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 |
|---|---|---|---|---|---|---|
How many three-digit whole numbers have no 7's and no 9's as digits? | 448 | 0.875 | 3,091.4375 | 2,362.785714 | 8,192 | |
Given a parallelogram \(ABCD\) with \(\angle B = 60^\circ\). Point \(O\) is the center of the circumcircle of triangle \(ABC\). Line \(BO\) intersects the bisector of the exterior angle \(\angle D\) at point \(E\). Find the ratio \(\frac{BO}{OE}\). | 1/2 | 0 | 8,192 | -1 | 8,192 | |
A lateral face of a regular triangular pyramid $SABC$ is inclined to the base plane $ABC$ at an angle $\alpha = \operatorname{arctg} \frac{3}{4}$. Points $M, N, K$ are midpoints of the sides of the base $ABC$. The triangle $MNK$ serves as the lower base of a rectangular prism. The edges of the upper base of the prism i... | 16 | 0 | 8,192 | -1 | 8,192 | |
Solve for $n$: $|n + 6| = 2 - n$. | -2 | 1 | 1,973.1875 | 1,973.1875 | -1 | |
What is $\frac{2+4+6}{1+3+5} - \frac{1+3+5}{2+4+6}$? | \frac{7}{12} | 1. **Calculate the sums in the numerators and denominators:**
- The sum of the numerators in the first fraction: $2 + 4 + 6 = 12$.
- The sum of the denominators in the first fraction: $1 + 3 + 5 = 9$.
- The sum of the numerators in the second fraction: $1 + 3 + 5 = 9$.
- The sum of the denominators in the s... | 1 | 1,992.4375 | 1,992.4375 | -1 |
What is the minimum possible value for $y$ in the equation $y = x^2 + 12x + 5$? | -31 | 1 | 2,149.8125 | 2,149.8125 | -1 | |
Van was asked to add 46 to 37 and round the result to the nearest ten. He figured that since he was going to have to round anyway, he might as well do it before adding the numbers. He rounded 46 to 50 and 37 to 40, thus getting an answer of $50+40=90$. This answer is not correct. What is the correct answer? | 80 | 0.75 | 2,550.25 | 2,065.583333 | 4,004.25 | |
A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? | \sqrt{40} | 0.0625 | 3,297.9375 | 2,085 | 3,378.8 | |
If $a(x+b)=3 x+12$ for all values of $x$, what is the value of $a+b$? | 7 | Since $a(x+b)=3 x+12$ for all $x$, then $a x+a b=3 x+12$ for all $x$. Since the equation is true for all $x$, then the coefficients on the left side must match the coefficients on the right side. Therefore, $a=3$ and $a b=12$, which gives $3 b=12$ or $b=4$. Finally, $a+b=3+4=7$. | 1 | 490.5 | 490.5 | -1 |
A train travelling at constant speed takes five seconds to pass completely through a tunnel which is $85 \mathrm{~m}$ long, and eight seconds to pass completely through a second tunnel which is $160 \mathrm{~m}$ long. What is the speed of the train? | 25 | 0.875 | 2,035.0625 | 2,194.142857 | 921.5 | |
Nine positive integers $a_1,a_2,...,a_9$ have their last $2$ -digit part equal to $11,12,13,14,15,16,17,18$ and $19$ respectively. Find the last $2$ -digit part of the sum of their squares. | 85 | 1 | 4,011.125 | 4,011.125 | -1 | |
Given that the sequence $\left\{a_{n}\right\}$ has a period of 7 and the sequence $\left\{b_{n}\right\}$ has a period of 13, determine the maximum value of $k$ such that there exist $k$ consecutive terms satisfying
\[ a_{1} = b_{1}, \; a_{2} = b_{2}, \; \cdots , \; a_{k} = b_{k} \] | 91 | 0.1875 | 7,422.25 | 5,127.666667 | 7,951.769231 | |
Bicycle license plates in Flatville each contain three letters. The first is chosen from the set $\{C,H,L,P,R\},$ the second from $\{A,I,O\},$ and the third from $\{D,M,N,T\}.$
When Flatville needed more license plates, they added two new letters. The new letters may both be added to one set or one letter may be added... | 40 | 0.8125 | 5,063.875 | 4,668.076923 | 6,779 | |
There is a graph with 30 vertices. If any of 26 of its vertices with their outgoiing edges are deleted, then the remained graph is a connected graph with 4 vertices.
