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 |
|---|---|---|---|---|---|---|
Circles \(\omega_{1}\) and \(\omega_{2}\) intersect at points \(A\) and \(B\). Segment \(PQ\) is tangent to \(\omega_{1}\) at \(P\) and to \(\omega_{2}\) at \(Q\), and \(A\) is closer to \(PQ\) than \(B\). Point \(X\) is on \(\omega_{1}\) such that \(PX \parallel QB\), and point \(Y\) is on \(\omega_{2}\) such that \(... | 2 - \sqrt{3} | 0 | 7,868.5625 | -1 | 7,868.5625 | |
Given $| \mathbf{e} |=1$, and it satisfies $|\mathbf{a} + \mathbf{e}|=|\mathbf{a} - 2\mathbf{e}|$, then the projection of vector $\mathbf{a}$ in the direction of $\mathbf{e}$ is | \frac{1}{2} | 0.9375 | 2,693.9375 | 2,327.4 | 8,192 | |
Let $a$, $b$, and $c$ be positive real numbers. What is the smallest possible value of $(a+b+c)\left(\frac{1}{a+b}+\frac{1}{a+c}+\frac{1}{b+c}\right)$? | \frac{9}{2} | 0.75 | 5,655.875 | 4,876.916667 | 7,992.75 | |
In the sequence $5, 8, 15, 18, 25, 28, \cdots, 2008, 2015$, how many numbers have a digit sum that is an even number? (For example, the digit sum of 138 is $1+3+8=12$) | 202 | 0.0625 | 8,085.75 | 6,509 | 8,190.866667 | |
Given an ellipse M: $$\frac {y^{2}}{a^{2}}+ \frac {x^{2}}{b^{2}}=1$$ (where $a>b>0$) whose eccentricity is the reciprocal of the eccentricity of the hyperbola $x^{2}-y^{2}=1$, and the major axis of the ellipse is 4.
(1) Find the equation of ellipse M;
(2) If the line $y= \sqrt {2}x+m$ intersects ellipse M at points A a... | \sqrt {2} | 0 | 7,604.9375 | -1 | 7,604.9375 | |
Given the function $f(x)=x^{3}+3x^{2}-9x+3.$ Find:
(I) The interval(s) where $f(x)$ is increasing;
(II) The extreme values of $f(x)$. | -2 | 1 | 2,553.875 | 2,553.875 | -1 | |
Find all integer $n$ such that the following property holds: for any positive real numbers $a,b,c,x,y,z$, with $max(a,b,c,x,y,z)=a$ , $a+b+c=x+y+z$ and $abc=xyz$, the inequality $$a^n+b^n+c^n \ge x^n+y^n+z^n$$ holds. | n \ge 0 |
We are given the conditions \( \max(a, b, c, x, y, z) = a \), \( a + b + c = x + y + z \), and \( abc = xyz \). We need to find all integer \( n \) such that the inequality
\[
a^n + b^n + c^n \ge x^n + y^n + z^n
\]
holds for any positive real numbers \( a, b, c, x, y, z \).
We claim that the answer is all \( n \ge 0 ... | 0 | 8,192 | -1 | 8,192 |
For what real value of $u$ is $\frac{-15+\sqrt{205}}{8}$ a root of $4x^2 + 15x + u$? | \frac{5}{4} | 1 | 3,334.5625 | 3,334.5625 | -1 | |
Let $x \neq y$ be positive reals satisfying $x^3+2013y=y^3+2013x$ , and let $M = \left( \sqrt{3}+1 \right)x + 2y$ . Determine the maximum possible value of $M^2$ .
*Proposed by Varun Mohan* | 16104 | 0.6875 | 6,748.6875 | 6,092.636364 | 8,192 | |
Winnie has 45 cherry lollipops, 116 wintergreen lollipops, 4 grape lollipops, and 229 shrimp cocktail lollipops. With no regard to flavor, Winnie gives each of her 11 nearest and dearest friends some lollipops. Each friend gets the same number of lollipops and she hands out as many lollipops as possible. How many lo... | 9 | 1 | 2,015.5625 | 2,015.5625 | -1 | |
Let $P(n)$ represent the product of all non-zero digits of a positive integer $n$. For example: $P(123) = 1 \times 2 \times 3 = 6$ and $P(206) = 2 \times 6 = 12$. Find the value of $P(1) + P(2) + \cdots + P(999)$. | 97335 | 0.125 | 8,030.9375 | 6,903.5 | 8,192 | |
If $a<b<c<d<e$ are consecutive positive integers such that $b+c+d$ is a perfect square and $a+b+c+d+e$ is a perfect cube, what is the smallest possible value of $c$? | 182 | Let's denote the consecutive positive integers as $a, b, c, d, e$ such that $a < b < c < d < e$. Given that $b = a+1$, $c = a+2$, $d = a+3$, and $e = a+4$, we can express the conditions in the problem using these variables.
