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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 \( xy \). |
Find all real solutions to the system:
\[ x^3 - 3x = y \]
\[ y^3 - 3y = z \]
\[ z^3 - 3z = x \] |
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 quadrilateral \(ABCD\). |
For what values of parameter \( m \) does the equation \( x^2 + (m+6)|x| + 2m + 9 = 0 \) have two distinct solutions? |
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. |
Given hexagon $ABCDEF$ with $AB=FA=14$ and $BC=CD=DE=EF=4$, determine the length of $AD$. |
Given $a, b, m, n \in \mathbb{N}$, solve the system of equations:
\[ a^2 + b^2 = m^2 - n^2 \]
\[ ab = 2mn \] |
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, you can use more than one divisor \( d \) on the left-hand side. |
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? |
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 \( abc = 1 \). |
Solve the equation \(2x^2 - 73y^2 = 1\) in the set of natural numbers \(\mathbb{N}\). |
Determine the number of squares whose vertices are lattice points on the xy-plane such that \(0 \leq x, y \leq 2023\). |
Determine the general term of the sequence $(a_n)_{n\ge 1}$ given by $a_0=0$, $a_1=b$, and the recurrence relation $$a_{n+1}=a_n\sqrt{1+a_{n-1}^2}+a_{n-1}\sqrt{1+a_n^2}.$$ |
Let \( n \) be a positive integer. On the lattice plane, \( k \) particles are located on points, and they can move in three directions: left, right, or down. Each particle will mark the path it goes through. Determine the least \( k \) (in terms of \( n \)) such that all edges between any two adjacent points \((x_1, y... |
A bug is on a cube. Every minute, it moves to another vertex randomly. What is the probability that it winds up on the original vertex after $n$ moves? |
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the equation
\[ yf(x) - xf(2y) = 8xy(x^2 - y^2). \] |
Find all positive integers \( x \) and \( y \) such that
\[ 5xy \sqrt{(x^2 + y^2)^3} = a^5 + b^5 + c^5 + d^5 \]
where \( a, b, c, d \in \mathbb{N} \). |
Given a real number $\alpha$, find all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying the following property:
\[ f(x^2 + f(y) - y) = (f(x))^2 + \alpha f(y). \] |
Find all integers \( n \) such that \( 2^{n-1}n + 1 \) is a perfect square. |
Find all $k \in \mathbb{R}_{>-2}$ such that the inequality $\frac{x}{x+y+kz} + \frac{y}{y+z+kx} + \frac{z}{z+x+ky} \ge \frac{3}{k+2}$ holds for all $x, y, z \geq 0$. |
An octagon with side lengths 3, 3, 11, 11, 15, 15, 15, 15 cm is inscribed in a circle. What is the area of the octagon in cm²? |
Triangle ABC has integer-length sides, and AC = 2007. The internal bisector of angle BAC meets BC at D. Given that AB = CD, determine the lengths of AB and BC. |
Let \( f(x) = (x^{1999} - 1)^{1999} \). Find \( f^{(1999)}(x) \), the 1999th derivative of \( f(x) \). |
Sketch the curve $C$ given by $x^{3}+y^{3} = axy$ where $x \geq 0$, $y \geq 0$, and $a > 0$. |
Find the sum \( A = 1 \times 5 + 2 \times 55 + 3 \times 555 + \cdots + n \times \overbrace{555\cdots5}^{n} \). |
If $f:\mathbb{R}\rightarrow \mathbb{R}$ and $f(f(x))=x^2-x-1$ for all $x \in \mathbb{R}$, then find $f(0)$. |
Find all integers \( a \) such that there exists an integer \( k \ge 3 \) for which the sequence \( (a, 2a, \ldots, ka) \) can be rearranged so that the sum of any two consecutive numbers in the sequence is a perfect square. |
Given a function \( f(x) \) such that \( x, y \in \mathbb{Q} \), find all solutions to the functional equation \( f(x+y) + f(x-y) = 2f(x) + 2f(y) \). |
Two players, A and B, play a game starting with a positive integer \( N_0 \) on the blackboard. A erases \( N_0 \) and writes \( N_1 \in \{N_0 - 1, \lfloor N_0 / 3 \rfloor\} \). B erases \( N_1 \) and writes \( N_2 \in \{N_1 - 1, \lfloor N_1 / 3 \rfloor\} \). They continue until 0 appears on the blackboard. The player ... |
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \),
\[
(f(x+y))^2 = (f(x))^2 + 2f(yx) + (f(y))^2.
