $70^\circ$. What is the degree measure of $\angle ADB$? | 140 | 1. **Identify Equal Angles**: Given that $BD = DC$, triangle $BDC$ is isosceles. Therefore, the base angles are equal, i.e., $\angle DBC = \angle DCB = 70^\circ$.
2. **Calculate $\angle BDC$**: In triangle $BDC$, the sum of the angles in any triangle is $180^\circ$. Thus,
\[
\angle BDC = 180^\circ - (\angle DBC ... | 0.25 | 6,981.5625 | 4,830.75 | 7,698.5 |
What is the second number in the row of Pascal's triangle that has 43 numbers? | 42 | 1 | 2,291.5 | 2,291.5 | -1 | |
Given positive numbers $m$ and $n$ that satisfy $m^2 + n^2 = 100$, find the maximum or minimum value of $m + n$. | 10\sqrt{2} | 0.8125 | 6,582.8125 | 6,460.076923 | 7,114.666667 | |
The measures of the interior angles of a convex polygon of $n$ sides are in arithmetic progression. If the common difference is $5^{\circ}$ and the largest angle is $160^{\circ}$, then $n$ equals: | 16 | 1. **Identify the formula for the sum of interior angles of a polygon**: The sum of the interior angles of a polygon with $n$ sides is given by the formula:
\[
S = 180^\circ (n-2)
\]
2. **Set up the arithmetic sequence**: Given that the angles are in arithmetic progression with a common difference of $5^\circ... | 0 | 3,011.25 | -1 | 3,011.25 |
Sixteen wooden Cs are placed in a 4-by-4 grid, all with the same orientation, and each is to be colored either red or blue. A quadrant operation on the grid consists of choosing one of the four two-by-two subgrids of Cs found at the corners of the grid and moving each C in the subgrid to the adjacent square in the subg... | 1296 | For each quadrant, we have three distinct cases based on the number of Cs in each color: - Case 1: all four the same color: 2 configurations (all red or all blue) - Case 2: 3 of one color, 1 of the other: 2 configurations (three red or three blue) - Case 3: 2 of each color: 2 configurations (red squares adjacent or opp... | 0 | 8,097.5625 | -1 | 8,097.5625 |
A cube, all of whose surfaces are painted, is cut into $1000$ smaller cubes of the same size. Find the expected value $E(X)$, where $X$ denotes the number of painted faces of a small cube randomly selected. | \frac{3}{5} | 0.5625 | 4,760.6875 | 4,783.555556 | 4,731.285714 | |
Compute $\frac{x^8+12x^4+36}{x^4+6}$ when $x=5$. | 631 | 1 | 1,875.75 | 1,875.75 | -1 | |
A certain electronic device contains three components, with probabilities of failure for each component being $0.1, 0.2, 0.3$, respectively. If the probabilities of the device failing when one, two, or three components fail are $0.25, 0.6, 0.9$, respectively, find the probability that the device fails. | 0.1601 | 0.0625 | 6,764.125 | 6,759 | 6,764.466667 | |
Let $ABC$ be a right triangle where $\measuredangle A = 90^\circ$ and $M\in (AB)$ such that $\frac{AM}{MB}=3\sqrt{3}-4$ . It is known that the symmetric point of $M$ with respect to the line $GI$ lies on $AC$ . Find the measure of $\measuredangle B$ . | 30 | 0 | 8,192 | -1 | 8,192 | |
Maurice travels to work either by his own car (and then due to traffic jams, he is late in half the cases) or by subway (and then he is late only one out of four times). If on a given day Maurice arrives at work on time, he always uses the same mode of transportation the next day as he did the day before. If he is late... | 2/3 | 0.0625 | 7,703 | 4,942 | 7,887.066667 | |
For a nonnegative integer $n$, let $r_7(3n)$ represent the remainder when $3n$ is divided by $7$. Determine the $22^{\text{nd}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_7(3n)\le 4~.$$ | 29 | 0.125 | 7,778.125 | 5,545 | 8,097.142857 | |