1. **Sum of $b$, $c$, and $d$ is a perfect square:**
\[
b + c + d = (a+1) + (a+2) + (a+3)... | 0 | 6,419.25 | -1 | 6,419.25 |
If $b$ is an even multiple of $7786$, find the greatest common divisor of $8b^2 + 85b + 200$ and $2b + 10$. | 10 | 0.0625 | 7,897.3125 | 5,979 | 8,025.2 | |
In a joint maritime search and rescue exercise between China and foreign countries, the Chinese side participated with 4 ships and 3 airplanes; the foreign side with 5 ships and 2 airplanes. If 2 units are selected from each group (either 1 airplane or 1 ship counts as one unit, and all ships and airplanes are distinct... | 180 | 0.6875 | 4,742.125 | 3,615.636364 | 7,220.4 | |
Let $ABC$ be a right triangle with $\angle A=90^{\circ}$. Let $D$ be the midpoint of $AB$ and let $E$ be a point on segment $AC$ such that $AD=AE$. Let $BE$ meet $CD$ at $F$. If $\angle BFC=135^{\circ}$, determine $BC/AB$. | \frac{\sqrt{13}}{2} | Let $\alpha=\angle ADC$ and $\beta=\angle ABE$. By exterior angle theorem, $\alpha=\angle BFD+\beta=$ $45^{\circ}+\beta$. Also, note that $\tan \beta=AE/AB=AD/AB=1/2$. Thus, $$1=\tan 45^{\circ}=\tan (\alpha-\beta)=\frac{\tan \alpha-\tan \beta}{1+\tan \alpha \tan \beta}=\frac{\tan \alpha-\frac{1}{2}}{1+\frac{1}{2} \tan ... | 0 | 8,050.0625 | -1 | 8,050.0625 |
Triangle \(\triangle P N R\) has side lengths \(P N=20, N R=18\), and \(P R=19\). Consider a point \(A\) on \(P N\). \(\triangle N R A\) is rotated about \(R\) to \(\triangle N^{\prime} R A^{\prime}\) so that \(R, N^{\prime}\), and \(P\) lie on the same line and \(A A^{\prime}\) is perpendicular to \(P R\). Find \(\fra... | \frac{19}{18} | Denote the intersection of \(P R\) and \(A A^{\prime}\) be \(D\). Note \(R A^{\prime}=R A\), so \(D\), being the altitude of an isosceles triangle, is the midpoint of \(A A^{\prime}\). Thus, \(\angle A R D=\angle A^{\prime} R D=\angle N R A\) so \(R A\) is the angle bisector of \(P N R\) through \(R\). By the angle bis... | 0 | 7,972.375 | -1 | 7,972.375 |
Five brothers equally divided an inheritance from their father. The inheritance included three houses. Since three houses could not be divided into 5 parts, the three older brothers took the houses, and the younger brothers were compensated with money. Each of the three older brothers paid 800 rubles, and the younger b... | 2000 | 0.5625 | 4,512.1875 | 3,608.555556 | 5,674 | |
Take 3 segments randomly, each shorter than a unit. What is the probability that these 3 segments can form a triangle? | 1/2 | 0 | 8,192 | -1 | 8,192 | |
The number $695$ is to be written with a factorial base of numeration, that is, $695=a_1+a_2\times2!+a_3\times3!+ \ldots a_n \times n!$ where $a_1, a_2, a_3 ... a_n$ are integers such that $0 \le a_k \le k,$ and $n!$ means $n(n-1)(n-2)...2 \times 1$. Find $a_4$ | 3 | To solve this problem, we need to express the number 695 in a factorial base. This means we need to find coefficients $a_1, a_2, a_3, \ldots, a_n$ such that:
\[ 695 = a_1 + a_2 \times 2! + a_3 \times 3! + \ldots + a_n \times n! \]
where $0 \leq a_k \leq k$ for each $k$.
1. **Find the largest factorial smaller than 695... | 0.9375 | 3,330 | 3,005.866667 | 8,192 |
Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP > CP$. Let $O_{1}$ and $O_{2}$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB = 12$ and $\angle O_{1}PO_{2} = 120^{\circ}$, then $AP = \sqrt{a} + \sqrt{b}$, where $a$ and $b$ are positive integers. Find $a + b$.
Quickest Metho... | 96 | Both $O_1$ and $O_2$ lie on the perpendicular bisector of $AB$.
Claim: $O_1O_2=12$ and $O_1P=O_2P$.
Proof. Translate $O_1$ and $P$ $12$ units down, and let their images be $O_1'$ and $P'$, respectively. Note that $\triangle ABP\cong\triangle DCP'$. Additionally, \[\angle CP'D = \angle BPA = 180^{\circ} - \angle BPC = 1... | 0.6875 | 5,529 | 5,086.818182 | 6,501.8 |
The number $a+\sqrt{b}$ and its radical conjugate have a sum of $-4$ and a product of $1$. Find $a+b$. | 1 | 1 | 1,319.125 | 1,319.125 | -1 | |
Solve the congruence $15x + 3 \equiv 9 \pmod{21}$ for $x$, and express the solution as $x \equiv a \pmod{m}$, where $a < m$ and find $a + m$. | 13 | 0.6875 | 2,668.875 | 2,263.636364 | 3,560.4 | |
Given two intersecting circles O: $x^2 + y^2 = 25$ and C: $x^2 + y^2 - 4x - 2y - 20 = 0$, which intersect at points A and B, find the length of the common chord AB. | \sqrt{95} | 1 | 4,569.3125 | 4,569.3125 | -1 | |
In triangle $\triangle ABC$, $a$, $b$, and $c$ are the opposite sides of angles $A$, $B$, and $C$ respectively, and $\dfrac{\cos B}{\cos C}=-\dfrac{b}{2a+c}$.
