sungyub/dapo-math-17k-verl huggingface dataset:
user: Find all 2-digit numbers $n$ having the property:

The number $n^2$ is a 4-digit number of the form $\overline{xxyy}$.

Answer: 88
user: Let $x$ be the answer to this problem. For what real number $a$ is the answer to this problem also $a-x$?

Answer: 0
user: How many distinct triangles are there with prime side lengths and perimeter $100$?

Answer: 0
user: In English class, you have discovered a mysterious phenomenon: if you spend $n$ hours on an essay, your score on the essay will be $100\left( 1-4^{-n} \right)$ points if $2n$ is an integer, and $0$ otherwise. For example, if you spend $30$ minutes on an essay you will get a score of $50$, but if you spend $35$ minutes on the essay you somehow do not earn any points.

It is 4 AM, your English class starts at 8:05 AM the same day, and you have four essays due at the start of class. If you can only work on one essay at a time, what is the maximum possible average of your essay scores?

Answer: 75
user: In how many distinct ways can the letters of $\text{STANTON}$ be arranged?

Answer: 1260
user: A teacher suggests four possible books for students to read. Each of the six students selects one of the four books. How many ways can these selections be made if each of the books is read by at least one student?

Answer: 1560
user: Let $f(x)$ be a function mapping real numbers to real numbers. Given that $f(f(x)) = \frac{1}{3x}$, and $f(2) = \frac{1}{9}$, find $f\left(\frac{1}{6}\right)$.

Answer: 3
user: Suppose \( xy - 5x + 2y = 30 \), where \( x \) and \( y \) are positive integers. Find the sum of all possible values of \( x \).

Answer: 31
user: Let $a$ and $b$ be positive integers satisfying $3a < b$ and $a^2 + ab + b^2 = (b + 3)^2 + 27.$ Find the minimum possible value of $a + b.$

Answer: 25
user: In the quadrilateral $ABCD$, angles $\angle B$ and $\angle C$ are both $120^\circ$. The side lengths are given as $AB = CD = 1$ and $CB = 4$. Find the length of $AD$.

Answer: 5
user: How many ways can you mark 8 squares of an $8 \times 8$ chessboard so that no two marked squares are in the same row or column, and none of the four corner squares is marked? (Rotations and reflections are considered different.)

Answer: 21600
user: For all positive integer triples $(m, n, k)$ such that $|m^k - n!| \le n$, find the maximum value of $\frac{n}{m}$.

Answer: 2
user: Triangle $ABC$ has side lengths $AB = 14$, $BC = 15$, and $CA = 13$. We draw a circle with diameter $AB$ such that it intersects $BC$ again at $D$ and intersects $CA$ again at $E$. If the circumradius of $\triangle CDE$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are coprime positive integers, determine $100m+n$.

Answer: 3308
user: Let $x, y, z$ be real numbers such that $x+y+z=-2$ and
\[
(x^2+xy+y^2)(y^2+yz+z^2) + (y^2+yz+z^2)(z^2+zx+x^2) + (z^2+zx+x^2)(x^2+xy+y^2) = 625+ \frac{3}{4}(xy+yz+zx)^2.
\]
Compute $|xy+yz+zx|$.

Answer: 14
user: For every positive integer $n$, we define $a_{n}$ as the last digit of the sum $1 + 2 + \cdots + n$. Compute $a_{1} + a_{2} + \cdots + a_{1992}$.

Answer: 6984
user: Let $k$ be the product of every third positive integer from $2$ to $2006$, that is, \[ k = 2 \cdot 5 \cdot 8 \cdot 11 \cdots 2006. \] Find the number of zeros at the right end of the decimal representation for $k$.

Answer: 168
user: Find all positive integers $n \geq 1$ such that there exists a pair $(a, b)$ of positive integers, satisfying the following conditions:
$a^2 + b + 3$ is not divisible by the cube of any prime.
\( n = \frac{ab + 3b + 8}{a^2 + b + 3} \)

Answer: 2
user: Kara rolls a six-sided die. If on the first roll she rolls an $n$, she rolls the die $n-1$ more times. She then computes that the product of all her rolls, including the first, is $8$. How many distinct sequences of rolls could Kara have rolled?

Answer: 4
user: What is the largest integer $n$ such that $n$ is divisible by every integer less than $\sqrt[3]{n}$?

Answer: 420
user: Antoine, Benoît, Claude, Didier, Étienne, and Françoise go to the cinema together to see a movie. The six of them want to sit in a single row of six seats. However, Antoine, Benoît, and Claude are mortal enemies and refuse to sit next to either of the other two. How many different seating arrangements are possible?