There is a list of seven numbers. The average of the first four numbers is $5$, and the average of the last four numbers is $8$. If the average of all seven numbers is $6\frac{4}{7}$, then the number common to both sets of four numbers is | 6 | 1. **Calculate the total of the first four numbers**:
Given that the average of the first four numbers is $5$, the sum of these numbers is:
\[
4 \times 5 = 20
\]
2. **Calculate the total of the last four numbers**:
Given that the average of the last four numbers is $8$, the sum of these numbers is:
... | 0.75 | 3,266.1875 | 2,431.833333 | 5,769.25 |
Given the set $A=\{2,3,4,8,9,16\}$, if $a\in A$ and $b\in A$, the probability that the event "$\log_{a}b$ is not an integer but $\frac{b}{a}$ is an integer" occurs is $\_\_\_\_\_\_$. | \frac{1}{18} | 0.3125 | 7,006.875 | 6,297.6 | 7,329.272727 | |
The amount $2.5$ is split into two nonnegative real numbers uniformly at random, for instance, into $2.143$ and $.357$, or into $\sqrt{3}$ and $2.5-\sqrt{3}$. Then each number is rounded to its nearest integer, for instance, $2$ and $0$ in the first case above, $2$ and $1$ in the second. What is the probability that th... | \frac{3}{5} | Let's denote the two parts into which $2.5$ is split as $x$ and $2.5 - x$. We need to find the probability that the sum of the nearest integers to $x$ and $2.5 - x$ equals $3$.
1. **Identify the rounding conditions**:
- $x$ rounds to $0$ if $x < 0.5$.
- $x$ rounds to $1$ if $0.5 \leq x < 1.5$.
- $x$ rounds t... | 0.0625 | 6,865.375 | 7,253 | 6,839.533333 |
On the sides \(A B, B C, C D\) and \(A D\) of the convex quadrilateral \(A B C D\) are points \(M, N, K\) and \(L\) respectively, such that \(A M: M B = 3: 2\), \(C N: N B = 2: 3\), \(C K = K D\) and \(A L: L D = 1: 2\). Find the ratio of the area of the hexagon \(M B N K D L\) to the area of the quadrilateral \(A B C ... | 4/5 | 0.0625 | 8,122.6875 | 7,083 | 8,192 | |
A square sheet of paper $ABCD$ is folded straight in such a way that point $B$ hits to the midpoint of side $CD$ . In what ratio does the fold line divide side $BC$ ? | 5/3 | 0.625 | 6,012.125 | 6,392.4 | 5,378.333333 | |
Two equal circles in the same plane cannot have the following number of common tangents. | 1 | To solve this problem, we need to consider the possible configurations of two equal circles in the same plane and determine the number of common tangents in each case. The configurations depend on the relative positions of the circles:
1. **Circles are separate (do not intersect)**:
- In this case, each circle will... | 0.5 | 4,501.1875 | 3,858.5 | 5,143.875 |
Let $\triangle ABC$ have sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$ respectively, given that $a^{2}+2b^{2}=c^{2}$, then $\dfrac {\tan C}{\tan A}=$ ______ ; the maximum value of $\tan B$ is ______. | \dfrac { \sqrt {3}}{3} | 0 | 7,880.875 | -1 | 7,880.875 | |
In a trapezoid $ABCD$ with bases $\overline{AB} \parallel \overline{CD}$ and $\overline{BC} \perp \overline{CD}$, suppose that $CD = 10$, $\tan C = 2$, and $\tan D = 1$. Calculate the length of $AB$ and determine the area of the trapezoid. | 300 | 0 | 8,192 | -1 | 8,192 | |
Alice drew a regular $2021$-gon in the plane. Bob then labeled each vertex of the $2021$-gon with a real number, in such a way that the labels of consecutive vertices differ by at most $1$. Then, for every pair of non-consecutive vertices whose labels differ by at most $1$, Alice drew a diagonal connecting them. Let $d... | 2018 |
To solve this problem, we need to find the least possible number of diagonals, \( d \), that Alice can draw given Bob's labeling constraints on the vertices of a regular 2021-gon.