\] |
If the incentre of $\triangle ABC$ is the midpoint of the median $AD$, find $\cos A$. |
Solve the equation $x^5 - x + 1 = y^2$ for integer values of $x$ and $y$. |
Find even numbers \(a_1\), \(a_2\), \(a_3\), \(a_4\) such that they form an arithmetic sequence and satisfy the equation:
\[ (a_2 + a_3 + a_4)(a_1 + a_4) = \left(\frac{a_1 + a_2}{2}\right)^3 \] |
Let $A$ be the set of $n$-tuples of non-negative integers such that the sum of its elements equals $b$. Given $a \in A$, find a simple form for the function $f$ such that $f(a) = |\{b \in A : b > a\}|$. |
Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin \( p_0 = (0,0) \) facing east and walks one unit, arriving at \( p_1 = (1,0) \). For \( n = 1, 2, 3, \ldots \), right after arriving at the point \( p_n \), if Aaron can turn \( 90^\circ \) left and walk one unit to an... |
Solve the equation for positive integers \( m \) and \( n \):
\[
\left \lfloor \frac{m^2}{n} \right \rfloor + \left \lfloor \frac{n^2}{m} \right \rfloor = \left \lfloor \frac{m}{n} + \frac{n}{m} \right \rfloor + mn
\] |
A thief is trying to unlock a security password of length $n$ using numbers from the set $\{1, 2, \ldots, n\}$. If a number in the sequence is correct, it lights up, but if a number is incorrect, all previously lit numbers turn off. Determine the minimum number of attempts the thief must make to guarantee unlocking the... |
Given a parabola passing through the points \( A \left(1, \frac{1-\cos \theta}{\sin \theta}\right) \) and \( B \left(-1, \frac{1+\cos \theta}{\sin \theta}\right) \) with the directrix being the \( x \)-axis, find the loci of the focus and vertex of this parabola as \( \theta \) varies. |
Find all positive integers \( n \) such that \( n = d_6^2 + d_7^2 - 1 \), where \( 1 = d_1 < d_2 < \ldots < d_k = n \) are all the positive divisors of \( n \). |
Find all functions \( f: \mathbb{N} \rightarrow \mathbb{Z} \) such that \( |f(k)| \leq k \) for all positive integers \( k \) and there is a prime number \( p > 2024 \) which satisfies both of the following conditions:
1) For all \( a \in \mathbb{N} \), we have \( af(a+p) = af(a) + pf(a) \),
2) For all \( a \in \mathbb... |
Find the 73rd digit from the end of the number $\underset{2012 \ \text{digits}}{\underbrace{111\dots 1}}^2$. |
What is the probability that for random real numbers \(a\), \(b\), and \(c\), the inequality \((ab)^3 + (ac)^3 + (bc)^3 \ge abc(a^3 + b^3 + c^3)\) holds? |
Point $O$ is the center of the circumscribed circle of triangle $ABC$. Point $X$ is selected on the circumscribed circle of triangle $BOC$ outside triangle $ABC$. On rays $XB$ and $XC$, behind points $B$ and $C$, points $Y$ and $Z$ are selected, respectively, such that $XY = XZ$. The circumscribed circle of triangle $A... |
Find $\sum_{k=1}^n a^{k^2}$. |
How many equilateral triangles are there on a Chinese checker board with 15 triangles along each side? |
How many solutions exist for the functions \( f, g: \mathbb{R} \to \mathbb{R} \) such that \( f(f(-x)) = f(x) \) for all real values of \( x \)? |
Let $n$ points $A_1, A_2, \ldots, A_n$ ($n > 2$) be considered in space, where no four points are coplanar. Each pair of points $A_i, A_j$ are connected by an edge. Find the maximal value of $n$ for which we can paint all edges by two colors – blue and red – such that the following conditions hold:
- Each edge is paint... |
The numbers $1, 2, \dots, 1999$ are written on the board. Two players take turns choosing $a, b$ from the board, erasing them, and writing one of $ab$, $a+b$, or $a-b$. The first player wants the last number on the board to be divisible by $1999$, while the second player wants to prevent this. Determine the winner. |
Express $\tan \frac{\pi}{48}$ in terms of $\sqrt{2}, \sqrt{3}$, and $\sqrt{6}$. |
Find the number of ways to express a natural number as the sum of distinct natural numbers less than or equal to it.
