problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
In rectangle $ABCD$, $AB=6$ and $AD=8$. Point $M$ is the midpoint of $\overline{AD}$. What is the area of $\triangle AMC$? | 12 |
1. **Understanding the Problem:**
- We have a rectangle $ABCD$ with $AB = 6$ and $AD = 8$.
- $M$ is the midpoint of $\overline{AD}$, so $AM = MD = \frac{AD}{2} = \frac{8}{2} = 4$.
- We need to find the area of $\triangle AMC$.
2. **Using the Triangle Area Formula:**
- The area $A$ of a triangle is given b... | 1 | 3,116.625 | 3,116.625 | -1 |
In a circle, parallel chords of lengths 8, 15, and 17 determine central angles of $\gamma$, $\delta$, and $\gamma + \delta$ radians, respectively, where $\gamma + \delta < \pi$. If $\cos \gamma$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denomi... | 32 | 0 | 6,368.8125 | -1 | 6,368.8125 | |
Solve for $x$: $2(3^x) = 162$. | 4 | 1 | 2,062.4375 | 2,062.4375 | -1 | |
Given the functions $f(x)=x^{2}+px+q$ and $g(x)=x+\frac{1}{x^{2}}$ on the interval $[1,2]$, determine the maximum value of $f(x)$. | 4 - \frac{5}{2} \sqrt[3]{2} + \sqrt[3]{4} | 0 | 7,156.875 | -1 | 7,156.875 | |
Given a right triangle \(ABC\) with a right angle at \(C\), a circle is drawn with diameter \(BC\) of length 26. A tangent \(AP\) from point \(A\) to this circle (distinct from \(AC\)) is drawn. The perpendicular \(PH\) dropped onto segment \(BC\) intersects segment \(AB\) at point \(Q\). Find the area of triangle \(BP... | 24 | 0.125 | 7,860.9375 | 5,728.5 | 8,165.571429 | |
A fair coin is flipped eight times in a row. Let $p$ be the probability that there is exactly one pair of consecutive flips that are both heads and exactly one pair of consecutive flips that are both tails. If $p=\frac{a}{b}$, where $a, b$ are relatively prime positive integers, compute $100a+b$. | 1028 | Separate the sequence of coin flips into alternating blocks of heads and tails. Of the blocks of heads, exactly one block has length 2, and all other blocks have length 1. The same statement applies to blocks of tails. Thus, if there are $k$ blocks in total, there are $k-2$ blocks of length 1 and 2 blocks of length 2, ... | 0 | 8,192 | -1 | 8,192 |
In the sum shown, each of the letters \( D, O, G, C, A \), and \( T \) represents a different digit.
$$
\begin{array}{r}
D O G \\
+C A T \\
\hline 1000
\end{array}
$$
What is the value of \( D + O + G + C + A + T \)? | 28 | 0.75 | 6,741.6875 | 6,258.25 | 8,192 | |
Let $a, b$ be positive reals with $a>b>\frac{1}{2} a$. Place two squares of side lengths $a, b$ next to each other, such that the larger square has lower left corner at $(0,0)$ and the smaller square has lower left corner at $(a, 0)$. Draw the line passing through $(0, a)$ and $(a+b, 0)$. The region in the two squares ... | \frac{5}{3} | Let $t=\frac{a}{b} \in(1,2)$; we will rewrite the sum $a+b$ as a function of $t$. The area condition easily translates to $\frac{a^{2}-a b+2 b^{2}}{2}=2013$, or $b^{2}\left(t^{2}-t+2\right)=4026 \Longleftrightarrow b=\sqrt{\frac{4026}{t^{2}-t+2}}$. Thus $a+b$ is a function $f(t)=(1+t) \sqrt{\frac{4026}{t^{2}-t+2}}$ of ... | 0 | 7,920.375 | -1 | 7,920.375 |
Let $A B$ be a segment of length 2 with midpoint $M$. Consider the circle with center $O$ and radius $r$ that is externally tangent to the circles with diameters $A M$ and $B M$ and internally tangent to the circle with diameter $A B$. Determine the value of $r$. | \frac{1}{3} | Let $X$ be the midpoint of segment $A M$. Note that $O M \perp M X$ and that $M X=\frac{1}{2}$ and $O X=\frac{1}{2}+r$ and $O M=1-r$. Therefore by the Pythagorean theorem, we have $$O M^{2}+M X^{2}=O X^{2} \Longrightarrow(1-r)^{2}+\frac{1}{2^{2}}=\left(\frac{1}{2}+r\right)^{2}$$ which we can easily solve to find that $... | 0.875 | 3,993.3125 | 3,613.357143 | 6,653 |
The angle bisectors \( A L_{1} \) and \( B L_{2} \) of triangle \( A B C \) intersect at point \( I \). It is known that \( A I : I L_{1} = 3 \) and \( B I : I L_{2} = 2 \). Find the ratio of the sides of triangle \( A B C \). | 3:4:5 | 0.25 | 5,680.0625 | 4,091.5 | 6,209.583333 | |
What is the perimeter of $\triangle UVZ$ if $UVWX$ is a rectangle that lies flat on a horizontal floor, a vertical semi-circular wall with diameter $XW$ is constructed, point $Z$ is the highest point on this wall, and $UV=20$ and $VW=30$? | 86 | The perimeter of $\triangle UVZ$ equals $UV+UZ+VZ$.