### Step 1: Understanding the Problem
Alice has a regular 2021-gon, and Bob labels each vertex with a real number such that the labels of... | 0 | 8,192 | -1 | 8,192 |
Player A and player B are two basketball players shooting from the same position independently, with shooting accuracies of $\dfrac{1}{2}$ and $p$ respectively, and the probability of player B missing both shots is $\dfrac{1}{16}$.
- (I) Calculate the probability that player A hits at least one shot in two attempts.
- ... | \dfrac{3}{8} | 0.6875 | 4,662.125 | 3,452.818182 | 7,322.6 | |
The eccentricity of the ellipse $\frac {x^{2}}{9}+ \frac {y^{2}}{4+k}=1$ is $\frac {4}{5}$. Find the value of $k$. | 21 | 0.125 | 6,787.0625 | 5,538.5 | 6,965.428571 | |
The probability of getting rain on any given day in August in Beach Town is \(\frac{1}{5}\). What is the probability that it rains on at most 3 days in the first week of August? | 0.813 | 0 | 7,452.8125 | -1 | 7,452.8125 | |
Trapezoid $ABCD$ has $\overline{AB}\parallel\overline{CD}, BC=CD=43$, and $\overline{AD}\perp\overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP=11$, the length of $AD$ can be written in the form $m\sqrt{n}$, wh... | 194 | 1. **Identify the properties of the trapezoid**: Given that $ABCD$ is a trapezoid with $\overline{AB}\parallel\overline{CD}$ and $BC=CD=43$. Also, $\overline{AD}\perp\overline{BD}$, which implies that $\triangle ABD$ is a right triangle.
2. **Intersection and midpoint properties**: Let $O$ be the intersection of the d... | 0 | 8,192 | -1 | 8,192 |
Find the product of all constants \(t\) such that the quadratic \(x^2 + tx + 12\) can be factored in the form \((x+a)(x+b)\), where \(a\) and \(b\) are integers. | 530816 | 0 | 5,883.25 | -1 | 5,883.25 | |
The Cookie Monster encounters a cookie whose boundary is the equation $x^2+y^2 + 21 = 4x + 18 y$ and is very confused. He wants to know if this cookie is a lunch-sized cookie or a snack-sized cookie. What is the radius of this cookie? | 8 | 1 | 2,469.3125 | 2,469.3125 | -1 | |
What is the smallest positive perfect square that is divisible by both 2 and 3? | 36 | 1 | 1,551.875 | 1,551.875 | -1 | |
Two positive integers differ by 6 and their product is 135. What is the larger integer? | 15 | 1 | 1,813.5 | 1,813.5 | -1 | |
The desired number is greater than 400 and less than 500. Find it if the sum of its digits is 9 and it is equal to 47/36 of the number obtained by reversing its digits. | 423 | 0.875 | 4,678.5625 | 4,176.642857 | 8,192 | |
Let $n$ be the integer such that $0 \le n < 31$ and $3n \equiv 1 \pmod{31}$. What is $\left(2^n\right)^3 - 2 \pmod{31}$?
Express your answer as an integer from $0$ to $30$, inclusive. | 6 | 0.9375 | 4,391.6875 | 4,138.333333 | 8,192 | |
For a real number $y$, find the maximum value of
\[
\frac{y^6}{y^{12} + 3y^9 - 9y^6 + 27y^3 + 81}.