Example: For 6, the representations are:
6 = 6
6 = 2 + 4
6 = 1 + 5
6 = 1 + 2 + 3 |
What is the first non-zero digit in $80!$? |
Find the greatest value of the real number \( c \) such that for every positive integers \( m \) and \( n \), there always exists a real number \( x \) satisfying \( \sin(mx) + \sin(nx) \geq c \). |
Given \( x, y, z \in [1, 2] \), find the maximum and minimum values of the expression:
\[ P = \frac{10x}{yz} + \frac{11y}{xz} + \frac{12z}{xy} \] |
Find all polynomials \( P \) with real coefficients such that for all \( x \in \mathbb{R} \),
\[ xP(x-a) = (x-b)P(x), \]
where \( a \) and \( b \) are given real numbers with \( a \neq 0 \). |
What is the probability that no ant will encounter another, either en route or at the next vertex, when ants at each vertex of a cube simultaneously crawl along an edge to the next vertex, each choosing its path randomly? (Express your answer as a reduced fraction.) |
Let \( n \geq 2 \) be an integer. Find the smallest real number \( \lambda \) such that for any positive real numbers \( x_1, x_2, \ldots, x_n \), the following inequality holds:
\[ x_1^2 + \left(\frac{x_1 + x_2}{2}\right)^2 + \cdots + \left(\frac{x_1 + x_2 + \cdots + x_n}{n}\right)^2 \leq \lambda (x_1^2 + x_2^2 + \cdo... |
Nine cyclists \(A_1, A_2, A_3, A_4, A_5, A_6, A_7, A_8, A_9\) are cycling in that order. They want to change positions such that the leader changes and no one is behind the same person. In how many ways can they do this? |
Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting, where a number $n$ is defined as [i]interesting[/i] if 2018 divides $d(n)$ (the number of positive divisors of $n$). |
Find all non-trivial solutions to the continuous function \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(f(x) + 1) = f(x + 1) \). |
Consider \( n \) points \( P_1, P_2, \ldots, P_n \) on the plane. Determine the largest angle \( x \) for which there exists a point \( P \) outside the convex polygon formed by these points such that \( \angle P_iPP_{i+1} \geq x \) for \( i = 1, 2, \ldots, n \). |
Let \( a, b, c \) be nonnegative real numbers, no two of which are zero. Determine the greatest value of the expression
\[ P = \frac{a\sqrt{k^2b^2 + c^2} + b\sqrt{k^2c^2 + a^2} + c\sqrt{k^2a^2 + b^2}}{(a + b + c)^2} \]
where \( k \) is a real number. |
Find all functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) satisfying
\[ f(f(n)) = n + 2013 \]
for all \( n \in \mathbb{N} \). |
Find the number of positive integers \( n < 3^8 \) such that the number of positive integers \( k \) (where \( 1 \leq k \leq \frac{n}{3} \)) for which \( \frac{n!}{(n-3k)! \cdot k! \cdot 3^{k+1}} \) is not an integer is exactly 216. |
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) satisfying the functional equation
\[ f(x + f(y)) - f(x) = (x + f(y))^4 - x^4 \]
for all \( x, y \in \mathbb{R} \). |
In triangle \(ABC\), if \(a = kb\) and \(\angle A = k \angle B\), determine the range of values for \(k\). |
In the sequence $\frac{3}{4}, \frac{3}{9}, \frac{3}{16}, \ldots, \frac{3}{n^2}, \ldots, \frac{3}{10000}$, determine the number of terms that cannot be expressed in the form $\frac{1}{a} + \frac{1}{b}$ where $a$ and $b$ are natural numbers. |