We know that $UV=20$. We need to calculate $UZ$ and $VZ$.
Let $O$ be the point on $XW$ directly underneath $Z$.
Since $Z$ is the highest point on the semi-circle and $XW$ is the diameter, then $O$ is the centre of the semi-circle.
We join $UO, VO, UZ$, and $VZ... | 0 | 6,132.875 | -1 | 6,132.875 |
Given that the line y=kx+b is tangent to the graph of the function f(x)=1/2 x^2 + ln x, find the minimum value of k-b. | \frac{7}{2} | 1 | 3,612.125 | 3,612.125 | -1 | |
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 | More directly than Solution 2, we have \[DE=BC\left(\frac{h_a-r}{h_a}\right)=20\left(1-\frac{r}{\frac{[ABC]}{\frac{BC}{2}}}\right)=20\left(1-\frac{10r}{sr}\right)=20\left(1-\frac{10}{\frac{63}{2}}\right)=\frac{860}{63}\implies \boxed{923}.\] | 0.125 | 8,031.5625 | 7,664.5 | 8,084 |
Find the number of real zeros of $x^{3}-x^{2}-x+2$. | 1 | Let $f(x)=x^{3}-x^{2}-x+2$, so $f^{\prime}(x)=3 x^{2}-2 x-1$. The slope is zero when $3 x^{2}-2 x-1=0$, where $x=-\frac{1}{3}$ and $x=1$. Now $f\left(\frac{1}{3}\right)>0$ and $f(1)>0$, so there are no zeros between $x=-\frac{1}{3}$ and $x=1$. Since \lim _{x \rightarrow+\infty} f(x)>0$, there are no zeros for $x>1$. Si... | 0.9375 | 5,681.625 | 5,514.266667 | 8,192 |
In the diagram, \( PR \) and \( QS \) meet at \( X \). Also, \(\triangle PQX\) is right-angled at \(Q\) with \(\angle QPX = 62^\circ\) and \(\triangle RXS\) is isosceles with \( RX = SX \) and \(\angle XSR = y^\circ\). The value of \( y \) is: | 76 | 0.25 | 5,256.9375 | 4,051.25 | 5,658.833333 | |
The points $P,$ $Q,$ and $R$ are represented by the complex numbers $z,$ $(1 + i) z,$ and $2 \overline{z},$ respectively, where $|z| = 1.$ When $P,$ $Q$, and $R$ are not collinear, let $S$ be the fourth vertex of the parallelogram $PQSR.$ What is the maximum distance between $S$ and the origin of the complex plane? | 3 | 0.8125 | 5,329.8125 | 4,669.307692 | 8,192 | |
Given an ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with a focal length of $2$, and point $Q( \frac{a^{2}}{ \sqrt{a^{2}-b^{2}}},0)$ on the line $l$: $x=2$.
(1) Find the standard equation of the ellipse $C$;
(2) Let $O$ be the coordinate origin, $P$ a moving point on line $l$, and $l'$ a line ... | \frac{ \sqrt{2}}{2} | 0 | 7,418.8125 | -1 | 7,418.8125 | |
In her school grade, Misha is the 30th highest and also the 40th lowest in her academic performance. How many students are there in total in Misha's grade? | 69 | 0.875 | 292.0625 | 292.928571 | 286 | |
The digits 2, 3, 5 and 7 are arranged randomly to form a four-digit number. What is the probability that the number is odd? Express your answer as a common fraction. | \frac{3}{4} | 1 | 1,454.5 | 1,454.5 | -1 | |
On a ring road, there are three cities: $A$, $B$, and $C$. It is known that the path from $A$ to $C$ along the arc not containing $B$ is three times longer than the path through $B$. The path from $B$ to $C$ along the arc not containing $A$ is four times shorter than the path through $A$. By what factor is the path fro... | 19 | 0.0625 | 6,557.9375 | 4,899 | 6,668.533333 | |
Inside a square with side length 10, two congruent equilateral triangles are drawn such that they share one side and each has one vertex on a vertex of the square. What is the side length of the largest square that can be inscribed in the space inside the square and outside of the triangles?
