sungyub/dapo-math-17k-verl huggingface dataset:
user: Calculate the single-digit result obtained from the repeated sum of the digits of the number $2^{2004}$. Follow these steps:
Calculate the sum $S$ of all digits of $2^{2004}$.
Calculate the sum of the digits of $S$ to obtain $R$.
Continue calculating the sum of the digits until a single-digit number is reached.

For example, for $2^7 = 128$, we have:
- $S = 1 + 2 + 8 = 11$
- $R = 1 + 1 = 2$
Thus, the single-digit result for $2^7$ is $2$.

Answer: 1
user: Farmer John owns 2013 cows. Some cows are enemies of each other, and Farmer John wishes to divide them into as few groups as possible such that each cow has at most 3 enemies in her group. Each cow has at most 61 enemies. Compute the smallest integer $G$ such that, no matter which enemies they have, the cows can always be divided into at most $G$ such groups.

Answer: 16
user: The real numbers $x$, $y$, and $z$ satisfy the system of equations:
\[
\begin{align*}
 x^2 + 27 &= -8y + 10z, \\
 y^2 + 196 &= 18z + 13x, \\
 z^2 + 119 &= -3x + 30y.
\end{align*}
\]
Find $x + 3y + 5z$.

Answer: 108
user: Suppose $A_1, A_2, A_3, \ldots, A_{20}$ is a 20-sided regular polygon. How many non-isosceles (scalene) triangles can be formed whose vertices are among the vertices of the polygon, but the sides are not the sides of the polygon?

Answer: 640
user: Determine the number of angles $\theta$ between $0$ and $2\pi$, excluding integer multiples of $\pi/2$, such that $\sin \theta$, $\cos \theta$, and $\tan \theta$ form a geometric sequence in some order.

Answer: 4
user: For each positive integer $k$, let $a_k$ be the greatest integer not exceeding $\sqrt{k}$ and let $b_k$ be the greatest integer not exceeding $\sqrt[3]{k}$. Calculate \(\sum_{k=1}^{2003} (a_k-b_k)\).

Answer: 40842
user: Let $a, b, c$ be the roots of the cubic $x^3 + 3x^2 + 5x + 7$. Given that $P$ is a cubic polynomial such that $P(a) = b + c$, $P(b) = c + a$, $P(c) = a + b$, and $P(a + b + c) = -16$, find $P(0)$.

Answer: 11
user: What are the last two digits of the decimal representation of $21^{2006}$?

Answer: 21
user: We are given the following function $f$, which takes a list of integers and outputs another list of integers. (Note that here the list is zero-indexed.)

\[
\begin{array}{l}
\textbf{FUNCTION} \ f(A) \\
\quad \textbf{FOR} \ i=1,\ldots, \operatorname{length}(A)-1: \\
\quad\quad A[i]\leftarrow A[A[i]] \\
\quad\quad A[0]\leftarrow A[0]-1 \\
\quad \textbf{RETURN} \ A
\end{array}
\]

Suppose the list $B$ is equal to $[0,1,2,8,2,0,1,7,0]$. In how many entries do $B$ and $f(B)$ differ?

Answer: 4
user: Let $\alpha$ denote $\cos^{-1}(\tfrac{2}{3})$. The recursive sequence $a_0, a_1, a_2, \ldots$ satisfies $a_0 = 1$ and, for all positive integers $n$, 
$$a_n = \dfrac{\cos(n\alpha) - (a_1a_{n-1} + \cdots + a_{n-1}a_1)}{2a_0}.$$ 
Suppose that the series 
$$\sum_{k=0}^\infty\dfrac{a_k}{2^k}$$ 
can be expressed uniquely as $\tfrac{p\sqrt{q}}{r}$, where $p$ and $r$ are coprime positive integers and $q$ is not divisible by the square of any prime. Find the value of $p+q+r$.

Answer: 23
user: Initially, each cell in a $29 \times 29$ table contains the number $0$. A movement consists of selecting a $5 \times 5$ sub-table and adding $+1$ to every cell within this sub-table. Determine the maximum value of $n$ such that, after performing $1000$ movements, there exist four cells whose centers form the vertices of a square, and the sum of the values in these four cells is at least $n$. 

**Note:** The sides of the square do not necessarily have to be parallel to the sides of the table.

Answer: 120
user: Two knights are placed on distinct squares of an $8 \times 8$ chessboard, where each square is a unit square. The knights are said to attack each other if the distance between the centers of the squares on which the knights are placed is $\sqrt{5}$. In how many ways can two identical knights be placed on distinct squares of the chessboard such that they do NOT attack each other?