(1) Find the measure of angle $B$;
(2) If $b=\sqrt {13}$ and $a+c=4$, find the area of $\triangle ABC$. | \dfrac{3\sqrt{3}}{4} | 0 | 4,331.625 | -1 | 4,331.625 | |
The lattice shown is continued for $9$ rows. What will be the third number in the $9$th row? \begin{tabular}{rccccc}
Row 1: & 1 & 2 & 3 & 4 & 5 \\
Row 2: & 6 & 7 & 8 & 9 & 10 \\
Row 3: & 11 & 12 & 13 & 14 & 15 \\
Row 4: & 16 & 17 & 18 & 19 & 20
\end{tabular} | 43 | 1 | 3,563.625 | 3,563.625 | -1 | |
For positive integer $n$, let $s(n)$ denote the sum of the digits of $n$. Find the smallest positive integer satisfying $s(n) = s(n+864) = 20$. | 695 | First of all, notice that the smallest $n$ with $s(n) = 20$ is $299$. Also, if $s(n + 864) = 20$, $s(n - 136) = 19$ (because subtracting $1000$ from the number removes the $1$ in the thousands place). After checking $s(n - 136)$ for various $n$ with $s(n) = 20$, we see that we need to have a carry when subtracting $136... | 0 | 8,192 | -1 | 8,192 |
Let triangle $ABC$ be a right triangle in the xy-plane with a right angle at $C_{}$. Given that the length of the hypotenuse $AB$ is $60$, and that the medians through $A$ and $B$ lie along the lines $y=x+3$ and $y=2x+4$ respectively, find the area of triangle $ABC$. | 400 | We first seek to find the angle between the lines $y = x + 3$ and $y = 2x + 4$. [asy] import graph; size(150); Label f; f.p=fontsize(6); xaxis(-8,8,Ticks(f, 2.0)); yaxis(-8,8,Ticks(f, 2.0)); real f(real x) { return (x + 3); } real g( real x){ return (2x + 4); } draw(graph(f,-8,5),red+linewidth(1)); draw(graph(g,-6,2),b... | 0 | 8,192 | -1 | 8,192 |
Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2009,0),(2009,2010),$ and $(0,2010)$. What is the probability that $x > 7y$? Express your answer as a common fraction. | \frac{287}{4020} | 0.1875 | 7,424.375 | 4,219.333333 | 8,164 | |
Given the coordinates of the foci of an ellipse are $F_{1}(-1,0)$, $F_{2}(1,0)$, and a line perpendicular to the major axis through $F_{2}$ intersects the ellipse at points $P$ and $Q$, with $|PQ|=3$.
$(1)$ Find the equation of the ellipse;
$(2)$ A line $l$ through $F_{2}$ intersects the ellipse at two distinct points ... | \frac {9}{16}\pi | 0 | 8,192 | -1 | 8,192 | |
Let $S = \{(x,y) | x = 1, 2, \ldots, 1993, y = 1, 2, 3, 4\}$. If $T \subset S$ and there aren't any squares in $T.$ Find the maximum possible value of $|T|.$ The squares in T use points in S as vertices. | 5183 |
Let \( S = \{(x,y) \mid x = 1, 2, \ldots, 1993, y = 1, 2, 3, 4\} \). We aim to find the maximum possible value of \( |T| \) for a subset \( T \subset S \) such that there are no squares in \( T \).
To solve this, we need to ensure that no four points in \( T \) form the vertices of a square. The key observation is th... | 0 | 8,192 | -1 | 8,192 |
Suppose \(\frac{1}{2} \leq x \leq 2\) and \(\frac{4}{3} \leq y \leq \frac{3}{2}\). Determine the minimum value of
$$
\frac{x^{3} y^{3}}{x^{6}+3 x^{4} y^{2}+3 x^{3} y^{3}+3 x^{2} y^{4}+y^{6}}.