What is the smallest number of the edges in the initial graph with 30 vertices? | 405 | 0 | 7,939.375 | -1 | 7,939.375 | |
What is the sum of the first ten positive multiples of $13$? | 715 | 1 | 1,966.0625 | 1,966.0625 | -1 | |
Given that $x > 0$, $y > 0$, and $x + 2y = 2$, find the minimum value of $xy$. | \frac{1}{2} | 0 | 7,381.0625 | -1 | 7,381.0625 | |
Given the sets:
$$
U = \{1, 2, 3, 4, 5\}, \quad I = \{X \mid X \subseteq U\}
$$
If two different elements \( A \) and \( B \) are randomly selected from the set \( I \), what is the probability that \( A \cap B \) contains exactly three elements? | 5/62 | 0.125 | 7,364.875 | 4,842.5 | 7,725.214286 | |
A barrel with a height of 1.5 meters is completely filled with water and sealed with a lid. The mass of the water in the barrel is 1000 kg. A long, thin vertical tube with a cross-section of $1 \mathrm{~cm}^{2}$ is inserted into the lid of the barrel and completely filled with water. Find the length of the tube if it i... | 1.5 | 0.625 | 5,810.0625 | 4,380.9 | 8,192 | |
Let the rational number $p/q$ be closest to but not equal to $22/7$ among all rational numbers with denominator $< 100$ . What is the value of $p - 3q$ ? | 14 | 0.0625 | 8,179.25 | 7,988 | 8,192 | |
Calculate the number of five-digit numbers formed from the digits 1, 2, 3, 4, 5 without repeating any digits, and where the digits 1 and 2 are not adjacent to the digit 5. | 36 | 0.4375 | 6,949.6875 | 5,352.428571 | 8,192 | |
At the end of the year 2000, Steve had $\$100$ and Wayne had $\$10000$. At the end of each following year, Steve had twice as much money as he did at the end of the previous year and Wayne had half as much money as he did at the end of the previous year. At the end of which year did Steve have more money than Wayne for... | 2004 | We make a table of the total amount of money that each of Steve and Wayne have at the end of each year. After the year 2000, each entry in Steve's column is found by doubling the previous entry and each entry in Wayne's column is found by dividing the previous entry by 2. We stop when the entry in Steve's column is lar... | 0.9375 | 4,329.75 | 4,072.266667 | 8,192 |
Given the function \( f(x) = 5(x+1)^{2} + \frac{a}{(x+1)^{5}} \) for \( a > 0 \), find the minimum value of \( a \) such that \( f(x) \geqslant 24 \) when \( x \geqslant 0 \). | 2 \sqrt{\left(\frac{24}{7}\right)^7} | 0 | 8,155.4375 | -1 | 8,155.4375 | |
Given the ranges $-6 \leq x \leq -2$ and $0 \leq y \leq 4$, determine the largest possible value of the expression $\frac{x+y}{x}$. | \frac{1}{3} | 0 | 4,378.125 | -1 | 4,378.125 | |
The lighting power increased by
\[ \Delta N = N_{\text {after}} - N_{\text {before}} = 300\, \text{BT} - 240\, \text{BT} = 60\, \text{BT} \] | 60 | 0.5625 | 1,348.6875 | 1,141.111111 | 1,615.571429 | |
If $a$ and $b$ are the roots of the equation $x^2-2cx-5d = 0$ , $c$ and $d$ are the roots of the equation $x^2-2ax-5b=0$ , where $a,b,c,d$ are distinct real numbers, what is $a+b+c+d$ ? | 30 | 0.6875 | 7,247.375 | 6,818 | 8,192 | |
A circle has radius $52$ and center $O$ . Points $A$ is on the circle, and point $P$ on $\overline{OA}$ satisfies $OP = 28$ . Point $Q$ is constructed such that $QA = QP = 15$ , and point $B$ is constructed on the circle so that $Q$ is on $\overline{OB}$ . Find $QB$ .