[asy]
size(100);
pair A, ... | 5-\frac{5\sqrt{3}}{3} | 0 | 8,192 | -1 | 8,192 | |
Let $\{a_{n}\}$ be a sequence with the sum of its first $n$ terms denoted as $S_{n}$, and ${S}_{n}=2{a}_{n}-{2}^{n+1}$. The sequence $\{b_{n}\}$ satisfies ${b}_{n}=log_{2}\frac{{a}_{n}}{n+1}$, where $n\in N^{*}$. Find the maximum real number $m$ such that the inequality $(1+\frac{1}{{b}_{2}})•(1+\frac{1}{{b}_{4}})•⋯•(1... | \frac{3}{4} | 0.0625 | 8,152.625 | 7,562 | 8,192 | |
The sequence ${a_n}$ satisfies $a_1=1$, $a_{n+1} \sqrt { \frac{1}{a_{n}^{2}}+4}=1$. Let $S_{n}=a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}$. If $S_{2n+1}-S_{n}\leqslant \frac{m}{30}$ holds for any $n\in\mathbb{N}^{*}$, find the minimum value of the positive integer $m$. | 10 | 0.125 | 8,164.125 | 7,969 | 8,192 | |
Along an alley, 75 trees consisting of maples and larches were planted in a single row. It is known that there are no two maples with exactly 5 trees between them. What is the maximum number of maples that could have been planted along the alley? | 39 | 0 | 8,192 | -1 | 8,192 | |
Find the greatest common divisor of $8!$ and $(6!)^2.$ | 1440 | 0 | 3,293.5 | -1 | 3,293.5 | |
The digits of a three-digit number form a geometric progression with distinct terms. If this number is decreased by 200, the resulting three-digit number has digits that form an arithmetic progression. Find the original three-digit number. | 842 | 0.5625 | 6,930.8125 | 5,949.888889 | 8,192 | |
Two concentric circles have radii of $12$ meters and $24$ meters respectively. A rabbit starts at point X and runs along a sixth of the circumference of the larger circle, followed by a straight line to the smaller circle, then along a third of the circumference of the smaller circle, and finally across the diameter of... | 16\pi + 36 | 0.125 | 5,488.625 | 5,273.5 | 5,519.357143 | |
Express $\frac{3}{8}$ as a decimal. | 0.375 | 1 | 1,542.625 | 1,542.625 | -1 | |
In quadrilateral $ABCD$ , we have $AB = 5$ , $BC = 6$ , $CD = 5$ , $DA = 4$ , and $\angle ABC = 90^\circ$ . Let $AC$ and $BD$ meet at $E$ . Compute $\dfrac{BE}{ED}$ . | 5/4 | 0 | 8,143.125 | -1 | 8,143.125 | |
Anna thinks of an integer that is not a multiple of three, not a perfect square, and the sum of its digits is a prime number. What could the integer be? | 14 | Since 12 and 21 are multiples of 3 (12 = 4 \times 3 and 21 = 7 \times 3), the answer is not 12 or 21. 16 is a perfect square (16 = 4 \times 4) so the answer is not 16. The sum of the digits of 26 is 8, which is not a prime number, so the answer is not 26. Since 14 is not a multiple of three, 14 is not a perfect square,... | 0 | 5,989.0625 | -1 | 5,989.0625 |
As shown in the figure, triangle $ABC$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of triangle $ABC$. | 315 | This problem can be done using mass points. Assign B a weight of 1 and realize that many of the triangles have the same altitude. After continuously using the formulas that state (The sum of the two weights) = (The middle weight), and (The weight $\times$ side) = (Other weight) $\times$ (The other side), the problem yi... | 0 | 8,089.875 | -1 | 8,089.875 |
There are $270$ students at Colfax Middle School, where the ratio of boys to girls is $5 : 4$. There are $180$ students at Winthrop Middle School, where the ratio of boys to girls is $4 : 5$. The two schools hold a dance and all students from both schools attend. What fraction of the students at the dance are girls? | \frac{22}{45} | 1. **Calculate the number of girls at Colfax Middle School:**
- The total number of students at Colfax Middle School is $270$.
- The ratio of boys to girls at Colfax Middle School is $5:4$. Let the number of boys be $5x$ and the number of girls be $4x$.
- The sum of boys and girls is equal to the total number ... | 0.9375 | 2,792.6875 | 2,432.733333 | 8,192 |
In the diagram, \( P Q = 19 \), \( Q R = 18 \), and \( P R = 17 \). Point \( S \) is on \( P Q \), point \( T \) is on \( P R \), and point \( U \) is on \( S T \) such that \( Q S = S U \) and \( U T = T R \). The perimeter of \(\triangle P S T\) is equal to: | 36 | 0.1875 | 7,983.9375 | 7,082.333333 | 8,192 | |
Given that the sum of the coefficients of the expansion of $(2x-1)^{n}$ is less than the sum of the binomial coefficients of the expansion of $(\sqrt{x}+\frac{1}{2\sqrt[4]{x}})^{2n}$ by $255$.