$$ | 27/1081 | 0.25 | 7,816.125 | 7,306 | 7,986.166667 | |
Let $f(x) = \displaystyle \frac{1}{ax+b}$ where $a$ and $b$ are nonzero constants. Find all solutions to $f^{-1}(x) = 0$. Express your answer in terms of $a$ and/or $b$. | \frac1b | 1 | 2,034.5 | 2,034.5 | -1 | |
If $e^{i \alpha} + e^{i \beta} = \frac{1}{4} + \frac{3}{7} i,$ then find $e^{-i \alpha} + e^{-i \beta}.$ | \frac{1}{4} - \frac{3}{7} i | 1 | 4,241 | 4,241 | -1 | |
Let the numbers x and y satisfy the conditions $\begin{cases} x^2 + y^2 - xy = 2
x^4 + y^4 + x^2y^2 = 8 \end{cases}$ The value of $P = x^8 + y^8 + x^{2014}y^{2014}$ is: | 48 | 0.5 | 6,445.5 | 4,699 | 8,192 | |
Use the Horner's method to calculate the value of the polynomial $f(x) = 5x^5 + 2x^4 + 3.5x^3 - 2.6x^2 + 1.7x - 0.8$ when $x=1$ and find the value of $v_3$. | 7.9 | 0.8125 | 3,229.375 | 3,384.384615 | 2,557.666667 | |
Given the functions $f(x)=x^{2}-2x+2$ and $g(x)=-x^{2}+ax+b- \frac {1}{2}$, one of their intersection points is $P$. The tangent lines $l_{1}$ and $l_{2}$ to the functions $f(x)$ and $g(x)$ at point $P$ are perpendicular. Find the maximum value of $ab$. | \frac{9}{4} | 0.3125 | 7,243.25 | 5,156 | 8,192 | |
In the polar coordinate system, let the point on the circle $ \begin{cases}x= \frac{ \sqrt{6}}{2}\cos \theta \\ y= \frac{ \sqrt{6}}{2}\sin \theta \end{cases} (\theta \text{ is a parameter}) $ have a distance $d$ from the line $ρ( \sqrt{7}\cos θ-\sin θ)= \sqrt{2}$. Find the maximum value of $d$. | \frac{ \sqrt{6}}{2} + \frac{1}{2} | 0 | 6,499.3125 | -1 | 6,499.3125 | |
If the tangent line of the curve $y=\ln x$ at point $P(x_{1}, y_{1})$ is tangent to the curve $y=e^{x}$ at point $Q(x_{2}, y_{2})$, then $\frac{2}{{x_1}-1}+x_{2}=$____. | -1 | 0.5625 | 7,545.6875 | 7,043 | 8,192 | |
In the triangle \(ABC\), points \(K\), \(L\), and \(M\) are taken on sides \(AB\), \(BC\), and \(AD\) respectively. It is known that \(AK = 5\), \(KB = 3\), \(BL = 2\), \(LC = 7\), \(CM = 1\), and \(MA = 6\). Find the distance from point \(M\) to the midpoint of \(KL\). | \frac{1}{2} \sqrt{\frac{3529}{21}} | 0 | 8,192 | -1 | 8,192 | |
$|2+i^{2}+2i^{2}|=$ | \sqrt{5} | 0 | 1,528.8125 | -1 | 1,528.8125 | |
Find the number of six-digit palindromes. | 900 | 0.9375 | 2,535.5 | 2,158.4 | 8,192 | |
The lengths of the three altitudes of a triangle are in the ratio \(3 : 4 : 6\). If the radius of the inscribed circle of the triangle is 1, find the radius of the circumcircle of the triangle. | 16/5 | 1 | 5,099.375 | 5,099.375 | -1 | |
Given a circle circumscribed around triangle \(FDC\), a tangent \(FK\) is drawn such that \(\angle KFC = 58^\circ\). Points \(K\) and \(D\) lie on opposite sides of line \(FC\) as shown in the diagram. Find the acute angle between the angle bisectors of \(\angle CFD\) and \(\angle FCD\). Provide your answer in degrees. | 61 | 0.75 | 5,467.5 | 4,559.333333 | 8,192 | |
Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 pet rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms? | 64 | 1. **Calculate the total number of students in all classrooms**:
Each classroom has $18$ students. Since there are $4$ classrooms, the total number of students is:
\[
18 \times 4 = 72 \text{ students}
\]
2. **Calculate the total number of rabbits in all classrooms**:
Each classroom has $2$ rabbits. Sin... | 1 | 351.3125 | 351.3125 | -1 |
In a cube $ABCDEFGH$, the coordinates of vertices are set in a conventional cube alignment with $A(0, 0, 0)$, $B(2, 0, 0)$, $C(2, 0, 2)$, $D(0, 0, 2)$, $E(0, 2, 0)$, $F(2, 2, 0)$, $G(2, 2, 2)$, and $H(0, 2, 2)$. Let $M$ and $N$ be the midpoints of the segments $\overline{EB}$ and $\overline{HD}$, respectively. Determin... | \frac{17}{2304} | 0 | 8,192 | -1 | 8,192 | |
The polynomial $2x^3 + bx + 7$ has a factor of the form $x^2 + px + 1.$ Find $b.$ | -\frac{45}{2} | 1 | 3,031.125 | 3,031.125 | -1 | |
The numbers 1 through 25 are written on 25 cards with one number on each card. Sara picks one of the 25 cards at random. What is the probability that the number on her card will be a multiple of 2 or 5? Express your answer as a common fraction. | \frac{3}{5} | 1 | 2,788 | 2,788 | -1 | |
In the United States, coins have the following thicknesses: penny, 1.55 mm; nickel, 1.95 mm; dime, 1.35 mm; quarter, 1.75 mm. If a stack of these coins is exactly 14 mm high, how many coins are in the stack? | 8 | 0.125 | 8,080 | 7,296 | 8,192 | |
