text stringlengths 386 113k |
|---|
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Let \(a, b, c, d\) be four positive integers. The integer \(n\) is the smallest positive integer such that the sum of every rational number of the form \(\frac{1}{a^2 + b^3 + c^8 + d^n}\) does not exceed a... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find all functions \( f \colon \mathbb{R} \to \mathbb{R} \) such that
\[
(f(x) + f(y))(f^2(y) - f(y)f(z) + f^2(z)) = xf(x^2 + z^2) - (y - z)f(yz) + y^3 - f(x)f(y)f(z)
\]
for all \( x, y, z \in \mathbb{R} ... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Determine all prime numbers $p$ and all positive integers $x$ and $y$ such that \[x^3 + y^3 = p(xy + p)\]<|im_end|>
<|im_start|>assistant
<think>
We need solutions. Rewrite: x^3+y^3 = (x+y)(x^2 - xy + y^2)... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Let \( m \) and \( n \) be positive integers with \( m \leq n \). Two friends, \( A \) and \( B \), play a cooperative game where \( A \) starts with \( a = 0 \). Player \( B \) then reduces \( a + m \) (m... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find a recurrence relation for the number of ways to distribute $n$ identical coins to $k$ different people so that each person receives between two and four coins.<|im_end|>
<|im_start|>assistant
<think>
... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Given that \( p \) is a prime number and \( x \) and \( y \) are natural numbers such that \(\frac{p^x - 1}{p-1} = 2^y\), find the number of divisors of \( xy \).<|im_end|>
<|im_start|>assistant
<think>
We... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Given the sequence of natural numbers \(a_n\) such that \(a_{n+3} = a_{n+2}(a_{n+1} + a_{n})\), find \(a_7\) if \(a_6 = 8820\).<|im_end|>
<|im_start|>assistant
<think>
We need a_n natural numbers satisfyin... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Is there a triangle ABC where the three sides are consecutive integers and one angle is three times another angle? Can a more general conclusion be made?<|im_end|>
<|im_start|>assistant
<think>
We need tri... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find all real solutions to the system:
\[ x^3 - 3x = y \]
\[ y^3 - 3y = z \]
\[ z^3 - 3z = x \]<|im_end|>
<|im_start|>assistant
<think>
We need real solutions to cyclic system. Let f(t)=t^3-3t. Then equati... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
What is the total number of squares that can be drawn on an \( m \times n \) chessboard?<|im_end|>
<|im_start|>assistant
<think>
We need formula: sum_{k=1}^{min(m,n)} (m - k +1)(n - k +1). Compute closed f... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find all functions \( f, g : \mathbb{R}^+ \to \mathbb{R}^+ \) such that \( f \) is injective and:
\[ f(g(x) + y) = f(x) + g(y) \]
for all positive numbers \( x \) and \( y \).<|im_end|>
<|im_start|>assista... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Let an integer quadrilateral be a quadrilateral whose sides are all of integer length, whose diagonals form $4$ segments of integer length and whose area is also an integer. Find the area of an integer qua... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Given the function \( f(x) = \frac{2x}{1 + x^2} \) and positive numbers \( x_k > 0 \) for \( k = 1, 2, \ldots, n \), determine the minimum and maximum values of the expression \( \sum_{k=1}^{n} f^{-1}(x_k)... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Given a quadrilateral \(ABCD\) with midpoints \(M\) and \(N\) of sides \(AB\) and \(CD\) respectively, and given the lengths \(AB = a\), \(BC = b\), \(CD = c\), \(DA = d\), and \(MN = l\), construct the qu... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find all positive integers \( a \) and \( b \) for which
\[ \left\lfloor \frac{a^2}{b} \right\rfloor + \left\lfloor \frac{b^2}{a} \right\rfloor = \left\lfloor \frac{a^2 + b^2}{ab} \right\rfloor + ab. \]
H... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Given \( x, y, z > 0 \) such that \( 2x + 4y + 7z = 2xyz \), find the minimum of \( S = x + y + z \).<|im_end|>
<|im_start|>assistant
<think>