*Proposed by Justin Hsieh* | 11 | 0.9375 | 4,229.5 | 3,965.333333 | 8,192 | |
Circle $\omega_1$ with radius 3 is inscribed in a strip $S$ having border lines $a$ and $b$ . Circle $\omega_2$ within $S$ with radius 2 is tangent externally to circle $\omega_1$ and is also tangent to line $a$ . Circle $\omega_3$ within $S$ is tangent externally to both circles $\omega_1$ and $... | \frac{9}{8} | 0.6875 | 5,943.125 | 5,065.272727 | 7,874.4 | |
Let $x,$ $y,$ and $z$ be positive real numbers satisfying the system of equations: \begin{align*} \sqrt{2x-xy} + \sqrt{2y-xy} &= 1 \\ \sqrt{2y-yz} + \sqrt{2z-yz} &= \sqrt2 \\ \sqrt{2z-zx} + \sqrt{2x-zx} &= \sqrt3. \end{align*} Then $\left[ (1-x)(1-y)(1-z) \right]^2$ can be written as $\frac{m}{n},$ where $m$ and $n$ ar... | 33 | Let $1-x=a;1-y=b;1-z=c$, rewrite those equations
$\sqrt{(1-a)(1+b)}+\sqrt{(1+a)(1-b)}=1$;
$\sqrt{(1-b)(1+c)}+\sqrt{(1+b)(1-c)}=\sqrt{2}$
$\sqrt{(1-a)(1+c)}+\sqrt{(1-c)(1+a)}=\sqrt{3}$
square both sides, get three equations:
$2ab-1=2\sqrt{(1-a^2)(1-b^2)}$
$2bc=2\sqrt{(1-b^2)(1-c^2)}$
$2ac+1=2\sqrt{(1-c^2)(1-a^2)}$
Get... | 0 | 8,192 | -1 | 8,192 |
The diagonals of a rhombus are $10$ inches and $24$ inches. What is the perimeter of the rhombus, in inches? | 52 | 1 | 1,246.9375 | 1,246.9375 | -1 | |
Points \( P \) and \( Q \) are located on the sides \( AB \) and \( AC \) of triangle \( ABC \) such that \( AP:PB = 1:4 \) and \( AQ:QC = 3:1 \). Point \( M \) is chosen randomly on side \( BC \). Find the probability that the area of triangle \( ABC \) exceeds the area of triangle \( PQM \) by no more than two times.... | 13/40 | 0.8125 | 5,298.75 | 5,143.384615 | 5,972 | |
Give an example of an expression consisting of ones, parentheses, the symbols "+", and "×" such that:
- Its value is equal to 11
- If in this expression all "+" signs are replaced with "×" signs, and all "×" signs are replaced with "+" signs, it will still result in 11. | 1+1+1+1+1+1+1+1+1+1+1 | 0 | 8,192 | -1 | 8,192 | |
If $N$, written in base $2$, is $11000$, the integer immediately preceding $N$, written in base $2$, is: | 10111 | To find the integer immediately preceding $N$ in binary, we need to subtract $1$ from $N$.
1. **Convert $N$ from binary to decimal**:
The binary number $11000_2$ can be converted to decimal as follows:
\[
1 \cdot 2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 0 \cdot 2^0 = 16 + 8 + 0 + 0 + 0 = 24
\]
2. ... | 0.875 | 3,918.6875 | 3,788.428571 | 4,830.5 |
Given that $F$ is the right focus of the hyperbola $C: x^{2}- \frac {y^{2}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,6 \sqrt {6})$, when the perimeter of $\triangle APF$ is minimized, the ordinate of point $P$ is ______. | 2 \sqrt {6} | 0 | 8,021 | -1 | 8,021 | |
In triangle \(ABC\), the height \(BD\) is equal to 11.2 and the height \(AE\) is equal to 12. Point \(E\) lies on side \(BC\) and \(BE : EC = 5 : 9\). Find side \(AC\). | 15 | 0.3125 | 7,318.875 | 5,398 | 8,192 | |
Determine all integers $k$ such that there exists infinitely many positive integers $n$ [b]not[/b] satisfying
\[n+k |\binom{2n}{n}\] | k \neq 1 |
Determine all integers \( k \) such that there exist infinitely many positive integers \( n \) not satisfying
\[
n + k \mid \binom{2n}{n}.
\]
We claim that all integers \( k \neq 1 \) satisfy the desired property.
First, recall that \(\frac{1}{n + 1} \binom{2n}{n}\) is the \( n \)-th Catalan number. Since the Catal... | 0 | 8,192 | -1 | 8,192 |
The function $f(x)$ satisfies
\[f(xy) = xf(y)\]for all real numbers $x$ and $y.$ If $f(1) = 25,$ find $f(79).$ | 1975 | 1 | 1,718 | 1,718 | -1 | |
A right circular cone is sliced into five pieces by planes parallel to its base. All of these pieces have the same height. What is the ratio of the volume of the third-largest piece to the volume of the largest piece? | \frac{19}{61} | 0.3125 | 7,402.8125 | 6,538.6 | 7,795.636364 | |
If $m$ and $n$ are the roots of $x^2+mx+n=0$, $m \ne 0$, $n \ne 0$, then the sum of the roots is: | -1 | 1. **Identify the coefficients and roots of the polynomial:**
Given the quadratic equation $x^2 + mx + n = 0$, where $m$ and $n$ are the roots of the equation.