A frog is placed at the origin on the number line, and moves according to the following rule: in a given move, the frog advances to either the closest point with a greater integer coordinate that is a multiple of 3, or to the closest point with a greater integer coordinate that is a multiple of 13. A move sequence is a... | 169 | Another way would be to use a table representing the number of ways to reach a certain number
$\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c|c|c} 0 & 3 & 6 & 9 & 12 & 13 & 15 & 18 & 21 & 24 & 26 & 27 & 30 & 33 & 36 \\ \hline 1 & 1 & 1 & 1 & 1 & 5 & 6 & 6 & 6 & 6 & 29 & 35 & 35 & 35 & 35 \\ \end{tabular}$
How we came with ... | 0 | 8,092.875 | -1 | 8,092.875 |
Let $f(x)=x^2-2x$. What is the value of $f(f(f(f(f(f(-1))))))$? | 3 | 1 | 1,948.125 | 1,948.125 | -1 | |
Evaluate the expression $2x^{2}+3x^{2}$ when $x=2$. | 20 | When $x=2$, we obtain $2x^{2}+3x^{2}=5x^{2}=5 \cdot 2^{2}=5 \cdot 4=20$. | 1 | 405.125 | 405.125 | -1 |
Find the sum of all positive integers $b < 1000$ such that the base-$b$ integer $36_{b}$ is a perfect square and the base-$b$ integer $27_{b}$ is a perfect cube. | 371 | The conditions are: \[3b+6 = n^2\] \[2b+7 = m^3\] We can see $n$ is multiple is 3, so let $n=3k$, then $b= 3k^2-2$. Substitute $b$ into second condition and we get $m^3=3(2k^2+1)$. Now we know $m$ is both a multiple of 3 and odd. Also, $m$ must be smaller than 13 for $b$ to be smaller than 1000. So the only two possibl... | 0.375 | 7,299 | 5,810.666667 | 8,192 |
Given the line $y=kx+1$, the maximum length of the chord intercepted by the ellipse $\frac{x^{2}}{4}+y^{2}=1$ as $k$ varies, determine the maximum length of the chord. | \frac{4\sqrt{3}}{3} | 0 | 7,486 | -1 | 7,486 | |
A circle is circumscribed around a unit square \(ABCD\), and a point \(M\) is selected on the circle.
What is the maximum value that the product \(MA \cdot MB \cdot MC \cdot MD\) can take? | 0.5 | 0.0625 | 8,080.3125 | 6,598 | 8,179.133333 | |
Medians $\overline{DP}$ and $\overline{EQ}$ of $\triangle DEF$ are perpendicular. An altitude $\overline{ER}$ from $E$ to line $DF$ is such that $ER = 15$. Given $DP= 27$ and $EQ = 36$, determine the length of ${DF}$. | 45 | 0 | 8,052.0625 | -1 | 8,052.0625 | |
Eugene swam on Sunday, Monday, and Tuesday. On Monday, he swam for 30 minutes. On Tuesday, he swam for 45 minutes. His average swim time over the three days was 34 minutes. For how many minutes did he swim on Sunday? | 27 | Since Eugene swam three times and had an average swim time of 34 minutes, he swam for \( 3 \times 34 = 102 \) minutes in total. Since he swam for 30 minutes and 45 minutes on Monday and Tuesday, then on Sunday, he swam for \( 102 - 30 - 45 = 27 \) minutes. | 1 | 958.3125 | 958.3125 | -1 |
In the geometric sequence $\{a_{n}\}$, $a_{20}$ and $a_{60}$ are the two roots of the equation $(x^{2}-10x+16=0)$. Find the value of $\frac{{{a}\_{30}}\cdot {{a}\_{40}}\cdot {{a}\_{50}}}{2}$. | 32 | 0.8125 | 5,647.125 | 5,059.846154 | 8,192 | |
Two numbers have a difference of 3 and a sum of 31. What is the larger of the two numbers? | 17 | 1 | 1,749.375 | 1,749.375 | -1 | |
There are two equilateral triangles with a vertex at $(0, 1)$ , with another vertex on the line $y = x + 1$ and with the final vertex on the parabola $y = x^2 + 1$ . Find the area of the larger of the two triangles. | 26\sqrt{3} + 45 | 0 | 8,173.5 | -1 | 8,173.5 | |
Let \( T = 1 + 2 - 3 - 4 + 5 + 6 - 7 - 8 + \cdots + 2023 + 2024 - 2025 - 2026 \). What is the residue of \( T \), modulo 2027? | 2026 | 0 | 6,455.625 | -1 | 6,455.625 | |
Convex quadrilateral $B C D E$ lies in the plane. Lines $E B$ and $D C$ intersect at $A$, with $A B=2$, $A C=5, A D=200, A E=500$, and $\cos \angle B A C=\frac{7}{9}$. What is the largest number of nonoverlapping circles that can lie in quadrilateral $B C D E$ such that all of them are tangent to both lines $B E$ and $... | 5 | Let $\theta=\angle B A C$, and $\cos \theta=\frac{7}{9}$ implies $\cos \frac{\theta}{2}=\sqrt{\frac{1+\frac{7}{9}}{2}}=\frac{2 \sqrt{2}}{3} ; \sin \frac{\theta}{2}=\frac{1}{3} ; B C=$ $\sqrt{4+25-2(2)(5) \frac{7}{9}}=\frac{11}{3}$. Let $O_{1}$ be the excircle of $\triangle A B C$ tangent to lines $A B$ and $A C$, and l... | 0 | 8,135 | -1 | 8,135 |