We consider maps on two spheres, say the earth and the moon. Each region on the earth is a country, and has a corresponding region or colony on the moon. We desire to color the countries and colonies so that:
1) Each country receives the same color as its colony, and
2) If two countries or colonies share a common borde... |
What is the sum of the rational terms in the expansion of $\left(\sqrt{2} + 3\sqrt{3} + 6\sqrt{5}\right)^{10}$? |
In the series from 2006 to 4012, find the summation of the maximum odd divisor of every number. |
Let \(ABC\) be an equilateral triangle. \(P\) is a point in the interior of the triangle such that the square root of the distance between \(P\) and one of the sides of the triangle equals the sum of the square roots of the distances between \(P\) and the other two sides. Find the locus of \(P\). |
Solve the equation \[\sqrt{x^2 + x + 19} + \sqrt{7x^2 + 22x + 28} + \sqrt{3x^2 + 43x + 37} = 3\sqrt{3}(x + 3).\] |
Let \( \triangle ABC \) be an isosceles triangle with \( AB = AC \). Suppose that the angle bisector of \( \angle B \) meets \( AC \) at \( D \) and that \( BC = BD + AD \). Determine \( \angle A \). |
Find all functions \( f: \mathbb{Q} \rightarrow \mathbb{Q} \) such that \( 4f(x)f(y) + \frac{1}{2} = f(2xy + \frac{1}{2}) + f(x - y) \). |
Given a point inside an equilateral triangle and the distances \(a\), \(b\), and \(c\) from the point to each vertex, can we determine the angles between the lines connecting the point to each vertex? |
Determine all functions \( f: \mathbb{N} \to \mathbb{R} \) such that for all integers \( n \geq 1 \),
\[ \log_{n+1} f(n) = \log_{f(n+2)} (n+3). \] |
Solve the equation $x^3 + 1 = 3\sqrt[3]{2x - 1}$. |
Let \( T \) be an acute triangle. Inscribe a rectangle \( R \) in \( T \) with one side along a side of \( T \). Then inscribe a rectangle \( S \) in the triangle formed by the side of \( R \) opposite the side on the boundary of \( T \), and the other two sides of \( T \), with one side along the side of \( R \). For ... |
How many permutations of the sequence $(1, 2, \ldots, n)$ satisfy the condition $s_{i} \leq s_{\lfloor i/2 \rfloor}$ for all $2 \leq i \leq n$, where $s_{i}$ is the value at the $i$-th position in the permutation? |
For all acute angles $\alpha, \beta, \gamma$ such that $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 1$, find the minimum value of $\frac{\sin \alpha + \sin \beta + \sin \gamma}{\cos \alpha + \cos \beta + \cos \gamma}$. |
Solve the equation \((2x^3 + x - 3)^3 = 3 - x^3\). |
Find all positive integers $(a, b, c)$ such that $a^2 + b^2 = 2c^2$. |
A banker has a set of similarly looking golden coins, one of which is counterfeit and lighter in weight. Using a balance without weights, an expert must determine the counterfeit coin with the condition that each coin participates in no more than two weighings. What is the largest number of coins the banker could have ... |
How many positive good numbers are smaller than $1000$? A number $a$ is called a good number if $a^2$ can be divided into some perfect squares (more than 1 and none of them can be 0). |