2. **Apply Vieta's formulas:**
Vieta's formulas state that for a quadratic equation $ax^2 + bx + c = 0$ with roots $r$ and $s$, the sum of the roots ... | 1 | 2,365.0625 | 2,365.0625 | -1 |
Queen High School has $1500$ students, and each student takes $6$ classes per day. Each teacher teaches $5$ classes, with each class having $25$ students and $1$ teacher. How many teachers are there at Queen High School? | 72 | 0.9375 | 3,303.875 | 3,260.133333 | 3,960 | |
Let \[f(x) = \left\{
\begin{array}{cl}
-x - 3 & \text{if } x \le 1, \\
\frac{x}{2} + 1 & \text{if } x > 1.
\end{array}
\right.\]Find the sum of all values of $x$ such that $f(x) = 0$. | -3 | 1 | 2,588.625 | 2,588.625 | -1 | |
Observing the equations:<br/>$1\times 3+1=4=2^{2}$;<br/>$2\times 4+1=9=3^{2}$;<br/>$3\times 5+1=16=4^{2}$;<br/>$4\times 6+1=25=5^{2}$;<br/>$\ldots $<br/>$(1)7\times 9+1=( $______)$^{2}$;<br/>$(2)$ Using the pattern you discovered, calculate: $(1+\frac{1}{1×3})×(1+\frac{1}{2×4})×(1+\frac{1}{3×5})×⋅⋅⋅×(1+\frac{1}{198×200... | \frac{199}{100} | 0.625 | 5,520.8125 | 4,478.6 | 7,257.833333 | |
The sum of the areas of all triangles whose vertices are also vertices of a $1$ by $1$ by $1$ cube is $m + \sqrt{n} + \sqrt{p},$ where $m, n,$ and $p$ are integers. Find $m + n + p.$
| 348 | 0 | 8,156.6875 | -1 | 8,156.6875 | |
Find the distance between the points $(2,2)$ and $(-1,-1)$. | 3\sqrt{2} | 1 | 2,214.5 | 2,214.5 | -1 | |
In triangle $ABC$ we have $AB = 25$, $BC = 39$, and $AC=42$. Points $D$ and $E$ are on $AB$ and $AC$ respectively, with $AD = 19$ and $AE = 14$. What is the ratio of the area of triangle $ADE$ to the area of the quadrilateral $BCED$? | \frac{19}{56} | 1. **Identify Similar Triangles and Use Ratios**:
Given that $D$ and $E$ are on $AB$ and $AC$ respectively, with $AD = 19$ and $AE = 14$, and $AB = 25$ and $AC = 42$. We introduce point $F$ on $AC$ such that $DE \parallel BF$. This implies $\triangle ADE \sim \triangle ABF$ by AA similarity (Angle-Angle).
2. **Calc... | 0.6875 | 6,927.375 | 6,352.545455 | 8,192 |
In the rectangular table shown below, the number $1$ is written in the upper-left hand corner, and every number is the sum of the any numbers directly to its left and above. The table extends infinitely downwards and to the right.
\[
\begin{array}{cccccc}
1 & 1 & 1 & 1 & 1 & \cdots
1 & 2 & 3 & 4 & 5 & \c... | 19 | 0 | 8,187.4375 | -1 | 8,187.4375 | |
Calculate the value for the expression $\sqrt{25\sqrt{15\sqrt{9}}}$. | 5\sqrt{15} | 0.125 | 5,050.125 | 902 | 5,642.714286 | |
Let $A$, $B$, and $C$ be three sets such that:
-A$|-|A|=100$, $|B|=50$, and $|C|=48$,
-The number of elements belonging to exactly one of the three sets is twice the number of elements belonging to exactly two of the sets.
-The number of elements belonging to exactly one of the three sets is three times the number o... | 22 | 0.5 | 6,584.375 | 4,976.75 | 8,192 | |
How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 100\}$ have a perfect square factor other than one? | 41 | 0 | 7,536.25 | -1 | 7,536.25 | |
Given the ellipse $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $\frac{\sqrt{2}}{2}$, and it passes through point $M(-2, 0)$.