Define a function $g(x),$ for positive integer values of $x,$ by \[g(x) = \left\{\begin{aligned} \log_3 x & \quad \text{ if } \log_3 x \text{ is an integer} \\ 1 + g(x + 2) & \quad \text{ otherwise}. \end{aligned} \right.\] Compute $g(50).$ | 20 | 0.0625 | 7,960.75 | 4,492 | 8,192 | |
An $n$-digit positive integer is cute if its $n$ digits are an arrangement of the set $\{1,2,...,n\}$ and its first $k$ digits form an integer that is divisible by $k$, for $k = 1,2,...,n$. For example, $321$ is a cute $3$-digit integer because $1$ divides $3$, $2$ divides $32$, and $3$ divides $321$. How many cute $6$... | 4 | 1. **Define the problem and constraints**: We need to find a 6-digit integer $abcdef$ such that:
- The digits are a permutation of $\{1, 2, 3, 4, 5, 6\}$.
- For each $k = 1, 2, \ldots, 6$, the number formed by the first $k$ digits is divisible by $k$.
2. **Divisibility by 1**: Any number is divisible by 1, so $a... | 0 | 8,192 | -1 | 8,192 |
Triangle $ABC$ has $AB=21$, $AC=22$ and $BC=20$. Points $D$ and $E$ are located on $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{DE}$ is parallel to $\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$. Then $DE=m/n$, where $m$ and $n$ are relatively prime positive... | 923 | 0.125 | 7,819.375 | 6,428 | 8,018.142857 | |
A palindrome is a number, word, or text that reads the same backward as forward. How much time in a 24-hour day display palindromes on a clock, showing time from 00:00:00 to 23:59:59? | 144 | 0 | 8,192 | -1 | 8,192 | |
Suppose point \(P\) is inside triangle \(ABC\). Let \(AP, BP\), and \(CP\) intersect sides \(BC, CA\), and \(AB\) at points \(D, E\), and \(F\), respectively. Suppose \(\angle APB=\angle BPC=\angle CPA, PD=\frac{1}{4}, PE=\frac{1}{5}\), and \(PF=\frac{1}{7}\). Compute \(AP+BP+CP\). | \frac{19}{12} | The key is the following lemma: Lemma: If \(\angle X=120^{\circ}\) in \(\triangle XYZ\), and the bisector of \(X\) intersects \(YZ\) at \(T\), then \(\frac{1}{XY}+\frac{1}{XZ}=\frac{1}{XT}\). Proof of the Lemma. Construct point \(W\) on \(XY\) such that \(\triangle XWT\) is equilateral. We also have \(TW \parallel XZ\)... | 0 | 8,192 | -1 | 8,192 |
The largest value of the real number $k$ for which the inequality $\frac{1+\sin x}{2+\cos x} \geqslant k$ has a solution. | \frac{4}{3} | 0.3125 | 7,315.625 | 5,646.4 | 8,074.363636 | |
Rodney uses the following clues to try to guess a secret number:
- It is a two-digit integer.
- The tens digit is odd.
- The units digit is even.
- The number is greater than 75.
What is the probability that Rodney will guess the correct number if he guesses a number that meets all these criteria? Express your answer ... | \frac{1}{7} | 0.1875 | 1,450.8125 | 4,535.333333 | 739 | |
Given that the focus of the parabola $C: y^{2}=4x$ is $F$, two lines $l_{1}$ and $l_{2}$ are drawn passing through point $F$. Line $l_{1}$ intersects the parabola $C$ at points $A$ and $B$, while line $l_{2}$ intersects the parabola $C$ at points $M$ and $N$. If the product of the slopes of $l_{1}$ and $l_{2}$ is $-1$,... | 16 | 0.5 | 7,315.1875 | 6,438.375 | 8,192 | |
For $\{1, 2, 3, \ldots, n\}$ and each of its non-empty subsets a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. For example, the alternating sum for $\{1, 2, 3, 6,9\}$ is $9-6+3-2+... | 448 | 0.0625 | 8,066 | 6,176 | 8,192 | |
Calculate $\int_{0}^{1} \frac{\sin x}{x} \, dx$ with an accuracy of 0.01. | 0.94 | 0 | 7,454.5625 | -1 | 7,454.5625 | |
Given that $a$, $b$, $c$, and $d$ are distinct positive integers, and $abcd = 441$, calculate the value of $a+b+c+d$. | 32 | 0.8125 | 4,982.125 | 4,241.384615 | 8,192 | |
Cameron writes down the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect cube, and all the multiples of 30 between them. How many integers are in Cameron's list? | 871 | 0.875 | 4,590.125 | 4,075.571429 | 8,192 | |
Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra \$2.50 to cover her portion of the total bill. What was the total bill? | $120 | 1. **Define the variables**: Let $m$ be the total bill that the 8 friends were supposed to split equally. Therefore, each friend's share would have been $\frac{m}{8}$.