\] | \frac{1}{27} | 0.4375 | 7,758.8125 | 7,201.857143 | 8,192 | |
Given $$a_{n}= \frac {n(n+1)}{2}$$, remove all the numbers in the sequence $\{a_n\}$ that can be divided by 2, and arrange the remaining numbers in ascending order to form the sequence $\{b_n\}$. Find the value of $b_{21}$. | 861 | 0.0625 | 7,999.6875 | 5,115 | 8,192 | |
The increasing sequence \( T = 2, 3, 5, 6, 7, 8, 10, 11, \ldots \) consists of all positive integers which are not perfect squares. What is the 2012th term of \( T \)? | 2057 | 0.9375 | 4,871.4375 | 4,650.066667 | 8,192 | |
A list of $2018$ positive integers has a unique mode, which occurs exactly $10$ times. What is the least number of distinct values that can occur in the list? | 225 | 1. **Understanding the Problem:**
- We have a list of $2018$ positive integers.
- The mode (most frequently occurring number) appears exactly $10$ times.
- We need to find the least number of distinct values in the list.
2. **Setting Up the Equation:**
- To minimize the number of distinct values, we should... | 0.6875 | 5,813.75 | 4,732.727273 | 8,192 |
Given two circles $x^{2}+y^{2}=4$ and $x^{2}+y^{2}-2y-6=0$, find the length of their common chord. | 2\sqrt{3} | 0.9375 | 3,268 | 2,939.733333 | 8,192 | |
In triangle $XYZ$, $E$ lies on $\overline{YZ}$ and $G$ lies on $\overline{XY}$. Let $\overline{XE}$ and $\overline{YG}$ intersect at $Q.$
If $XQ:QE = 5:2$ and $GQ:QY = 3:4$, find $\frac{XG}{GY}.$ | \frac{4}{3} | 0 | 8,107.9375 | -1 | 8,107.9375 | |
A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $$ 1$ to $$ 9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a particular day, the dig... | 420 | 0 | 8,085.0625 | -1 | 8,085.0625 | |
C is the complex numbers. \( f : \mathbb{C} \to \mathbb{R} \) is defined by \( f(z) = |z^3 - z + 2| \). What is the maximum value of \( f \) on the unit circle \( |z| = 1 \)? | \sqrt{13} | 0.4375 | 7,557.625 | 6,742 | 8,192 | |
Convert -630° to radians. | -\frac{7\pi}{2} | 0.6875 | 341.625 | 332.454545 | 361.8 | |
In an $11 \times 11$ table, integers from 0 to 10 are placed (naturally, numbers can repeat, and not necessarily all listed numbers occur). It is known that in every $3 \times 2$ or $2 \times 3$ rectangle, the sum of the numbers is 10. Find the smallest possible value of the sum of the numbers in the entire table. | 200 | 0 | 8,192 | -1 | 8,192 | |
Find $n$ such that $2^6 \cdot 3^3 \cdot n = 10!$. | 2100 | 1 | 2,957 | 2,957 | -1 | |
Given the geometric sequence $\{a_{n}\}$, $a_{2}$ and $a_{18}$ are the two roots of the equation $x^{2}+15x+16=0$, find the value of $a_{3}a_{10}a_{17}$. | -64 | 0.4375 | 6,654.9375 | 5,973.428571 | 7,185 | |
If five people are selected at random from a group of ten men and five women, what is the probability that at least one woman is selected? Express your answer as a common fraction. | \frac{917}{1001} | 0 | 4,181.875 | -1 | 4,181.875 | |
A *palindromic table* is a $3 \times 3$ array of letters such that the words in each row and column read the same forwards and backwards. An example of such a table is shown below.