(I) Find the standard equation of ellipse $C$;
(II) Let line $l$ with a slope of $1$ intersect ellipse $C$ at points $A(x\_1, y\_1)$ and $B(x\_2, y\_2)... | S = \sqrt{10} | 0.3125 | 7,806.3125 | 7,531.8 | 7,931.090909 | |
A $37$-gon $Q_1$ is drawn in the Cartesian plane, and the sum of the $x$-coordinates of its $37$ vertices equals $185$. The midpoints of the sides of $Q_1$ form a second $37$-gon, $Q_2$. Then, the midpoints of the sides of $Q_2$ form a third $37$-gon, $Q_3$. Find the sum of the $x$-coordinates of the vertices of $Q_3$. | 185 | 0.9375 | 3,876.6875 | 3,589 | 8,192 | |
The function $f(x)$ satisfies
\[3f(x) - 2 f \left( \frac{1}{x} \right) = x\]for all $x \neq 0.$ Find $f(4).$ | \frac{5}{2} | 0.9375 | 2,981.4375 | 2,634.066667 | 8,192 | |
There are 100 houses in a row on a street. A painter comes and paints every house red. Then, another painter comes and paints every third house (starting with house number 3) blue. Another painter comes and paints every fifth house red (even if it is already red), then another painter paints every seventh house blue, a... | 52 | House $n$ ends up red if and only if the largest odd divisor of $n$ is of the form $4 k+1$. We have 25 values of $n=4 k+1 ; 13$ values of $n=2(4 k+1)$ (given by $k=0,1,2, \ldots, 12$ ); 7 values of $n=4(4 k+1)(k=0,1, \ldots, 6) ; 3$ values of $n=8(4 k+1)(k=0,1,2) ; 2$ of the form $n=16(4 k+1$ ) (for $k=0,1)$; 1 of the ... | 0 | 8,192 | -1 | 8,192 |
A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? | 2\sqrt{10} | 1 | 2,771.375 | 2,771.375 | -1 | |
Given vectors $\overrightarrow{a}=m \overrightarrow{i}+5 \overrightarrow{j}- \overrightarrow{k}, \overrightarrow{b}=3 \overrightarrow{i}+ \overrightarrow{j}+r \overrightarrow{k}$, if $\overrightarrow{a}//\overrightarrow{b}$, find the values of real numbers $m=$ \_\_\_\_\_\_ and $r=$ \_\_\_\_\_\_. | -\frac{1}{5} | 1 | 1,528.125 | 1,528.125 | -1 | |
Find the phase shift of the graph of \( y = \cos(5x - \frac{\pi}{2}) \). | \frac{\pi}{10} | 1 | 2,795.4375 | 2,795.4375 | -1 | |
What is the largest number, with its digits all different, whose digits add up to 16? | 643210 | 0 | 7,862.5625 | -1 | 7,862.5625 | |
Simplify $(2x - 5)(x + 7) - (x + 5)(2x - 1)$. | -30 | 1 | 1,886.3125 | 1,886.3125 | -1 | |
In a new arcade game, the "monster" is the shaded region of a semicircle with radius 2 cm as shown in the diagram. The mouth, which is an unshaded piece within the semicircle, subtends a central angle of 90°. Compute the perimeter of the shaded region.
A) $\pi + 3$ cm
B) $2\pi + 2$ cm
C) $\pi + 4$ cm
D) $2\pi + 4$ cm
E... | \pi + 4 | 0 | 4,267.625 | -1 | 4,267.625 | |
Given an ellipse $E: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a>b>0)$ with an eccentricity of $\frac{\sqrt{2}}{2}$ and upper vertex at B. Point P is on E, point D is at (0, -2b), and the maximum area of △PBD is $\frac{3\sqrt{2}}{2}$.