Answer: 1848
user: Let $N = \overline{abcd}$ be a positive integer with four digits. We define the **plátano power** of $N$, denoted as $p(N) = \overline{\alpha_1\alpha_2\ldots\alpha_k}$, to be the smallest positive integer that can be inserted between the numbers $\overline{ab}$ and $\overline{cd}$ such that the new number $\overline{ab\alpha_1\alpha_2\ldots\alpha_kcd}$ is divisible by $N$. Determine the value of $p(2025)$.

Answer: 45
user: On a $4 \times 4$ board, the numbers from $1$ to $16$ are written, one in each box. Andres and Pablo choose four numbers each. Andres chooses the biggest number from each row, and Pablo chooses the biggest number from each column. The same number can be chosen by both. Then, all chosen numbers are removed from the board. What is the greatest possible sum of the numbers left on the board?

Answer: 78
user: In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\alpha$, $\beta$, and $\alpha + \beta$ radians, respectively, where $\alpha + \beta < \pi$. If $\cos \alpha$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?

Answer: 49
user: How many subsets $A$ of $ \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \} $ have the property that no two elements of $A$ sum to $11$?

Answer: 243
user: How many integers between $2$ and $100$ have only odd numbers in their prime factorizations?

Answer: 49
user: A hollow box, shaped like a rectangular prism and with negligible thickness, has a volume of $108$ cubic units. The top of the box is removed, exposing the faces on the inside of the box. What is the minimum possible value for the sum of the areas of the faces on the outside and inside of the box?

Answer: 216
user: Say that an ordered triple $(a, b, c)$ is pleasing if:

$a$, $b$, and $c$ are in the set $\{ 1, 2, \dots, 17 \}$.
Both $b - a$ and $c - b$ are greater than $3$, and at least one of them is equal to $4$.

How many pleasing triples are there?

Answer: 81
user: Suppose that $10$ mathematics teachers gather at a circular table with $25$ seats to discuss the upcoming mathematics competition. Each teacher is assigned a unique integer ID number between $1$ and $10$, and the teachers arrange themselves in such a way that teachers with consecutive ID numbers are not separated by any other teacher (IDs $1$ and $10$ are considered consecutive). In addition, each pair of teachers is separated by at least one empty seat. Given that seating arrangements obtained by rotation are considered identical, how many ways are there for the teachers to sit at the table?

Answer: 4004
user: Andy wishes to open an electronic lock with a keypad containing all digits from $0$ to $9$. He knows that the password registered in the system is $2469$. Unfortunately, he is also aware that exactly two different buttons (but he does not know which ones) $\underline{a}$ and $\underline{b}$ on the keypad are broken - when $\underline{a}$ is pressed, the digit $b$ is registered in the system, and when $\underline{b}$ is pressed, the digit $a$ is registered in the system. Find the least number of attempts Andy needs to surely be able to open the lock.

Answer: 31
user: Let $S$ be a set. We say $S$ is $D^\ast$-finite if there exists a function $f : S \to S$ such that for every nonempty proper subset $Y \subsetneq S$, there exists a $y \in Y$ such that $f(y) \notin Y$. The function $f$ is called a witness of $S$. How many witnesses does $\{0,1,\cdots,5\}$ have?

Answer: 120
user: If $\prod_{i=6}^{2021} (1-\tan^2((2^i)^\circ))$ can be written in the form $a^b$ for positive integers $a,b$ with $a$ squarefree, find $a+b$.

Answer: 2018
user: Jeremy wrote all the three-digit integers from 100 to 999 on a blackboard. Then Allison erased each of the 2700 digits Jeremy wrote and replaced each digit with the square of that digit. Thus, Allison replaced every 1 with a 1, every 2 with a 4, every 3 with a 9, every 4 with a 16, and so forth. The proportion of all the digits Allison wrote that were ones is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 87
user: For each integer from 1 to 2019, calculate the product of its digits. Compute the sum of all 2019 products obtained.

Answer: 184320
user: Let $m > n$ be positive integers such that $3(3mn - 2)^2 - 2(3m - 3n)^2 = 2019$. Find $3m + n$.

Answer: 46
user: For positive integers $1 \le n \le 100$, let
\[ f(n) = \sum_{i=1}^{100} i \left\lvert i-n \right\rvert. \]
Compute $f(54) - f(55)$.