We need minimize sum given constraint. Use AM-GM? Rearr: 2xyz -... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(f(x) + yz) = x^3 + f(y)f(z) \) for all real numbers \( x, y, z \).<|im_end|>
<|im_start|>assistant
<think>
We look for functions. Try li... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
In triangle \(ABC\), the altitude, angle bisector, and median from \(C\) divide angle \(C\) into four equal angles. Find angle \(B\).<|im_end|>
<|im_start|>assistant
<think>
We need condition: altitude, an... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
For what values of parameter \( m \) does the equation \( x^2 + (m+6)|x| + 2m + 9 = 0 \) have two distinct solutions?<|im_end|>
<|im_start|>assistant
<think>
We consider cases x≥0 and x<0. For x≥0: equatio... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Determine the minimum value of $\frac{IA^3+IB^3+IC^3}{AB^3+BC^3+CA^3}$, where $ABC$ is any triangle, and $I$ is the incenter.<|im_end|>
<|im_start|>assistant
<think>
We need min of sum of cubes distances f... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Given hexagon $ABCDEF$ with $AB=FA=14$ and $BC=CD=DE=EF=4$, determine the length of $AD$.<|im_end|>
<|im_start|>assistant
<think>
We have a hexagon with consecutive sides: AB=14, BC=4, CD=4, DE=4, EF=4, FA... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
On a plane, consider a triangle \(ABC\), a circle \(\Gamma\), a point \(P \in \Gamma\), and a point \(Q\). The perpendicular line from \(P\) to \(BC\) intersects \(AQ\) at \(A'\), and similarly define \(B'... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Let $\triangle ABC$ have side lengths $AB=30$, $BC=32$, and $AC=34$. Point $X$ lies in the interior of $\overline{BC}$, and points $I_1$ and $I_2$ are the incenters of $\triangle ABX$ and $\triangle ACX$, ... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Given $a, b, m, n \in \mathbb{N}$, solve the system of equations:
\[ a^2 + b^2 = m^2 - n^2 \]
\[ ab = 2mn \]<|im_end|>
<|im_start|>assistant
<think>
We need integer natural solutions. From equations: (a+ b... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
For all integers \( n \) and \( d \) where \( d \) is a divisor of \( n \), do there exist integers \( y, z, t, s, a \) such that \( z \geq 2 \) and
\[ y^z \cdot n^t \cdot d^s + 1 = (n+1)^a \]
Additionally... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find (if they exist) natural numbers \( a_1, a_2, \ldots, a_{19} \) such that every natural number \( \leq 999999 \) can be expressed as the sum of \( a_{i_1}, \ldots, a_{i_k} \) with \( k \leq 19 \) and \... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
What are the properties of a convex hexagon such that each diagonal cuts off a triangle whose area is not less than $\frac{1}{6}$ of the area of the hexagon?<|im_end|>
<|im_start|>assistant
<think>
We need... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Solve the equation \(\frac{\sqrt{3}-1}{\sin x} + \frac{\sqrt{3}+1}{\cos x} = 4\sqrt{2}\).<|im_end|>
<|im_start|>assistant
<think>
We need solve for x (presumably real) where expression equals 4√2. Solve:
... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Using a compass and a ruler, construct a triangle $ABC$ given the lengths of sides $b$ and $c$, and the length of the segment $AI$, where $I$ is the incenter of the triangle.<|im_end|>
<|im_start|>assistan... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Hannusya, Petrus, and Mykolka each drew an isosceles triangle \(ABC\) with integer degree angles. The bases \(AC\) of these triangles are equal, and for each triangle, there is a point \(E\) on the ray \(B... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find all polynomials \( f(x,y,z) \) with real coefficients such that \( f\left(a + \frac{1}{a}, b + \frac{1}{b}, c + \frac{1}{c}\right) = 0 \) whenever \( abc = 1 \).<|im_end|>
<|im_start|>assistant
<think... |
<|im_start|>user
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Solve the equation \(2x^2 - 73y^2 = 1\) in the set of natural numbers \(\mathbb{N}\).<|im_end|>
<|im_start|>assistant
<think>
We need solutions to Pell equation 2x^2 -73 y^2 =1. Rewrite as 2x^2 -1 =73 y^2 ... |
End of preview. Expand in Data Studio
No dataset card yet
- Downloads last month
- 21