(I) Find the equation of E;
(II) If line DP intersects E at another point Q, and lin... | \frac{2}{3} | 0.625 | 6,781.875 | 5,935.8 | 8,192 | |
Krystyna has some raisins. After giving some away and eating some, she has 16 left. How many did she start with? | 54 | Working backwards, Krystyna had 36 raisins before eating 4, and 54 raisins initially. | 0 | 383.9375 | -1 | 383.9375 |
Louise is designing a custom dress and needs to provide her hip size in millimeters. If there are $12$ inches in a foot and $305$ millimeters in a foot, and Louise's hip size is $42$ inches, what size should she specify in millimeters? | 1067.5 | 0.125 | 896.625 | 2,113 | 722.857143 | |
Let the set \( M = \{1, 2, \cdots, 1000\} \). For any non-empty subset \( X \) of \( M \), let \( a_X \) represent the sum of the maximum and minimum numbers in \( X \). What is the arithmetic mean of all such \( a_X \)? | 1001 | 0.25 | 7,479.5 | 6,156.75 | 7,920.416667 | |
Given the point $M(m, m^2)$ and $N(n, n^2)$, where $m$ and $n$ are the two distinct real roots of the equation $\sin\theta \cdot x^2 + \cos\theta \cdot x - 1 = 0 (\theta \in R)$. If the maximum distance from a point on the circle $O: x^2 + y^2 = 1$ to the line $MN$ is $d$, and the positive real numbers $a$, $b$, and $c... | \frac{3}{2} | 0.25 | 7,690 | 6,620.5 | 8,046.5 | |
Given that the terminal side of angle $\alpha$ passes through point $P(-4a, 3a) (a \neq 0)$, find the value of $\sin \alpha + \cos \alpha - \tan \alpha$. | \frac{19}{20} | 0.0625 | 7,009.4375 | 8,192 | 6,930.6 | |
Let $M$ be the greatest five-digit number whose digits have a product of $180$. Find the sum of the digits of $M$. | 20 | 0 | 8,192 | -1 | 8,192 | |
For the fractional equation involving $x$, $\frac{x+m}{x-2}+\frac{1}{2-x}=3$, if it has a root with an increase, then $m=\_\_\_\_\_\_$. | -1 | 0.3125 | 7,184.3125 | 6,321.2 | 7,576.636364 | |
Let
\[\mathbf{A} = \renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{\sqrt{3}}{2} & 0 & -\frac{1}{2} \\ 0 & -1 & 0 \\ \frac{1}{2} & 0 & \frac{\sqrt{3}}{2} \end{pmatrix} \renewcommand{\arraystretch}{1}.\]Compute $\mathbf{A}^{2018}.$ | \begin{pmatrix} \frac{1}{2} & 0 & -\frac{\sqrt{3}}{2} \\ 0 & 1 & 0 \\ \frac{\sqrt{3}}{2} & 0 & \frac{1}{2} \end{pmatrix} | 0 | 7,361.875 | -1 | 7,361.875 | |
Let $l,$ $m,$ and $n$ be real numbers, and let $A,$ $B,$ $C$ be points such that the midpoint of $\overline{BC}$ is $(l,0,0),$ the midpoint of $\overline{AC}$ is $(0,m,0),$ and the midpoint of $\overline{AB}$ is $(0,0,n).$ Find
\[\frac{AB^2 + AC^2 + BC^2}{l^2 + m^2 + n^2}.\] | 8 | 0.75 | 6,193 | 5,526.666667 | 8,192 | |
The function $f(x)$ satisfies
\[f(x) - 2 f \left( \frac{1}{x} \right) = 4^x\]for all $x \neq 0.$ Find $f(2).$ | -\frac{20}{3} | 1 | 3,307.0625 | 3,307.0625 | -1 | |
In the store "Everything for School," three types of chalk packs are sold: regular, unusual, and excellent. Initially, the quantitative ratio of the types was 3:4:6. As a result of sales and deliveries from the warehouse, this ratio changed to 2:5:8. It is known that the number of packs of excellent chalk increased by ... | 390 | 0 | 3,995.625 | -1 | 3,995.625 | |
A fly is being chased by three spiders on the edges of a regular octahedron. The fly has a speed of $50$ meters per second, while each of the spiders has a speed of $r$ meters per second. The spiders choose their starting positions, and choose the fly's starting position, with the requirement that the fly must begi... | 25 | 0.0625 | 8,192 | 8,192 | 8,192 | |