Answer: 2080
user: The school store is running out of supplies, but it still has five items: one pencil (costing $1), one pen (costing $1), one folder (costing $2), one pack of paper (costing $3), and one binder (costing $4). If you have $10, in how many ways can you spend your money? (You don't have to spend all of your money, or any of it.)

Answer: 31
user: Alice is bored in class, so she thinks of a positive integer. Every second after that, she subtracts from her current number its smallest prime divisor, possibly itself. After 2022 seconds, she realizes that her number is prime. Find the sum of all possible values of her initial number.

Answer: 8093
user: The points $A$, $B$, $C$, $D$, and $E$ lie in one plane and have the following properties:

- $AB = 12$
- $BC = 50$
- $CD = 38$
- $AD = 100$
- $BE = 30$
- $CE = 40$

Find the length of the segment $ED$.

Answer: 74
user: If for the real numbers $x, y, z, k$ the following conditions are valid: $x \ne y \ne z \ne x$ and \[ x^3 + y^3 + k(x^2 + y^2) = y^3 + z^3 + k(y^2 + z^2) = z^3 + x^3 + k(z^2 + x^2) = 2008, \] find the product $xyz$.

Answer: 1004
user: Suppose $a$, $b$, and $c$ are real numbers that satisfy the equations:

$$a + b + c = 5$$

and

$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{5}.$$ 

Find the greatest possible value of $a^3 + b^3 + c^3$.

Answer: 125
user: Nine points form a grid of size $3 \times 3$. How many triangles can be formed with $3$ vertices at these points?

Answer: 76
user: In an $n \times m$ grid, 1 row and 1 column are colored blue, and the rest of the cells are white. If precisely $\frac{1}{2010}$ of the cells in the grid are blue, how many values are possible for the ordered pair $(n, m)$?

Answer: 96
user: In triangle $ABC$ with $AB=8$ and $AC=10$, the incenter $I$ is reflected across side $AB$ to point $X$ and across side $AC$ to point $Y$. Given that segment $XY$ bisects $AI$, compute $BC^2$.

Answer: 84
user: Evaluate $1 \oplus 2 \oplus \dots \oplus 987654321$ where $\oplus$ is bitwise exclusive OR.

($A \oplus B$ in binary has an $n$-th digit equal to $1$ if the $n$-th binary digits of $A$ and $B$ differ and $0$ otherwise. For example, $5 \oplus 9 = 0101_{2} \oplus 1001_{2} = 1100_2 = 12$ and $6 \oplus 7 = 110_2 \oplus 111_2 = 001_2 = 1$.)

Answer: 1
user: Given that $20N^2$ is a divisor of $11!$, what is the greatest possible integer value of $N$?

Answer: 72
user: Let $ABC$ be a right-angled triangle with $\angle B = 90^{\circ}$. The length of the altitude $BD$ is equal to $12$. What is the minimum possible length of $AC$, given that both $AC$ and the perimeter of triangle $ABC$ are integers?

Answer: 25
user: Line $DE$ cuts through triangle $ABC$, with $DF$ parallel to $BE$. Given that $BD = DF = 10$ and $AD = BE = 25$, find $BC$.

Answer: 50
user: A tetrahedron is a polyhedron composed of four triangular faces. Faces $ABC$ and $BCD$ of a tetrahedron $ABCD$ meet at an angle of $\frac{\pi}{6}$. The area of triangle $\triangle ABC$ is $120$. The area of triangle $\triangle BCD$ is $80$, and $BC = 10$. What is the volume of the tetrahedron? The volume of a tetrahedron is defined as one-third the area of its base times its height.

Answer: 320
user: If $f(x)$ is a monic quartic polynomial such that $f(-1) = -1$, $f(2) = -4$, $f(-3) = -9$, and $f(4) = -16$, find $f(1)$.

Answer: 23
user: In the diagram below, square $ABCD$ with side length 23 is cut into nine rectangles by two lines parallel to $\overline{AB}$ and two lines parallel to $\overline{BC}$. The areas of four of these rectangles are indicated in the diagram. Compute the largest possible value for the area of the central rectangle.

size(250);
 defaultpen (linewidth (0.7) + fontsize (10));
 draw ((0,0)--(23,0)--(23,23)--(0,23)--cycle);
 label("$A$", (0,23), NW);
 label("$B$", (23, 23), NE);
 label("$C$", (23,0), SE);
 label("$D$", (0,0), SW);
 draw((0,6)--(23,6));
 draw((0,19)--(23,19));
 draw((5,0)--(5,23));
 draw((12,0)--(12,23));
 label("13", (17/2, 21));
 label("111",(35/2,25/2));
 label("37",(17/2,3));
 label("123",(2.5,12.5));[/asy]

Answer: 180
user: For any positive integer $n$, let $f(n)$ be the maximum number of groups formed by a total of $n$ people such that the following holds: every group consists of an even number of members, and every two groups share an odd number of members. Compute $\sum_{n=1}^{2022} f(n) \mod 1000$.