$(1)$ Find all the rational terms of $x$ in the expansion of $(\sqrt{x}+\frac{1}{2\sqrt[4]{x}})^{2n}$;
$(2)$ If $(2x-1)^{n}... | 81 | 0.0625 | 8,133.75 | 8,192 | 8,129.866667 | |
$J K L M$ is a square. Points $P$ and $Q$ are outside the square such that triangles $J M P$ and $M L Q$ are both equilateral. The size, in degrees, of angle $P Q M$ is | 15 | 0.0625 | 7,493.375 | 6,981 | 7,527.533333 | |
Solve \[\frac{2x+4}{x^2+4x-5}=\frac{2-x}{x-1}\]for $x$. | -6 | 0.9375 | 2,831.0625 | 2,473.666667 | 8,192 | |
Suppose that the angles of triangle $ABC$ satisfy
\[\cos 3A + \cos 3B + \cos 3C = 1.\]Two sides of the triangle have lengths 10 and 13. Find the maximum length of the third side. | \sqrt{399} | 0.125 | 8,001.625 | 6,791.5 | 8,174.5 | |
Find the matrix $\mathbf{M}$ that triples the second row of a matrix. In other words,
\[\mathbf{M} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a & b \\ 3c & 3d \end{pmatrix}.\]If no such matrix $\mathbf{M}$ exists, then enter the zero matrix. | \begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix} | 0.75 | 4,872.9375 | 4,025.833333 | 7,414.25 | |
Given the function $f(x)=\sin (ωx+φ)(ω > 0,|φ|\leqslant \dfrac {π}{2})$, $y=f(x- \dfrac {π}{4})$ is an odd function, $x= \dfrac {π}{4}$ is the symmetric axis of the graph of $y=f(x)$, and $f(x)$ is monotonic in $(\dfrac {π}{14}, \dfrac {13π}{84})$, determine the maximum value of $ω$. | 11 | 0 | 8,192 | -1 | 8,192 | |
China has become the world's largest electric vehicle market. Electric vehicles have significant advantages over traditional vehicles in ensuring energy security and improving air quality. After comparing a certain electric vehicle with a certain fuel vehicle, it was found that the average charging cost per kilometer f... | 5000 | 0.125 | 7,646.4375 | 4,405.5 | 8,109.428571 | |
As shown in the diagram, \(E, F, G, H\) are the midpoints of the sides \(AB, BC, CD, DA\) of the quadrilateral \(ABCD\). The intersection of \(BH\) and \(DE\) is \(M\), and the intersection of \(BG\) and \(DF\) is \(N\). What is \(\frac{S_{\mathrm{BMND}}}{S_{\mathrm{ABCD}}}\)? | 1/3 | 0.0625 | 8,098.5 | 8,192 | 8,092.266667 | |
Given that $α$ and $β$ are both acute angles, and $\cos (α+β)=\dfrac{\sin α}{\sin β}$, the maximum value of $\tan α$ is \_\_\_\_. | \dfrac{ \sqrt{2}}{4} | 0 | 6,414.0625 | -1 | 6,414.0625 | |
Given $(3-2x)^{5}=a_{0}+a_{1}x+a_{2}x^{2}+…+a_{5}x^{5}$, find the value of $a_{0}+a_{1}+2a_{2}+…+5a_{5}$. | 233 | 0.6875 | 5,525.6875 | 4,830.454545 | 7,055.2 | |
In the sequence $\{a_n\}$, if $a_1=-2$ and for any $n\in\mathbb{N}^*$, $a_{n+1}=1+2a_n$, then the sum of the first $10$ terms of the sequence $\{a_n\}$ is ______. | -1033 | 0.875 | 4,738.5625 | 4,245.214286 | 8,192 | |
The diagram shows a polygon made by removing six $2\times 2$ squares from the sides of an $8\times 12$ rectangle. Find the perimeter of this polygon.
 | 52 | 0 | 8,185.5 | -1 | 8,185.5 | |
A regular 201-sided polygon is inscribed inside a circle with center $C$. Triangles are drawn by connecting any three of the 201 vertices of the polygon. How many of these triangles have the point $C$ lying inside the triangle? | 338350 | 0.25 | 7,867.6875 | 6,894.75 | 8,192 | |
How many ways are there to arrange numbers from 1 to 8 in circle in such way the adjacent numbers are coprime?
Note that we consider the case of rotation and turn over as distinct way. | 72 | 0 | 8,192 | -1 | 8,192 | |
In the multiplication problem below $A$, $B$, $C$, $D$ are different digits. What is $A+B$?
$\begin{array}{cccc} & A & B & A\\ \times & & C & D\\ \hline C & D & C & D\\ \end{array}$ | 1 | 1. **Identify the value of $A$:**
Given the multiplication problem:
\[
\begin{array}{cccc}
& A & B & A\\
\times & & C & D\\
\hline
C & D & C & D\\
\end{array}
\]
We observe that the product of $A$ and $D$ results in a number ending in $D$. This implies that $A \times D$ must be a num... | 0.875 | 4,832.5625 | 4,352.642857 | 8,192 |
Centered at each lattice point in the coordinate plane are a circle radius $\frac{1}{10}$ and a square with sides of length $\frac{1}{5}$ whose sides are parallel to the coordinate axes. The line segment from $(0,0)$ to $(1001, 429)$ intersects $m$ of the squares and $n$ of the circles. Find $m + n$.