\[ \begin{array}[h]{ccc}
O & M & O
N & M & N
O & M & O
\end{array} \]
How many palindromic tables are there that use only the le... | 16 | 0.5 | 5,853.0625 | 5,641.625 | 6,064.5 | |
In triangle \(ABC\), \(AC = 18 \, \text{cm}\) and \(BC = 21 \, \text{cm}\). Point \(K\) is the midpoint of side \(BC\), and point \(M\) is the midpoint of side \(AB\). Point \(N\) lies on side \(AC\) such that \(AN = 6 \, \text{cm}\). Additionally, \(MN = KN\). Find the length of side \(AB\). | 15 | 1 | 3,284.1875 | 3,284.1875 | -1 | |
The sides of a triangle are 5, 6, and 7. Find the area of the orthogonal projection of the triangle onto a plane that forms an angle equal to the smallest angle of the triangle with the plane of the triangle. | \frac{30 \sqrt{6}}{7} | 0 | 5,242.875 | -1 | 5,242.875 | |
Let $\theta=\frac{2\pi}{2015}$ , and suppose the product \[\prod_{k=0}^{1439}\left(\cos(2^k\theta)-\frac{1}{2}\right)\] can be expressed in the form $\frac{b}{2^a}$ , where $a$ is a non-negative integer and $b$ is an odd integer (not necessarily positive). Find $a+b$ .
*2017 CCA Math Bonanza Tiebreaker Round #3... | 1441 | 0.0625 | 8,096 | 6,656 | 8,192 | |
BoatWorks built 3 canoes in January of this year and then each subsequent calendar month they built twice the number of canoes they had built the previous month. How many total canoes were built by BoatWorks by the end of March of this year? | 21 | 1 | 690.875 | 690.875 | -1 | |
Find the probability that a randomly selected 8-digit number composed of 0s and 1s has the sum of the digits in even positions equal to the sum of the digits in odd positions. | 35/128 | 0.9375 | 4,438.0625 | 4,187.8 | 8,192 | |
Gilda has a bag of marbles. She gives $20\%$ of them to her friend Pedro. Then Gilda gives $10\%$ of what is left to another friend, Ebony. Finally, Gilda gives $25\%$ of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself? | 54 | 1. **Initial Amount of Marbles**: Let's denote the initial number of marbles Gilda has as $M$.
2. **Marbles Given to Pedro**: Gilda gives $20\%$ of her marbles to Pedro. Therefore, the number of marbles she gives to Pedro is $0.20M$. The number of marbles left with Gilda after giving to Pedro is:
\[
M - 0.20M = ... | 1 | 1,780.6875 | 1,780.6875 | -1 |
Find the greatest common divisor of $8!$ and $(6!)^2.$ | 5760 | 1 | 3,684.6875 | 3,684.6875 | -1 | |
If \( 6x + t = 4x - 9 \), what is the value of \( x + 4 \)? | -4 | 0 | 7,177.25 | -1 | 7,177.25 | |
Ten children were given 100 pieces of macaroni each on their plates. Some children didn't want to eat and started playing. With one move, one child transfers one piece of macaroni from their plate to each of the other children's plates. What is the minimum number of moves needed such that all the children end up with a... | 45 | 0 | 8,192 | -1 | 8,192 | |
Given a square region with a side length of 1 meter, and a total of 5120 beans within the square with 4009 beans within the inscribed circle, determine the approximate value of pi rounded to three decimal places. | 3.13 | 0 | 6,983.3125 | -1 | 6,983.3125 | |
Amy works for 36 hours per week for 10 weeks during the summer, making $\$3000$. If she works for 30 weeks during the school year at the same rate of pay and needs to make another $\$3000$, how many hours per week must she work? | 12 | 0.375 | 5,839.9375 | 4,262.666667 | 6,786.3 | |
Right $ \triangle ABC$ has $ AB \equal{} 3$ , $ BC \equal{} 4$ , and $ AC \equal{} 5$ . Square $ XYZW$ is inscribed in $ \triangle ABC$ with $ X$ and $ Y$ on $ \overline{AC}$ , $ W$ on $ \overline{AB}$ , and $ Z$ on $ \overline{BC}$ . What is the side length of the square?
[asy]size(200);defaultpen... | \frac {60}{37} | 0.0625 | 8,077.125 | 6,354 | 8,192 |
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