Answer: 242
user: In a chess tournament, each of two players has only one game played. After 2 rounds, 5 players left the tournament. At the end of the tournament, it was found that the total number of games played is 100. How many players were there at the start of the tournament?

Answer: 105
user: Let $y_0$ be chosen randomly from $\{0, 50\}$, let $y_1$ be chosen randomly from $\{40, 60, 80\}$, let $y_2$ be chosen randomly from $\{10, 40, 70, 80\}$, and let $y_3$ be chosen randomly from $\{10, 30, 40, 70, 90\}$. (In each choice, the possible outcomes are equally likely to occur.) Let $P$ be the unique polynomial of degree less than or equal to $3$ such that $P(0) = y_0$, $P(1) = y_1$, $P(2) = y_2$, and $P(3) = y_3$. What is the expected value of $P(4)$?

Answer: 107
user: Let $ABC$ be a triangle with sides $51$, $52$, and $53$. Let $\Omega$ denote the incircle of $\triangle ABC$. Draw tangents to $\Omega$ which are parallel to the sides of $ABC$. Let $r_1$, $r_2$, $r_3$ be the inradii of the three corner triangles so formed. Find the largest integer that does not exceed $r_1 + r_2 + r_3$.

Answer: 15
user: Convex equiangular hexagon $ABCDEF$ has $AB=CD=EF=1$ and $BC = DE = FA = 4$. Congruent and pairwise externally tangent circles $\gamma_1$, $\gamma_2$, and $\gamma_3$ are drawn such that $\gamma_1$ is tangent to side $\overline{AB}$ and side $\overline{BC}$, $\gamma_2$ is tangent to side $\overline{CD}$ and side $\overline{DE}$, and $\gamma_3$ is tangent to side $\overline{EF}$ and side $\overline{FA}$. The area of $\gamma_1$ can be expressed as $\frac{m\pi}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.

Answer: 14800
user: Let $A, B, C, \ldots, Z$ be 26 nonzero real numbers. Suppose that $T = TNYWR$. Compute the smallest possible value of \[ \left\lceil A^2 + B^2 + \cdots + Z^2 \right\rceil. \] (The notation $\left\lceil x \right\rceil$ denotes the least integer $n$ such that $n \geq x$.)

Answer: 5
user: Let $N$ be the number of $2015$-tuples of (not necessarily distinct) subsets $(S_1, S_2, \dots, S_{2015})$ of $\{1, 2, \dots, 2015 \}$ such that the number of permutations $\sigma$ of $\{1, 2, \dots, 2015 \}$ satisfying $\sigma(i) \in S_i$ for all $1 \le i \le 2015$ is odd. Let $k_2, k_3$ be the largest integers such that $2^{k_2} \mid N$ and $3^{k_3} \mid N$ respectively. Find $k_2 + k_3.$

Answer: 2030612
user: Let $ABC$ be a triangle with $AB = 2$ and $AC = 3$. Let $H$ be the orthocenter, and let $M$ be the midpoint of $BC$. Let the line through $H$ perpendicular to line $AM$ intersect line $AB$ at $X$ and line $AC$ at $Y$. Suppose that lines $BY$ and $CX$ are parallel. Then $[ABC]^2 = \frac{a + b\sqrt{c}}{d}$ for positive integers $a, b, c$, and $d$, where $\gcd(a, b, d) = 1$ and $c$ is not divisible by the square of any prime. Compute $1000a + 100b + 10c + d$.

Answer: 270382
user: In a triangle $ABC$, the median from $B$ to $CA$ is perpendicular to the median from $C$ to $AB$. If the median from $A$ to $BC$ is $30$, determine $\frac{BC^2 + CA^2 + AB^2}{100}$.