| 574 | 0 | 7,433.375 | -1 | 7,433.375 | |
An urn initially contains two red balls and one blue ball. George undertakes the operation of randomly drawing a ball and then adding two more balls of the same color from a box into the urn. This operation is done three times. After these operations, the urn has a total of nine balls. What is the probability that ther... | \frac{3}{10} | 0 | 8,192 | -1 | 8,192 | |
The expression $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. Find the value of $$\left\lfloor\frac{2002!}{2001!+2000!+1999!+\cdots+1!}\right\rfloor.$$ | 2000 | 2000 We break up 2002! = 2002(2001)! as $$2000(2001!)+2 \cdot 2001(2000!)=2000(2001!)+2000(2000!)+2002 \cdot 2000(1999!) >2000(2001!+2000!+1999!+\cdots+1!)$$ On the other hand, $$2001(2001!+2000!+\cdots+1!)>2001(2001!+2000!)=2001(2001!)+2001!=2002!$$ Thus we have $2000<2002!/(2001!+\cdots+1!)<2001$, so the answer is 20... | 0.0625 | 7,858.375 | 5,598 | 8,009.066667 |
Let \(P\) and \(P+2\) be both prime numbers satisfying \(P(P+2) \leq 2007\). If \(S\) represents the sum of such possible values of \(P\), find the value of \(S\). | 106 | 0.9375 | 4,092 | 4,098.866667 | 3,989 | |
Find the maximum constant \( k \) such that for \( x, y, z \in \mathbb{R}_+ \), the following inequality holds:
$$
\sum \frac{x}{\sqrt{y+z}} \geqslant k \sqrt{\sum x},
$$
where " \( \sum \) " denotes a cyclic sum. | \sqrt{\frac{3}{2}} | 0 | 8,019.375 | -1 | 8,019.375 | |
The graph of $y = f(x)$ is shown below.
[asy]
unitsize(0.5 cm);
real func(real x) {
real y;
if (x >= -3 && x <= 0) {y = -2 - x;}
if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}
if (x >= 2 && x <= 3) {y = 2*(x - 2);}
return(y);
}
int i, n;
for (i = -5; i <= 5; ++i) {
draw((i,-5)--(i,5),gray(0.7));
... | \text{D} | 0 | 4,556.75 | -1 | 4,556.75 | |
Triangle $ABC$ has side lengths $AB = 12$, $BC = 25$, and $CA = 17$. Rectangle $PQRS$ has vertex $P$ on $\overline{AB}$, vertex $Q$ on $\overline{AC}$, and vertices $R$ and $S$ on $\overline{BC}$. In terms of the side length $PQ = \omega$, the area of $PQRS$ can be expressed as the quadratic polynomial \[Area(PQRS) = \... | 161 | Proceed as in solution 1. When $\omega$ is equal to zero, $\alpha - \beta\omega=\alpha$ is equal to the altitude. This means that $25\beta$ is equal to $\frac{36}{5}$, so $\beta = \frac{36}{125}$, yielding $\boxed{161}$. | 0.8125 | 6,250.3125 | 5,802.230769 | 8,192 |
A dragon has 40 piles of gold coins, with the number of coins in any two piles differing. After the dragon plundered a neighboring city and brought back more gold, the number of coins in each pile increased by either 2, 3, or 4 times. What is the minimum number of different piles of coins that could result? | 14 | 0 | 8,192 | -1 | 8,192 | |
A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arit... | 34 | 1. **Define the sequences**: Let the quarterly scores for the Raiders be $a, ar, ar^2, ar^3$ and for the Wildcats be $b, b+d, b+2d, b+3d$. Given that the sequences are increasing and the Raiders' sequence is geometric while the Wildcats' sequence is arithmetic.
2. **Initial conditions**: The game was tied at the end o... | 0.625 | 5,597.4375 | 4,392.4 | 7,605.833333 |
An ant starts out at $(0,0)$. Each second, if it is currently at the square $(x, y)$, it can move to $(x-1, y-1),(x-1, y+1),(x+1, y-1)$, or $(x+1, y+1)$. In how many ways can it end up at $(2010,2010)$ after 4020 seconds? | $\binom{4020}{1005}^{2}$ | Note that each of the coordinates either increases or decreases the x and y coordinates by 1. In order to reach 2010 after 4020 steps, each of the coordinates must be increased 3015 times and decreased 1005 times. A permutation of 3015 plusses and 1005 minuses for each of $x$ and $y$ uniquely corresponds to a path the ... | 0 | 8,041.625 | -1 | 8,041.625 |
Given that $40\%$ of students initially answered "Yes", $40\%$ answered "No", and $20\%$ were "Undecided", and $60\%$ answered "Yes" after a semester, $30\%$ answered "No", and $10\%$ remained "Undecided", determine the difference between the maximum and minimum possible values of $y\%$ of students who changed their an... | 40\% | 0 | 7,905.6875 | -1 | 7,905.6875 | |
Maria ordered a certain number of televisions for the stock of a large store, paying R\$ 1994.00 per television. She noticed that in the total amount to be paid, the digits 0, 7, 8, and 9 do not appear. What is the smallest number of televisions she could have ordered? | 56 | 0.125 | 7,665.4375 | 5,410 | 7,987.642857 | |
If $P$ is the product of $n$ quantities in Geometric Progression, $S$ their sum, and $S'$ the sum of their reciprocals, then $P$ in terms of $S, S'$, and $n$ is | $(S/S')^{\frac{1}{2}n}$ | To solve the problem, we first need to understand the properties of a geometric progression (GP). Let the first term of the GP be $a$ and the common ratio be $r$. Then the $n$ terms of the GP are $a, ar, ar^2, \ldots, ar^{n-1}$.