2. **Adjust for Judi's share**: Since Judi did not pay, each of the remaining 7 friends paid an extra $2.50 to cover her share. Thus, each of these fri... | 0 | 1,543.5625 | -1 | 1,543.5625 |
Given the quadratic function \( y = ax^2 + bx + c \) where \( a \neq 0 \), its vertex is \( C \), and it intersects the x-axis at points \( A \) and \( B \). If triangle \( \triangle ABC \) is an acute triangle and \(\sin \angle BCA = \frac{4}{5}\), find the discriminant \(\Delta = b^2 - 4ac\). | 16 | 0.25 | 7,646.5 | 6,010 | 8,192 | |
How many primes less than $100$ have $7$ as the ones digit? | 6 | 1. **Identify the numbers with 7 as the ones digit under 100**:
The numbers are $7, 17, 27, 37, 47, 57, 67, 77, 87, 97$.
2. **Check each number for primality**:
- $7$ is prime.
- $17$ is prime.
- $27$ is not prime (divisible by $3$).
- $37$ is prime.
- $47$ is prime.
- $57$ is not prime (divisibl... | 1 | 2,479.875 | 2,479.875 | -1 |
The terms of the sequence $(a_i)$ defined by $a_{n + 2} = \frac {a_n + 2009} {1 + a_{n + 1}}$ for $n \ge 1$ are positive integers. Find the minimum possible value of $a_1 + a_2$. | 90 | If $a_{n} \ne \frac {2009}{a_{n+1}}$, then either \[a_{n} = \frac {a_{n}}{1} < \frac {a_{n} + 2009}{1 + a_{n+1}} < \frac {2009}{a_{n+1}}\]
or
\[\frac {2009}{a_{n+1}} < \frac {2009 + a_{n}}{a_{n+1} + 1} < \frac {a_{n}}{1} = a_{n}\]
All the integers between $a_{n}$ and $\frac {2009}{a_{n+1}}$ would be included in the s... | 0.0625 | 7,985.625 | 4,890 | 8,192 |
Let $C$ be a point not on line $AE$ and $D$ a point on line $AE$ such that $CD \perp AE.$ Meanwhile, $B$ is a point on line $CE$ such that $AB \perp CE.$ If $AB = 4,$ $CD = 8,$ and $AE = 5,$ then what is the length of $CE?$ | 10 | 0.5 | 6,376 | 4,560 | 8,192 | |
In the ancient Chinese mathematical masterpiece "The Mathematical Classic of Sunzi" Volume $26$, the $26$th question is: "There is an unknown quantity, when divided by $3$, the remainder is $2$; when divided by $5$, the remainder is $3; when divided by $7$, the remainder is $2$. What is the quantity?" The mathematical ... | \frac{2023}{4050} | 0.25 | 6,751.0625 | 3,752.75 | 7,750.5 | |
Which of the following numbers is not an integer? | $\frac{2014}{4}$ | 0 | 7,098.625 | -1 | 7,098.625 | |
Given 8 teams, of which 3 are weak teams, they are randomly divided into two groups, A and B, with 4 teams in each group. Find:
(1) The probability that one of the groups A or B has exactly 2 weak teams.
(2) The probability that group A has at least 2 weak teams. | \frac{1}{2} | 1 | 4,269.625 | 4,269.625 | -1 | |
If the sum of the binomial coefficients of the odd terms in the expansion of ${(x-\frac{2}{x})}^{n}$ is $16$, then the coefficient of $x^{3}$ in the expansion is ______. | -10 | 0.3125 | 7,439.4375 | 5,783.8 | 8,192 | |
How many ways are there to place $3$ indistinguishable red chips, $3$ indistinguishable blue chips, and $3$ indistinguishable green chips in the squares of a $3 \times 3$ grid so that no two chips of the same color are directly adjacent to each other, either vertically or horizontally? | 36 | We are tasked with placing $3$ indistinguishable red chips, $3$ indistinguishable blue chips, and $3$ indistinguishable green chips in a $3 \times 3$ grid such that no two chips of the same color are directly adjacent either vertically or horizontally.