Answer: 144
user: How many distinct sums can be made from adding together exactly 8 numbers that are chosen from the set $\{ 1, 4, 7, 10 \}$, where each number in the set is chosen at least once? 

For example, one possible sum is $1 + 1 + 1 + 4 + 7 + 7 + 10 + 10 = 41$.

Answer: 13
user: Let $\theta=\frac{2\pi}{2015}$, and suppose the product \[\prod_{k=0}^{1439}\left(\cos(2^k\theta)-\frac{1}{2}\right)\] can be expressed in the form $\frac{b}{2^a}$, where $a$ is a non-negative integer and $b$ is an odd integer (not necessarily positive). Find $a+b$.

Answer: 1441
user: A semicircle has diameter $\overline{AD}$ with $AD = 30$. Points $B$ and $C$ lie on $\overline{AD}$, and points $E$ and $F$ lie on the arc of the semicircle. The two right triangles $\vartriangle BCF$ and $\vartriangle CDE$ are congruent. The area of $\vartriangle BCF$ is $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m + n$.

Answer: 52
user: Compute the number of five-digit positive integers $\overline{vwxyz}$ for which 
\[ (10v+w) + (10w+x) + (10x+y) + (10y+z) = 100. \]

Answer: 164
user: The pentagon below has three right angles. Find its area.

\begin{asy}
size(150);
defaultpen(linewidth(1));
draw((4,10)--(0,10)--origin--(10,0)--(10,2)--cycle);
label("4",(2,10),N);
label("10",(0,5),W);
label("10",(5,0),S);
label("2",(10,1),E);
label("10",(7,6),NE);
\end{asy}

Answer: 76
user: Solomiya wrote the numbers $1, 2, \ldots, 2024$ on the board. In one move, she can erase any two numbers $a, b$ from the board and write the sum $a+b$ instead of each of them. After some time, all the numbers on the board became equal. What is the minimum number of moves Solomiya could make to achieve this?

Answer: 1012
user: A set of five-digit numbers $\{N_1, ..., N_k\}$ is such that any five-digit number, all of whose digits are in non-decreasing order, coincides in at least one digit with at least one of the numbers $N_1, ..., N_k$. Find the smallest possible value of $k$.

Answer: 9
user: A positive integer $n$ is called *nice* if it has at least 3 proper divisors and it is equal to the sum of its three largest proper divisors. For example, $6$ is *nice* because its largest three proper divisors are $3, 2, 1$ and $6 = 3 + 2 + 1$. Find the number of *nice* integers not greater than $3000$.

Answer: 200
user: Evaluate \[ \frac{1}{729} \sum_{a=1}^{9} \sum_{b=1}^{9} \sum_{c=1}^{9} \left( abc + ab + bc + ca + a + b + c \right). \]

Answer: 215
user: Let $ABC$ be a right triangle, right at $B$, and let $M$ be the midpoint of the side $BC$. Let $P$ be the point on the bisector of the angle $\angle BAC$ such that $PM$ is perpendicular to $BC$ (with $P$ being outside the triangle $ABC$). Determine the area of triangle $ABC$ if $PM = 1$ and $MC = 5$.

Answer: 120
user: Call a pair of integers $(a,b)$ primitive if there exists a positive integer $\ell$ such that $(a+bi)^\ell$ is real. Find the smallest positive integer $n$ such that less than $1\%$ of the pairs $(a, b)$ with $0 \le a, b \le n$ are primitive.

Answer: 299
user: If $a$, $b$, $c$, $d$, and $e$ are constants such that for every $x > 0$:
\[
\frac{5x^4 - 8x^3 + 2x^2 + 4x + 7}{(x + 2)^4} = a + \frac{b}{x + 2} + \frac{c}{(x + 2)^2} + \frac{d}{(x + 2)^3} + \frac{e}{(x + 2)^4} \, ,
\]
then what is the value of $a + b + c + d + e$?

Answer: 18
user: Suppose $a$ and $b$ are positive integers. Isabella and Vidur both fill up an $a \times b$ table. Isabella fills it up with numbers $1, 2, \ldots, ab$, putting the numbers $1, 2, \ldots, b$ in the first row, $b + 1, b + 2, \ldots, 2b$ in the second row, and so on. Vidur fills it up like a multiplication table, putting $ij$ in the cell in row $i$ and column $j$.