The curves $x^{2}+y^{2}=36$ and $y=x^{2}-7$ intersect at four points. Find the sum of the squares of the $x$-coordinates of these points. | 26 | If we use the system of equations to solve for $y$, we get $y^{2}+y-29=0$ (since $x^{2}=y+7$). The sum of the roots of this equation is -1. Combine this with $x^{2}=y+7$ to see that the sum of the square of the possible values of $x$ is $2 \cdot(-1+7 \cdot 2)=26$. | 0.9375 | 3,484.6875 | 3,490.733333 | 3,394 |
Chuck the llama is tied to the corner of a $2\text{ m}$ by $3\text{ m}$ shed on a $3\text{ m}$ leash. How much area (in square meters) does Chuck have in which to play if he can go only around the outside of the shed? [asy]
draw((0,0)--(15,0)--(15,10)--(0,10)--cycle,black+linewidth(1));
draw((15,10)--(27,19),black+line... | 7\pi | 0.375 | 7,179.875 | 5,878.166667 | 7,960.9 | |
Let \( x, y, z \) be positive numbers that satisfy the following system of equations:
\[
\begin{cases}
x^2 + xy + y^2 = 108 \\
y^2 + yz + z^2 = 9 \\
z^2 + xz + x^2 = 117
\end{cases}
\]
Find the value of the expression \( xy + yz + xz \). | 36 | 0.5 | 7,492.625 | 6,793.25 | 8,192 | |
Let $x$, $y$, and $z$ be positive real numbers such that $(x \cdot y) + z = (x + z) \cdot (y + z)$. What is the maximum possible value of $xyz$? | \frac{1}{27} | 1 | 2,736 | 2,736 | -1 | |
Luna writes down all the five-digit numbers that contain each of the digits 1, 2, 3, 4 and 5 exactly once. What is the smallest number in Luna's list that is divisible by 6? | 12354 | 0.6875 | 7,310.5 | 6,909.818182 | 8,192 | |
A force of $60 \mathrm{H}$ stretches a spring by 2 cm. The initial length of the spring is $14 \mathrm{~cm}$. How much work is required to stretch it to 20 cm? | 5.4 | 0.25 | 5,810.6875 | 2,156.25 | 7,028.833333 | |
Find the sum of all prime numbers $p$ which satisfy \[p = a^4 + b^4 + c^4 - 3\] for some primes (not necessarily distinct) $a$ , $b$ and $c$ . | 719 | 0.125 | 8,192 | 8,192 | 8,192 | |
Find the area of the triangle with vertices $(2, -3),$ $(1, 4),$ and $(-3, -2).$ | 17 | 0.9375 | 3,584.3125 | 3,277.133333 | 8,192 | |
If $4u-5v=23$ and $2u+4v=-8$, find the value of $u+v$. | -1 | 0.9375 | 2,149.4375 | 1,746.6 | 8,192 | |
Alicia has $n$ candies, where $n$ is a positive integer with three digits. If she buys $5$ more, she will have a multiple of $8$. If she loses $8$, she will have a multiple of $5$. What is the smallest possible value of $n$? | 123 | 0.9375 | 4,251.25 | 3,988.533333 | 8,192 | |
What is the units digit of the sum $1! + 2! + 3! + 4! + 5! + \cdots + 1000!$? | 3 | 1 | 2,135.8125 | 2,135.8125 | -1 | |
Petya is thinking of a four-digit number of the form \( \overline{20 * *} \).
Vasya consecutively checks whether the number chosen by Petya is divisible by 1, 3, 5, 7, 9, 11. If the number is divisible, Vasya pays Petya 1, 3, 5, 7, 9, or 11 rubles respectively. For example, for the number 2000, Vasya would pay Petya \... | 31 | 0.0625 | 8,135.0625 | 7,281 | 8,192 | |
A \( 5 \mathrm{~cm} \) by \( 5 \mathrm{~cm} \) pegboard and a \( 10 \mathrm{~cm} \) by \( 10 \mathrm{~cm} \) pegboard each have holes at the intersection of invisible horizontal and vertical lines that occur in \( 1 \mathrm{~cm} \) intervals from each edge. Pegs are placed into the holes on the two main diagonals of b... | 100 | 0 | 7,650.5 | -1 | 7,650.5 | |
Consider the decimal function denoted by $\{ x \} = x - \lfloor x \rfloor$ which represents the decimal part of a number $x$. Find the sum of the five smallest positive solutions to the equation $\{x\} = \frac{1}{\lfloor x \rfloor}$. Express your answer as a mixed number. | 21\frac{9}{20} | 0.75 | 3,559.625 | 3,654.25 | 3,275.75 | |
The graph of the function $y=g(x)$ is shown below. For all $x > 5$, it is true that $g(x) > 0.5$. If $g(x) = \frac{x^2}{Dx^2 + Ex + F}$, where $D, E,$ and $F$ are integers, then find $D+E+F$. Assume the function has vertical asymptotes at $x = -3$ and $x = 4$ and a horizontal asymptote below 1 but above 0.5. | -24 | 0.125 | 8,062.6875 | 7,157.5 | 8,192 | |