#### Step 1: Fixing the position of one chip
To simplify the probl... | 0 | 7,991.1875 | -1 | 7,991.1875 |
If α is in the interval (0, π) and $\frac{1}{2}\cos2α = \sin\left(\frac{π}{4} + α\right)$, then find the value of $\sin2α$. | -1 | 0.875 | 6,065.375 | 5,761.571429 | 8,192 | |
A child who does not understand English tries to spell the word "hello" using cards with the letters "e", "o", "h", "l", "l". How many possible incorrect arrangements can there be if the cards cannot be laid horizontally or upside down? | 59 | 0.9375 | 4,477.375 | 4,229.733333 | 8,192 | |
Expand the following expression: $16(2x+5)$ | 32x+80 | 1 | 603.375 | 603.375 | -1 | |
Find the value of $k$ for the ellipse $\frac{x^2}{k+8} + \frac{y^2}{9} = 1$ with an eccentricity of $\frac{1}{2}$. | -\frac{5}{4} | 0.5625 | 6,765.875 | 7,626.111111 | 5,659.857143 | |
For the graph $y = mx + 3$, determine the maximum value of $a$ such that the line does not pass through any lattice points for $0 < x \leq 50$ when $\frac{1}{3} < m < a$. | \frac{17}{51} | 0 | 8,135.625 | -1 | 8,135.625 | |
Twenty people, including \( A, B, \) and \( C \), sit randomly at a round table. What is the probability that at least two of \( A, B, \) and \( C \) sit next to each other? | 17/57 | 0 | 8,192 | -1 | 8,192 | |
In how many ways can 10 people be seated in a row of chairs if four of the people, Alice, Bob, Charlie, and Dana, refuse to sit in four consecutive seats? | 3507840 | 0.875 | 3,750.625 | 3,592.857143 | 4,855 | |
Given two numbers, a and b, are randomly selected within the interval (-π, π), determine the probability that the function f(x) = x^2 + 2ax - b^2 + π has a root. | \dfrac{3}{4} | 0.8125 | 5,276.3125 | 4,603.461538 | 8,192 | |
A mathematician is working on a geospatial software and comes across a representation of a plot's boundary described by the equation $x^2 + y^2 + 8x - 14y + 15 = 0$. To correctly render it on the map, he needs to determine the diameter of this plot. | 10\sqrt{2} | 0.5625 | 987.4375 | 1,039.555556 | 920.428571 | |
Let $S$ be a region in the plane with area 4. When we apply the matrix
\[\begin{pmatrix} 2 & -1 \\ 7 & 2 \end{pmatrix}\]to $S,$ we obtain the region $S'.$ Find the area of $S'.$ | 44 | 1 | 1,402.9375 | 1,402.9375 | -1 | |
$(MON 1)$ Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other ... | 1944 | 0.0625 | 8,047.1875 | 5,875 | 8,192 | |
Let $\alpha, \beta$, and $\gamma$ be three real numbers. Suppose that $\cos \alpha+\cos \beta+\cos \gamma =1$ and $\sin \alpha+\sin \beta+\sin \gamma =1$. Find the smallest possible value of $\cos \alpha$. | \frac{-1-\sqrt{7}}{4} | Let $a=\cos \alpha+i \sin \alpha, b=\cos \beta+i \sin \beta$, and $c=\cos \gamma+i \sin \gamma$. We then have $a+b+c=1+i$ where $a, b, c$ are complex numbers on the unit circle. Now, to minimize $\cos \alpha=\operatorname{Re}[a]$, consider a triangle with vertices $a, 1+i$, and the origin. We want $a$ as far away from ... | 0 | 7,920.9375 | -1 | 7,920.9375 |
All three vertices of $\triangle ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$? | 8 | 1. **Identify the Coordinates of Points on the Parabola**:
Since $\triangle ABC$ has all vertices on the parabola $y = x^2$ and $A$ is at the origin, the coordinates of $A$ are $(0,0)$. Let the coordinates of $B$ and $C$ be $(-a, a^2)$ and $(a, a^2)$ respectively, since $\overline{BC}$ is parallel to the $x$-axis a... | 1 | 2,391.5 | 2,391.5 | -1 |
Triangle $ABC$ is an isosceles triangle with side lengths of 25, 25 and 48 centimeters. What is the area of triangle $ABC$, in square centimeters? | 168 | 0.9375 | 3,227.5625 | 2,896.6 | 8,192 | |
I have 5 red plates and 4 blue plates. If I randomly select two plates to serve dinner on, what is the probability that they're both the same color? | \frac{4}{9} | 1 | 1,515.5 | 1,515.5 | -1 | |
What is the greatest common factor of the numbers 2750 and 9450? | 50 | 0.9375 | 2,875.375 | 2,520.933333 | 8,192 | |
A square with a perimeter of 36 is inscribed in a square with a perimeter of 40. What is the greatest distance between a vertex of the inner square and a vertex of the outer square?
A) $\sqrt{101}$
B) $9\sqrt{2}$
C) $8\sqrt{2}$
D) $\sqrt{90}$
E) $10$ | 9\sqrt{2} | 0 | 8,192 | -1 | 8,192 |
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