Isabella sums up the numbers in her grid, and Vidur sums up the numbers in his grid; the difference between these two quantities is $1200$. Compute $a + b$.

Answer: 21
user: Given positive integers $a$, $b$, and $c$ such that $a + b + c = 20$, determine the number of possible integer values for $\frac{a + b}{c}$.

Answer: 5
user: Real numbers $X_1, X_2, \dots, X_{10}$ are chosen uniformly at random from the interval $[0,1]$. If the expected value of $\min(X_1,X_2,\dots, X_{10})^4$ can be expressed as a rational number $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, what is $m+n$?

Answer: 1002
user: Given that the base-$17$ integer $\overline{8323a02421_{17}}$ (where $a$ is a base-$17$ digit) is divisible by $\overline{16_{10}}$, find $a$. Express your answer in base $10$.

Answer: 7
user: GM Bisain's IQ is so high that he can move around in $10$-dimensional space. He starts at the origin and moves in a straight line away from the origin, stopping after $3$ units. How many lattice points can he land on? A lattice point is one with all integer coordinates.

Answer: 88660
user: Determine the largest two-digit number $d$ with the following property: for any six-digit number $\overline{aabbcc}$, the number $d$ is a divisor of $\overline{aabbcc}$ if and only if $d$ is a divisor of the corresponding three-digit number $\overline{abc}$. 

The numbers $a \ne 0$, $b$, and $c$ need not be different.

Answer: 99
user: Find all positive integers $n$ that have 4 digits, all of which are perfect squares, and such that $n$ is divisible by 2, 3, 5, and 7.

Answer: 4410
user: Define the \textit{bigness} of a rectangular prism to be the sum of its volume, its surface area, and the lengths of all of its edges. Find the least integer $N$ for which there exists a rectangular prism with integer side lengths and \textit{bigness} $N$ and another one with integer side lengths and \textit{bigness} $N + 1$.

Answer: 55
user: Given that there are $24$ primes between $3$ and $100$, inclusive, determine the number of ordered pairs $(p, a)$ where $p$ is a prime number such that $3 \le p < 100$, and $1 \le a < p$. The condition is that the sum:
\[
a + a^2 + a^3 + \cdots + a^{(p-2)!}
\]
is not divisible by $p$.

Answer: 27
user: Alice starts with the number 0. She can apply 100 operations on her number. In each operation, she can either add 1 to her number or square her number. After applying all operations, her score is the minimum distance from her number to any perfect square. What is the maximum score she can attain?

Answer: 94
user: There are 16 cities participating in a football match. Each city has two teams: Team A and Team B. According to the rules, each team can compete with any other team from different cities at most once. After a few days, it is observed that the number of matches played by each team, except for Team A from City 1, is unique. Determine the number of matches that Team A from City 1 has played.

Answer: 15
user: If $x$ is a real number, let $\lfloor x \rfloor$ be the greatest integer that is less than or equal to $x$. If $n$ is a positive integer, let $S(n)$ be defined by
\[
 S(n) 
 = \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
 + 10 \left( n - 10^{\lfloor \log n \rfloor} 
 \cdot \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
 \right) \, .
\]
(All the logarithms are base 10.) How many integers $n$ from 1 to 2011 (inclusive) satisfy $S(S(n)) = n$?

Answer: 108
user: Find the number of subsets of $\{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\}$ where the sum of the elements in the subset is $49$.

Answer: 22
user: Pairwise distinct points $P_1,P_2,\ldots, P_{16}$ lie on the perimeter of a square with side length $4$ centered at $O$ such that $\lvert P_iP_{i+1} \rvert = 1$ for $i=1,2,\ldots, 16$. (We take $P_{17}$ to be the point $P_1$.) We construct points $Q_1,Q_2,\ldots,Q_{16}$ as follows: for each $i$, a fair coin is flipped. If it lands heads, we define $Q_i$ to be $P_i$; otherwise, we define $Q_i$ to be the reflection of $P_i$ over $O$. (So, it is possible for some of the $Q_i$ to coincide.) Let $D$ be the length of the vector $\overrightarrow{OQ_1} + \overrightarrow{OQ_2} + \cdots + \overrightarrow{OQ_{16}}$. Compute the expected value of $D^2$.

Answer: 88
user: Find the smallest positive integer $n$ such that $\underbrace{2^{2^{\cdots^{2}}}}_{n} > 3^{3^{3^3}}$. 