1. **Calculate the Product $P$:**
The product of the terms in the GP is:
\[
P = a... | 0 | 8,129.25 | -1 | 8,129.25 |
In a 4 by 4 grid, each of the 16 small squares measures 3 cm by 3 cm and is shaded. Four unshaded circles are then placed on top of the grid as shown. The area of the visible shaded region can be written in the form $A-B\pi$ square cm. What is the value $A+B$? | 180 | 0.0625 | 5,171.3125 | 3,848 | 5,259.533333 | |
Liam builds two snowmen using snowballs of radii 4 inches, 6 inches, and 8 inches for the first snowman. For the second snowman, he uses snowballs that are 75% of the size of each corresponding ball in the first snowman. Assuming all snowballs are perfectly spherical, what is the total volume of snow used in cubic inch... | \frac{4504.5}{3}\pi | 0 | 4,449.4375 | -1 | 4,449.4375 | |
There are five unmarked envelopes on a table, each containing a letter for a different person. If the mail is randomly distributed to these five people, with each person getting one letter, what is the probability that exactly three people receive the correct letter? | \frac{1}{12} | 0.9375 | 3,916.5625 | 3,631.533333 | 8,192 | |
A three-digit number \( \mathrm{abc} \) divided by the sum of its digits leaves a remainder of 1. The three-digit number \( \mathrm{cba} \) divided by the sum of its digits also leaves a remainder of 1. If different letters represent different digits and \( a > c \), then \( \overline{\mathrm{abc}} = \) ____. | 452 | 0.1875 | 7,987.875 | 7,103.333333 | 8,192 | |
900 cards are inscribed with all natural numbers from 1 to 900. Cards inscribed with squares of integers are removed, and the remaining cards are renumbered starting from 1.
Then, the operation of removing the squares is repeated. How many times must this operation be repeated to remove all the cards? | 59 | 0.3125 | 8,123.9375 | 7,974.2 | 8,192 | |
Semicircles of diameter 3 inches are lined up as shown. What is the area, in square inches, of the shaded region in an 18-inch length of this pattern? Express your answer in terms of \(\pi\). | \frac{27}{4}\pi | 0.9375 | 5,164 | 4,962.133333 | 8,192 | |
Given that Let \\(S_{n}\\) and \\(T_{n}\\) be the sums of the first \\(n\\) terms of the arithmetic sequences \\(\{a_{n}\}\\) and \\(\{b_{n}\}\\), respectively, and \\( \frac {S_{n}}{T_{n}}= \frac {n}{2n+1} (n∈N^{*})\\), determine the value of \\( \frac {a_{6}}{b_{6}}\\). | \frac{11}{23} | 0.8125 | 4,483.5625 | 3,627.769231 | 8,192 | |
A set of 36 square blocks is arranged into a 6 × 6 square. How many different combinations of 4 blocks can be selected from that set so that no two blocks are in the same row or column? | 5400 | 0.9375 | 4,282.25 | 4,021.6 | 8,192 | |
Determine the volume of the right rectangular parallelepiped whose edges are formed by the distances from the orthocenter to the vertices of a triangle, where the radius of the circumcircle $r = 2.35$ and the angles are: $\alpha = 63^{\circ} 18^{\prime} 13^{\prime \prime}, \beta = 51^{\circ} 42^{\prime} 19^{\prime \pri... | 12.2 | 0 | 8,023.6875 | -1 | 8,023.6875 | |
At a circular table, there are 5 people seated: Arnaldo, Bernaldo, Cernaldo, Dernaldo, and Ernaldo, each in a chair. Analyzing clockwise, we have:
I. There is 1 empty chair between Arnaldo and Bernaldo;
II. There are 5 chairs between Bernaldo and Cernaldo;
III. There are 4 chairs between Dernaldo and Ernaldo, almost... | 12 | 0.125 | 7,227.875 | 5,020 | 7,543.285714 | |
Evaluate the infinite geometric series: $$\frac{4}{3} - \frac{3}{4} + \frac{9}{16} - \frac{27}{64} + \dots$$ | \frac{64}{75} | 0.0625 | 7,553.875 | 1,636 | 7,948.4 | |
If \( n \) is a positive integer such that \( n^{6} + 206 \) is divisible by \( n^{2} + 2 \), find the sum of all possible values of \( n \). | 32 | 0.9375 | 5,229.4375 | 5,031.933333 | 8,192 | |
Find the number of positive integers with three not necessarily distinct digits, $abc$, with $a \neq 0$ and $c \neq 0$ such that both $abc$ and $cba$ are multiples of $4$. | 40 | A number is divisible by four if its last two digits are divisible by 4. Thus, we require that $10b + a$ and $10b + c$ are both divisible by $4$. If $b$ is odd, then $a$ and $c$ must both be $2 \pmod 4$ meaning that $a$ and $c$ are $2$ or $6$. If $b$ is even, then $a$ and $c$ must be $0 \pmod 4$ meaning that $a$ and $c... | 0.125 | 8,192 | 8,192 | 8,192 |
Each of the nine dots in this figure is to be colored red, white or blue. No two dots connected by a segment (with no other dots between) may be the same color. How many ways are there to color the dots of this figure?