Answer: 24
user: Equilateral triangle $ABC$ has an area of $1$. Points $A'$, $B'$, and $C'$ are the midpoints of sides $BC$, $CA$, and $AB$, respectively. Points $A''$, $B''$, and $C''$ are the midpoints of segments $B'C'$, $C'A'$, and $A'B'$, respectively. The area of trapezoid $BB''C''C$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine the value of $m+n$.

Answer: 21
user: Solve the following equation in positive numbers:
$$ (2a+1)(2a^2+2a+1)(2a^4+4a^3+6a^2+4a+1) = 828567056280801 $$

Answer: 100
user: A canister contains $2\frac{1}{2}$ cups of flour. Greg and Sally have a brownie recipe that calls for $1\frac{1}{3}$ cups of flour. They want to make $1\frac{1}{2}$ recipes of brownies. To the nearest whole percent, what percent of the flour in the canister would they use?

Answer: 80
user: In triangle $ABC$, where $AB = 28$, $AC = 36$, and $BC = 32$, let $D$ be the point on segment $BC$ such that $\angle BAD = \angle DAC$. Let $E$ be the unique point such that $DE \parallel AB$ and line $AE$ is tangent to the circumcircle of $\triangle ABC$. Find the length of segment $AE$.

Answer: 18
user: A collection of $n$ squares on the plane is called tri-connected if the following criteria are satisfied:

All the squares are congruent.
If two squares have a point $P$ in common, then $P$ is a vertex of each of the squares.
Each square touches exactly three other squares.

How many positive integers $n$ are there with $2018 \leq n \leq 3018$, such that there exists a collection of $n$ squares that is tri-connected?

Answer: 501
user: For how many integers $n$ with $1 \leq n \leq 2012$ is the product
\[
 \prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right)
\]
equal to zero?

Answer: 335
user: Find the least positive integer that cannot be represented as \( \frac{2^a - 2^b}{2^c - 2^d} \), where \( a, b, c, \) and \( d \) are positive integers.

Answer: 11
user: Given a square $ABCD$ with side length $10$. On sides $BC$ and $AD$ of this square, points $E$ and $F$ are selected such that they form a rectangle $ABEF$. Rectangle $KLMN$ is positioned such that its vertices $K, L, M$, and $N$ lie on segments $CD, DF, FE$, and $EC$, respectively. It is given that rectangles $ABEF$ and $KLMN$ are congruent with $AB = MN$. Find the length of segment $AL$.

Answer: 5
user: What is the number formed by the last three digits of $1201^{1202}$?

Answer: 401
user: A fighting game club has $2024$ members. One day, a game of Smash is played between some pairs of members such that every member has played against exactly $3$ other members. Each match has a winner and a loser. A member will be happy if they won in at least $2$ of the matches. What is the maximum number of happy members over all possible match-ups and all possible outcomes?

Answer: 1012
user: Convex quadrilateral $ABCD$ satisfies the following conditions: \( \angle{CAB} = \angle{ADB} = 30^{\circ} \), \( \angle{ABD} = 77^{\circ} \), \( BC = CD \), and \( \angle{BCD} = n^{\circ} \) for some positive integer \( n \). Compute \( n \).

Answer: 68
user: At a party, there were some couples attending. As they arrive, each person gets to talk with all the other persons who are already in the room. During the party, after all the guests arrive, groups of persons form such that:
No two persons forming a couple belong to the same group.
For each two persons that do not form a couple, there is one and only one group to which both belong.

Find the number of couples attending the party, knowing that there are fewer groups than persons at the party.

Answer: 4
user: Math teacher wrote in a table a polynomial $P(x)$ with integer coefficients and said:

"Today my daughter has a birthday. If in the polynomial $P(x)$ we have $x=a$ where $a$ is the age of my daughter, then $P(a)=a$ and $P(0)=p$ where $p$ is a prime number such that $p>a$."

How old is the daughter of the math teacher?

Answer: 1
user: Find the number of integers $n$ such that $1 \le n \le 2017$ and $(n-2)(n-0)(n-1)(n-7)$ is a multiple of $1001$.

Answer: 99
user: Let $p, q$ be two-digit numbers, neither of which are divisible by $10$. Let $r$ be the four-digit number formed by placing the digits of $p$ followed by the digits of $q$ in order. As $p$ and $q$ vary, a computer prints $r$ on the screen if $\gcd(p, q) = 1$ and $p + q$ divides $r$. Suppose that the largest number printed by the computer is $N$. Determine the number formed by the last two digits of $N$ (in the same order).