(The notation $\underbrace{2^{2^{\cdots^{2}}}}_{n}$ is used to denote a power tower with $n$ $2$’s. For example, $\underbrace{2^{2^{\cdots^{2}}}}_{4}$ would equal $2^{2^{2^2}}$.)

Answer: 5
user: Let $S$ be the set of numbers of the form $n^5 - 5n^3 + 4n$, where $n$ is an integer that is not a multiple of $3$. What is the largest integer that is a divisor of every number in $S$?

Answer: 360
user: Let $a$, $b$, $c$ be positive integers such that $29a + 30b + 31c = 366$. Find $19a + 20b + 21c$.

Answer: 246
user: Find the 1000th number, in ascending order, that is coprime to $105$.

Answer: 2186
user: For every three-digit natural number $n$ (where the leading digit of $n$ is nonzero), consider the number $n_0$ obtained by eliminating all zero digits from $n$. For example, if $n = 207$, then $n_0 = 27$. Determine the number of three-digit positive integers $n$ for which $n_0$ is a divisor of $n$ and $n_0 \neq n$.

Answer: 93
user: Let $ABCD$ be a rectangle with $AB=10$ and $BC=26$. Let $\omega_1$ be the circle with diameter $\overline{AB}$ and $\omega_2$ be the circle with diameter $\overline{CD}$. Suppose $\ell$ is a common internal tangent to $\omega_1$ and $\omega_2$ and that $\ell$ intersects $AD$ and $BC$ at $E$ and $F$ respectively. What is $EF$?

Answer: 26
user: Kris is asked to compute $\log_{10} (x^y)$, where $y$ is a positive integer and $x$ is a positive real number. However, they misread this as $(\log_{10} x)^y$ and compute this value. Despite the reading error, Kris still got the right answer. Given that $x > 10^{1.5}$, determine the largest possible value of $y$.

Answer: 4
user: Compute the number of ordered triples \((x, y, z)\) of integers satisfying the equation: 
\[ x^2 + y^2 + z^2 = 9. \]

Answer: 30
user: Suppose the polynomial $f(x) = x^{2014}$ is expressed as $f(x) = \sum^{2014}_{k=0} a_k \binom{x}{k}$ for some real numbers $a_0, a_1, \ldots, a_{2014}$. Find the largest integer $m$ such that $2^m$ divides $a_{2013}$.

Answer: 2004
user: Let $s_1, s_2, s_3$ be the three roots of the polynomial $x^3 + x^2 + \frac{9}{2}x + 9$.

Find the value of $a + b + c$ if the expression $\prod_{i=1}^{3}(4s^4_i + 81)$ can be written as $2^a3^b5^c$.

Answer: 16
user: For a positive integer $n$, an $n$-branch $B$ is an ordered tuple $(S_1, S_2, \dots, S_m)$ of nonempty sets (where $m$ is any positive integer) satisfying $S_1 \subset S_2 \subset \dots \subset S_m \subseteq \{1,2,\dots,n\}$. An integer $x$ is said to appear in $B$ if it is an element of the last set $S_m$. Define an $n$-plant to be an (unordered) set of $n$-branches $\{ B_1, B_2, \dots, B_k\}$, and call it perfect if each of $1$, $2$, \dots, $n$ appears in exactly one of its branches.

Let $T_n$ be the number of distinct perfect $n$-plants (where $T_0=1$), and suppose that for some positive real number $x$ we have the convergence \[ \ln \left( \sum_{n \ge 0} T_n \cdot \frac{\left( \ln x \right)^n}{n!} \right) = \frac{6}{29}. \] If $x = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$.

Answer: 76
user: Given that $20^{22} + 1$ has exactly four prime divisors $p_1 < p_2 < p_3 < p_4$, determine $p_1 + p_2$.

Answer: 490
user: Two ants crawl along the sides of the $49$ squares of a $7 \times 7$ board. Each ant passes through all $64$ vertices exactly once and returns to its starting point. What is the smallest possible number of sides covered by both ants?

Answer: 16
user: Alice, Bob, and Carol each independently roll a fair six-sided die and obtain the numbers $a, b, c$, respectively. They then compute the polynomial $f(x) = x^{3} + px^{2} + qx + r$ with roots $a, b, c$. If the expected value of the sum of the squares of the coefficients of $f(x)$ is $\frac{m}{n}$ for relatively prime positive integers $m, n$, find the remainder when $m+n$ is divided by 1000.