The integer $m$ is between $30$ and $80$ and is a multiple of $6$. When $m$ is divided by $8,$ the remainder is $2$. Similarly, when $m$ is divided by $5,$ the remainder is $2$. What is the value of $m$? | 42 | 1 | 2,613.625 | 2,613.625 | -1 | |
A non-square rectangle has integer dimensions. The number of square units in its area is numerically equal to the number of units in its perimeter. What is the number of units in the perimeter of this rectangle? | 18 | 1 | 1,828.6875 | 1,828.6875 | -1 | |
The whole numbers from 1 to 1000 are written. How many of these numbers have at least two 7's appearing side-by-side? | 19 | 0.1875 | 7,840.4375 | 6,317 | 8,192 | |
If $a\equiv 18\pmod{42}$ and $b\equiv 73\pmod{42}$, then for what integer $n$ in the set $\{100,101,102,\ldots,140,141\}$ is it true that $$a-b\equiv n\pmod{42}~?$$ | 113 | 1 | 2,574.0625 | 2,574.0625 | -1 | |
Let \( E(n) \) denote the largest integer \( k \) such that \( 5^{k} \) divides the product \( 1^{1} \cdot 2^{2} \cdot 3^{3} \cdot 4^{4} \cdots \cdots n^{n} \). What is the value of \( E(150) \)? | 2975 | 0 | 8,192 | -1 | 8,192 | |
Given real numbers $x$ and $y$ satisfying $x^{2}+2y^{2}-2xy=4$, find the maximum value of $xy$. | 2\sqrt{2} + 2 | 0 | 7,921.5625 | -1 | 7,921.5625 | |
The minimum positive period of $y=\tan(4x+ \frac{\pi}{3})$ is $\pi$. | \frac{\pi}{4} | 0.9375 | 4,258.1875 | 3,995.933333 | 8,192 | |
A digit is written to the right of the units digit of $757$. If the resulting four-digit number is divisible by $3$, how many possibilities are there for the digit that was written? | 3 | 1 | 2,403.25 | 2,403.25 | -1 | |
Regular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle of area $1.$ Point $P$ lies inside the circle so that the region bounded by $\overline{PA_1},\overline{PA_2},$ and the minor arc $\widehat{A_1A_2}$ of the circle has area $\tfrac{1}{7},$ while the region bounded by $\overline{PA_3},\overline{PA_4},$ an... | 504 | The actual size of the diagram doesn't matter. To make calculation easier, we discard the original area of the circle, $1$, and assume the side length of the octagon is $2$. Let $r$ denote the radius of the circle, $O$ be the center of the circle. Then $r^2= 1^2 + (\sqrt{2}+1)^2= 4+2\sqrt{2}$. Now, we need to find the ... | 0 | 8,192 | -1 | 8,192 |
A triangle has sides of lengths 40 units, 50 units, and 70 units. An altitude is dropped from the vertex opposite the side of length 70 units. Calculate the length of this altitude.
A) $\frac{40\sqrt{7}}{7}$ units
B) $\frac{80\sqrt{7}}{7}$ units
C) $\frac{120\sqrt{7}}{7}$ units
D) $\frac{160\sqrt{7}}{7}$ units | \frac{80\sqrt{7}}{7} | 0 | 8,192 | -1 | 8,192 | |
When finding the sum $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}$, the least common denominator used is | 420 | To find the least common denominator (LCD) for the sum $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}$, we need to determine the least common multiple (LCM) of the denominators: $2, 3, 4, 5, 6, 7$.
1. **Prime Factorization**:
- $2$ is prime.
- $3$ is prime.
- $4 = 2^2$.
- $5$ is p... | 1 | 2,458.125 | 2,458.125 | -1 |
Compute $0.18\div0.003.$ | 60 | 1 | 2,475.9375 | 2,475.9375 | -1 | |
Find \( n \) such that \( 2^3 \cdot 5 \cdot n = 10! \). | 45360 | 0 | 3,744.875 | -1 | 3,744.875 | |
Let $ABC$ be a triangle where $\angle$ **B=55** and $\angle$ **C = 65**. **D** is the mid-point of **BC**. Circumcircle of **ACD** and**ABD** cuts **AB** and**AC** at point **F** and **E** respectively. Center of circumcircle of **AEF** is**O**. $\angle$ **FDO** = ? | 30 | 0.0625 | 8,192 | 8,192 | 8,192 |
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