[asy]
draw((-75,0)--(-45,0)--(-60,26)--cycle);
draw((0,0)--(30,0)--(15,26)--cycle);
draw((75,0)--(10... | 54 | 0 | 8,096.25 | -1 | 8,096.25 | |
Let $p(x)$ be a monic quartic polynomial such that $p(1) = 2,$ $p(2) = 5,$ $p(3) = 10,$ and $p(4) = 17.$ Find $p(5).$ | 50 | 0.6875 | 4,907.125 | 3,414 | 8,192 | |
Given a fixed point A (3, 4), and point P is a moving point on the parabola $y^2=4x$, the distance from point P to the line $x=-1$ is denoted as $d$. Find the minimum value of $|PA|+d$. | 2\sqrt{5} | 0.5 | 8,047.25 | 7,902.5 | 8,192 | |
Homewood Middle School has 1200 students, and 730 of these students attend a summer picnic. If two-thirds of the girls in the school and one-half of the boys in the school attend the picnic, how many girls attend the picnic? (Assume that each student in the school is either a boy or a girl.) | 520 | 1 | 2,515.875 | 2,515.875 | -1 | |
For $x$ a real number, let $f(x)=0$ if $x<1$ and $f(x)=2 x-2$ if $x \geq 1$. How many solutions are there to the equation $f(f(f(f(x))))=x ?$ | 2 | Certainly 0,2 are fixed points of $f$ and therefore solutions. On the other hand, there can be no solutions for $x<0$, since $f$ is nonnegative-valued; for $0<x<2$, we have $0 \leq f(x)<x<2$ (and $f(0)=0$ ), so iteration only produces values below $x$, and for $x>2, f(x)>x$, and iteration produces higher values. So the... | 0.1875 | 8,118.625 | 7,800.666667 | 8,192 |
Find all square numbers $ S_1$ and $ S_2$ such that $ S_1 \minus{} S_2 \equal{} 1989.$ | $ (S_1,S_2)\in \{ (995^2,994^2), (333^2,330^2), (115^2,106^2), (83^2, 70^2), (67^2,50^2), (45^2, 6^2)\}$ |
Given the equation \( S_1 - S_2 = 1989 \), where \( S_1 \) and \( S_2 \) are square numbers, we seek to find all such pairs \((S_1, S_2)\). Let \( S_1 = a^2 \) and \( S_2 = b^2 \), where \( a > b \) are integers. Thus, we have:
\[
a^2 - b^2 = 1989.
\]
This can be factored using the difference of squares:
\[
(a - b)... | 0 | 5,273.75 | -1 | 5,273.75 |
Find all real numbers \( x \) that satisfy the equation
$$
\frac{x-2020}{1} + \frac{x-2019}{2} + \cdots + \frac{x-2000}{21} = \frac{x-1}{2020} + \frac{x-2}{2019} + \cdots + \frac{x-21}{2000},
$$
and simplify your answer(s) as much as possible. Justify your solution. | 2021 | 0.375 | 7,025.8125 | 5,368.833333 | 8,020 | |
Given the numbers: $8, a, b, 26, x$, where each of the first four numbers is the average of the two adjacent numbers, find the value of $x$. | 32 | 0.0625 | 4,546.375 | 1,022 | 4,781.333333 | |
The volume of a cylinder is $54\pi$ $\text{cm}^3$. How many cubic centimeters are in the volume of a cone with the same radius and height as the cylinder? Express your answer in terms of $\pi$. [asy]
import solids; currentprojection=orthographic(0,100,25); defaultpen(linewidth(0.8));
revolution cyl = cylinder((5,0,0),1... | 18\pi | 1 | 882.1875 | 882.1875 | -1 | |
Determine the number of ways to arrange the letters of the word PROOF. | 60 | 0.9375 | 1,685.125 | 1,526.333333 | 4,067 | |
Find the minimum value of the function $f(x)=\sum_{n=1}^{19}{|x-n|}$. | 90 | 0.8125 | 5,390.4375 | 4,743.923077 | 8,192 | |
Find all values of $x$ with $0 \le x < \pi$ that satisfy $\sin x - \cos x = 1$. Enter all the solutions, separated by commas. | \frac{\pi}{2} | 0.8125 | 5,213.625 | 5,063.384615 | 5,864.666667 | |
An $E$-shape is a geometric figure in the two-dimensional plane consisting of three rays pointing in the same direction, along with a line segment such that the endpoints of the rays all lie on the segment, the segment is perpendicular to all three rays, both endpoints of the segment are endpoints of rays. Suppose two ... | 11 | Define a $C$-shape to be an $E$-shape without the middle ray. Then, an $E$-shape consists of a ray and a $C$-shape. Two $C$-shapes can intersect at most 6 times, a $C$-shape and a ray can intersect at most 2 times, and two rays can intersect at most 1 time. Thus, the number of intersections of two $E$-shapes is at most... | 0.0625 | 7,245.75 | 5,361 | 7,371.4 |
The punch machines from before the flood punch some or even all of the nine numbered fields of a ticket. The inspectors request from the machine setter that the machine should not punch the same fields if someone places their ticket in reverse, instead of the prescribed orientation. How many such settings are possible ... | 448 | 0.125 | 6,176.25 | 6,608.5 | 6,114.5 | |
Kevin colors a ninja star on a piece of graph paper where each small square has area $1$ square inch. Find the area of the region colored, in square inches.