Answer: 551
user: Let $S \subset [0, 1]$ be a set of 5 points with $\{0, 1\} \subset S$. The graph of a real function $f : [0, 1] \to [0, 1]$ is continuous and increasing, and it is linear on every subinterval $I$ in $[0, 1]$ such that the endpoints but no interior points of $I$ are in $S$.

We want to compute, using a computer, the extreme values of 
\[ g(x, t) = \frac{f(x+t)-f(x)}{f(x)-f(x-t)} \]
for $x - t, x + t \in [0, 1]$. At how many points $(x, t)$ is it necessary to compute $g(x, t)$ with the computer?

Answer: 75
user: The positive integers $a$ and $b$ are such that the numbers $15a + 16b$ and $16a - 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

Answer: 231361
user: Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?

\[ \textbf{(A)}\ 37 \quad \textbf{(B)}\ 63 \quad \textbf{(C)}\ 117 \quad \textbf{(D)}\ 143 \quad \textbf{(E)}\ 163 \]

Answer: 143
user: Find the largest natural number $n$ for which there exist different sets $S_1, S_2, \ldots, S_n$ such that:

$|S_i \cup S_j| \leq 2004$ for each pair $1 \leq i, j \leq n$.
$S_i \cup S_j \cup S_k = \{1, 2, \ldots, 2008\}$ for each three integers $1 \leq i < j < k \leq n$.

Answer: 32
user: A marble is placed on each $33$ unit square of a $10 \times 10$ chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board?

Answer: 438
user: Let the roots of \(x^{2022} - 7x^{2021} + 8x^2 + 4x + 2\) be \(r_1, r_2, \ldots, r_{2022}\), the roots of \(x^{2022} - 8x^{2021} + 27x^2 + 9x + 3\) be \(s_1, s_2, \ldots, s_{2022}\), and the roots of \(x^{2022} - 9x^{2021} + 64x^2 + 16x + 4\) be \(t_1, t_2, \ldots, t_{2022}\). Compute the value of:
\[
\sum_{1 \le i,j \le 2022} r_i s_j + \sum_{1 \le i,j \le 2022} s_i t_j + \sum_{1 \le i,j \le 2022} t_i r_j.
\]

Answer: 191
user: For all positive integers $n$, let
\[ f(n) = \sum_{k=1}^n \varphi(k) \left\lfloor \frac{n}{k} \right\rfloor^2. \]
Compute $f(2019) - f(2018)$. Here, $\varphi(n)$ denotes the number of positive integers less than or equal to $n$ which are relatively prime to $n$.

Answer: 11431
user: For positive real numbers $x$ and $y$, let $f(x, y) = x^{\log_2 y}$. The sum of the solutions to the equation \[ 4096 f(f(x, x), x) = x^{13} \] can be written in simplest form as $\frac{m}{n}$. Compute $m + n$.

Answer: 177
user: A subset $X$ of $\{1, 2, 3, \ldots, 10000\}$ has the following property: If $a, b$ are distinct elements of $X$, then $ab \not\in X$. What is the maximal number of elements in $X$?

Answer: 9901
user: Neo has an infinite supply of red pills and blue pills. When he takes a red pill, his weight will double, and when he takes a blue pill, he will lose one pound. If Neo originally weighs one pound, what is the minimum number of pills he must take to make his weight \(2015\) pounds?

Answer: 13
user: Find the largest prime number less than $2008$ that is a divisor of some integer in the infinite sequence \[ \left\lfloor\frac{2008^1}{1}\right\rfloor, \left\lfloor\frac{2008^2}{2}\right\rfloor, \left\lfloor\frac{2008^3}{3}\right\rfloor, \left\lfloor\frac{2008^4}{4}\right\rfloor, \ldots. \]

Answer: 2003
user: Find a multiple of $2018$ whose decimal expansion's first four digits are $2017$.

Answer: 20171928
user: Compute the remainder when $2^{30}$ is divided by $1000$.

Answer: 824
user: Let $\alpha$ and $\beta$ be positive integers such that $\frac{43}{197} < \frac{\alpha}{\beta} < \frac{17}{77}$. Find the minimum possible value of $\beta$.

Answer: 32
user: A two-digit positive integer is $\textit{primeable}$ if one of its digits can be deleted to produce a prime number. A two-digit positive integer that is prime, yet not primeable, is $\textit{unripe}$. Compute the total number of unripe integers.