Answer: 13
user: Compute the integer $$\frac{2^{\left(2^5-2\right)/5-1}-2}{5}.$$

Answer: 6
user: Find $100m+n$ if $m$ and $n$ are relatively prime positive integers such that 
\[ \sum_{\substack{i,j \ge 0 \\ i+j \text{ odd}}} \frac{1}{2^i3^j} = \frac{m}{n}. \]

Answer: 504
user: Distinct prime numbers $p$, $q$, and $r$ satisfy the following equation:

$$2pqr + 50pq = 7pqr + 55pr = 8pqr + 12qr = A$$

for some positive integer $A$. What is the value of $A$?

Answer: 1980
user: A sequence $a_n$ of real numbers satisfies the following conditions:

- $a_1 = 1$
- $a_2 = 0$
- $a_n = (S_{n-1} + 1) S_{n-2}$ for all integers $n \geq 3$

where $S_k = a_1 + a_2 + \dots + a_k$ for positive integers $k$. 

Determine the smallest integer $m > 2$ such that $127$ divides $a_m$.

Answer: 10
user: Define the *hotel elevator cubic* as the unique cubic polynomial $P$ for which $P(11) = 11$, $P(12) = 12$, $P(13) = 14$, $P(14) = 15$. What is $P(15)$?

Answer: 13
user: Let $ABC$ be a right triangle with hypotenuse $\overline{AC}$ and circumcenter $O$. Point $E$ lies on $\overline{AB}$ such that $AE = 9$, $EB = 3$. Point $F$ lies on $\overline{BC}$ such that $BF = 6$, $FC = 2$. Now suppose $W, X, Y$, and $Z$ are the midpoints of $\overline{EB}$, $\overline{BF}$, $\overline{FO}$, and $\overline{OE}$, respectively. Compute the area of quadrilateral $WXYZ$.

Answer: 12
user: Given that $9 \times 10 \times 11 \times \cdots \times 15 = 32432400$, what is $1 \times 3 \times 5 \times \cdots \times 15$?

Answer: 2027025
user: Jenny places 100 pennies on a table, with 30 showing heads and 70 showing tails. She randomly selects 40 pennies and flips them over. This means if a selected penny was showing heads, it will now show tails, and vice versa. What is the expected number of pennies showing heads at the end?

Answer: 46
user: Find the number of triples \((a, b, c)\) of positive integers such that \(a + ab + abc = 11\).

Answer: 3
user: Let \( \gamma = \alpha \times \beta \) where \[ \alpha = \underbrace{999 \cdots 9}_{2010 \text{ '9's}} \] and \[ \beta = \underbrace{444 \cdots 4}_{2010 \text{ '4's}} \] Find the sum of digits of \( \gamma \).

Answer: 18090
user: How many positive five-digit integers have the product of their five digits equal to $900$?

Answer: 210
user: For some positive integer $n$, consider the usual prime factorization \[ n = \prod_{i=1}^{k} p_{i}^{e_{i}} = p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k}, \] where $k$ is the number of prime factors of $n$ and $p_{i}$ are the prime factors of $n$. Define $Q(n), R(n)$ by \[ Q(n) = \prod_{i=1}^{k} p_{i}^{p_{i}} \text{ and } R(n) = \prod_{i=1}^{k} e_{i}^{e_{i}}. \] For how many $1 \leq n \leq 70$ does $R(n)$ divide $Q(n)$?

Answer: 56
user: Triangles $OAB$, $OBC$, and $OCD$ are isosceles triangles with $\angle OAB = \angle OBC = \angle OCD = 90^\circ$. Find the area of triangle $OAB$ if the area of triangle $OCD$ is 12.

Answer: 3
user: A **palindromic table** is a $3 \times 3$ array of letters such that the words in each row and column read the same forwards and backwards. An example of such a table is shown below.
\[
\begin{array}{ccc}
 O & M & O \\
 N & M & N \\
 O & M & O
\end{array}
\]

How many palindromic tables are there that use only the letters $O$ and $M$? (The table may contain only a single letter.)

Answer: 16
user: A point $(x, y)$ in the first quadrant lies on a line with intercepts $(a, 0)$ and $(0, b)$, where $a, b > 0$. Rectangle $M$ has vertices $(0, 0)$, $(x, 0)$, $(x, y)$, and $(0, y)$, while rectangle $N$ has vertices $(x, y)$, $(x, b)$, $(a, b)$, and $(a, y)$. What is the ratio of the area of $M$ to that of $N$?

Answer: 1