 | 12 | 0.0625 | 5,716.875 | 3,872 | 5,839.866667 | |
Kolya's parents give him pocket money once a month based on the following criteria: for each A in math, he gets 100 rubles; for each B, he gets 50 rubles; for each C, they subtract 50 rubles; for each D, they subtract 200 rubles. If the total amount is negative, Kolya gets nothing. The math teacher assigns the quarterl... | 250 | 0 | 7,808.3125 | -1 | 7,808.3125 | |
Simplify $(2x^3)^3$. | 8x^9 | 1 | 1,594 | 1,594 | -1 | |
Let $\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. Determine the number of two-digit values of $x$ for which $\clubsuit(\clubsuit(x))=4$. | 10 | 1 | 4,394.875 | 4,394.875 | -1 | |
A 5 by 5 grid of unit squares is partitioned into 5 pairwise incongruent rectangles with sides lying on the gridlines. Find the maximum possible value of the product of their areas. | 2304 | The greatest possible value for the product is $3 \cdot 4 \cdot 4 \cdot 6 \cdot 8=2304$, achieved when the rectangles are $3 \times 1,1 \times 4,2 \times 2,2 \times 3,4 \times 2$. To see that this is possible, orient these rectangles so that the first number is the horizontal dimension and the second number is the vert... | 0 | 8,088.6875 | -1 | 8,088.6875 |
Define: For any three-digit natural number $m$, if $m$ satisfies that the tens digit is $1$ greater than the hundreds digit, and the units digit is $1$ greater than the tens digit, then this three-digit number is called an "upward number"; for any three-digit natural number $n$, if $n$ satisfies that the tens digit is ... | 531 | 0.125 | 7,568.3125 | 5,856.5 | 7,812.857143 | |
There are several white rabbits and gray rabbits. When 6 white rabbits and 4 gray rabbits are placed in a cage, there are still 9 more white rabbits remaining, and all the gray rabbits are placed. When 9 white rabbits and 4 gray rabbits are placed in a cage, all the white rabbits are placed, and there are still 16 gray... | 159 | 0 | 3,852.5 | -1 | 3,852.5 | |
For a positive integer $N$, we color the positive divisors of $N$ (including 1 and $N$ ) with four colors. A coloring is called multichromatic if whenever $a, b$ and $\operatorname{gcd}(a, b)$ are pairwise distinct divisors of $N$, then they have pairwise distinct colors. What is the maximum possible number of multichr... | 192 | First, we show that $N$ cannot have three distinct prime divisors. For the sake of contradiction, suppose $p q r \mid N$ for three distinct primes $p, q, r$. Then by the problem statement, $(p, q, 1),(p, r, 1)$, and $(q, r, 1)$ have three distinct colors, so $(p, q, r, 1)$ has four distinct colors. In addition, $(p q, ... | 0 | 8,192 | -1 | 8,192 |
New this year at HMNT: the exciting game of $R N G$ baseball! In RNG baseball, a team of infinitely many people play on a square field, with a base at each vertex; in particular, one of the bases is called the home base. Every turn, a new player stands at home base and chooses a number $n$ uniformly at random from \{0,... | \frac{409}{125} | For $i=0,1,2,3$, let $P_{i}$ be the probability that a player on the $i$-th base scores a point before strikeout (with zeroth base being the home base). We have the following equations: $$\begin{aligned} P_{0} & =\frac{1}{5}\left(P_{1}+P_{2}+P_{3}+1\right) \\ P_{1} & =\frac{1}{5}\left(P_{2}+P_{3}+1+1\right) \\ P_{2} & ... | 0 | 8,027.5625 | -1 | 8,027.5625 |
Use the method of random simulation experiments to estimate the probability of having exactly two days of rain in three days: First, use a calculator to generate random integers between \\(0\\) and \\(9\\), with \\(1\\), \\(2\\), \\(3\\), \\(4\\) representing rain, and \\(5\\), \\(6\\), \\(7\\), \\(8\\), \\(9\\), \\(0\... | 0.25 | 0.125 | 5,554.6875 | 5,688.5 | 5,535.571429 | |
What is the largest possible value of \(| |a_1 - a_2| - a_3| - \ldots - a_{1990}|\), where \(a_1, a_2, \ldots, a_{1990}\) is a permutation of \(1, 2, 3, \ldots, 1990\)? | 1989 | 0 | 8,192 | -1 | 8,192 | |
Given an ellipse with its focus on the $y$-axis $\frac{x^2}{m^2} + \frac{y^2}{4} = 1$ ($m > 0$) and eccentricity $e = \frac{1}{2}$, where $A$ is the right vertex of the ellipse and $P$ is any point on the ellipse. Find the maximum value of $|PA|$. | 2\sqrt{3} | 0.0625 | 6,572 | 6,477 | 6,578.333333 |
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