Answer: 5
user: If $57a + 88b + 125c \geq 1148$, where $a, b, c > 0$, what is the minimum value of
\[
a^3 + b^3 + c^3 + 5a^2 + 5b^2 + 5c^2?
\]

Answer: 466
user: Let $f(n)$ denote the largest odd factor of $n$, including possibly $n$ itself. Determine the value of \[ \frac{f(1)}{1} + \frac{f(2)}{2} + \frac{f(3)}{3} + \cdots + \frac{f(2048)}{2048}, \] rounded to the nearest integer.

Answer: 1365
user: Consider the following three lines in the Cartesian plane:

\[
\begin{cases}
\ell_1: & 2x - y = 7\\
\ell_2: & 5x + y = 42\\
\ell_3: & x + y = 14
\end{cases}
\]

Let \( f_i(P) \) correspond to the reflection of the point \( P \) across \( \ell_i \). Suppose \( X \) and \( Y \) are points on the \( x \) and \( y \) axes, respectively, such that \( f_1(f_2(f_3(X)))= Y \). Let \( t \) be the length of segment \( XY \); what is the sum of all possible values of \( t^2 \)?

Answer: 260
user: Find the maximal positive integer $p$ such that $5^7$ is the sum of $p$ consecutive positive integers.

Answer: 250
user: Let $d_1, d_2, \ldots , d_{k}$ be the distinct positive integer divisors of $6^8$. Find the number of ordered pairs $(i, j)$ such that $d_i - d_j$ is divisible by $11$.

Answer: 665
user: Let $\alpha$ be a root of $x^6-x-1$, and call two polynomials $p$ and $q$ with integer coefficients \textit{equivalent} if $p(\alpha)\equiv q(\alpha)\pmod{3}$. It is known that every such polynomial is equivalent to exactly one of $0, 1, x, x^2, \ldots, x^{727}$. Find the largest integer $n < 728$ for which there exists a polynomial $p$ such that $p^3 - p - x^n$ is equivalent to $0$.

Answer: 727
user: How many ways are there to fill in a $2 \times 2$ square grid with the numbers $1, 2, 3,$ and $4$ such that the numbers in any two grid squares that share an edge have an absolute difference of at most $2$?

Answer: 8
user: Find the $2016$th smallest positive integer that satisfies the congruence $x^x \equiv x \pmod{5}$.

Answer: 3360
user: How many positive integers $n$ are there such that $n+2$ divides $(n+18)^2$?

Answer: 7
user: A multiple choice test consists of 100 questions. If a student answers a question correctly, he will get 4 marks; if he answers a question wrongly, he will get \(-1\) mark. He will get 0 mark for an unanswered question. Determine the number of different total marks of the test. (A total mark can be negative.)

Answer: 495
user: Triangle $ABC$ satisfies $AB=104$, $BC=112$, and $CA=120$. Let $\omega$ and $\omega_A$ denote the incircle and $A$-excircle of $\triangle ABC$, respectively. There exists a unique circle $\Omega$ passing through $A$ which is internally tangent to $\omega$ and externally tangent to $\omega_A$. Compute the radius of $\Omega$.

Answer: 49
user: Let $\triangle ABC$ be an isosceles triangle with $AB=AC$ and $BC=14$. The triangle is inscribed in a circle $\omega$. Let $D$ be a point on the ray $BC$ such that $CD=6$. The intersection of $AD$ and $\omega$ is point $E$. Given that $AE=7$, find $AC^2$.

Answer: 105
user: Find the positive integer $k$ such that the roots of the polynomial $x^3 - 15x^2 + kx - 1105$ are three distinct collinear points in the complex plane.

Answer: 271
user: A right regular hexagonal prism has bases $ABCDEF$ and $A'B'C'D'E'F'$ with edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, and $FF'$, each perpendicular to both hexagons. The height of the prism is $5$ and the side length of the hexagons is $6$. A plane $P$ passes through points $A$, $C'$, and $E$. The area of the portion of $P$ contained within the prism can be expressed as $m\sqrt{n}$, where $n$ is not divisible by the square of any prime. Find $m+n$.

Answer: 323
user: How many triples of integers $(a, b, c)$ with $-10 \leq a, b, c \leq 10$ satisfy $a^2 + b^2 + c^2 = (a + b + c)^2$?

Answer: 103
user: For positive integers $n$, let $S_n$ be the set of integers $x$ such that $n$ distinct lines, with no three concurrent, can divide a plane into $x$ regions. For example, $S_2 = \{3, 4\}$ because the plane is divided into 3 regions if the two lines are parallel, and 4 regions otherwise. What is the minimum $i$ such that $S_i$ contains at least 4 elements?

Answer: 4