Solve the equation $\cos 4x - \tan 5x = \cot 5x - \sin 6x$ in the interval $(-\pi, \pi)$. |
Let \( a, b, c \in \mathbb{R}_+ \) such that \( a + b + c = 2005 \). Find the minimum value of the expression:
\[ E = a^{2006} + b^{2006} + c^{2006} + \frac{(ab)^{2004} + (bc)^{2004} + (ca)^{2004}}{(abc)^{2004}} \] |
Find all functions \( f: \mathbb{R} \longrightarrow \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \),
\[ f(2x + f(y)) = f(2x) + x f(2y) + f(f(y)). \] |
Find the formula for the expression:
\[ \sum_{i=1}^n (mi+1)^k \] |
What is the distribution of primes of the form $2^n + 1$? Specifically, how many such primes are there less than a given number $x$? |
Determine all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that, for any real numbers \( x \) and \( y \):
\[ f(x - f(y)) = f(f(y)) + x f(y) + f(x) - 1 \] |
Determine the maximum value of $\lambda$ such that if $f(x) = x^3 + ax^2 + bx + c$ is a cubic polynomial with all its roots nonnegative, then \[f(x) \geq \lambda(x - a)^3\] for all $x \geq 0$. Find the equality condition. |
Given two distinct circles $K_1$ and $K_2$ intersecting at points $A$ and $B$ where $AB$ is the diameter of $K_1$, and a point $P$ on $K_2$ and inside $K_1$, construct two points $C$ and $D$ on $K_1$ such that $CD$ is perpendicular to $AB$ and $\angle CPD$ is a right angle using only a T-square. |
Tom made two rectangles of $2 \times 6$ and $7 \times 8$ from several rectangular tiles measuring $1 \times 3$ and $1 \times 4$, but Jerry snatched and hid one tile from each of the rectangles. Will Tom be able to make a $5 \times 12$ rectangle from the remaining tiles? |
Given an isosceles trapezium ABCD with base AB and the intersection point of the diagonals E, M is the midpoint of AE, N is the midpoint of CD, and P is the foot of the perpendicular from E to AD. If MP is perpendicular to NP, find the angles of the trapezium. |
Given that \( \left | (x-y)(y-z)(z-x) \right | = 3 \), find the minimum value of \( P = (3x^2 + 4)(3y^2 + 4)(3z^2 + 4) \). |
Let $\triangle ABC$ have incenter $I$ and side lengths $AB = c$, $BC = a$, and $CA = b$ such that $b > c > a$. The incircle of the triangle touches $AB$ at $D$, and $E$ is a point on $AB$ such that $CD \parallel IE$. Find the length of $DE$. |
Construct a triangle given one of its vertices \( A \), its centroid \( G \), and its orthocenter \( H \). |
Find all continuous functions \( f: \mathbb{R}^+ \rightarrow \mathbb{R}^+ \) satisfying the equation \( f(f(x)) = 2xf(x) \). |
On a blackboard, all $(n+1)^2$ pairs of the type $(i, j)$ for $i, j \in \{0, 1, 2, \ldots, n\}$ are written. Every minute, Alex chooses two pairs $(x_1, y_1)$ and $(x_2, y_2)$, deletes them, and writes two times the pair $(\min\{x_1, x_2\}, \min\{y_1, y_2\})$. What is the minimum number of moves Alex should make so tha... |
Solve the equation \(x^2 + 5 = y^3\) in integers. |
Let $C_1 , C_2$ be two circles in the plane intersecting at two distinct points. Let $P$ be the midpoint of a variable chord $AB$ of $C_2$ with the property that the circle on $AB$ as diameter meets $C_1$ at a point $T$ such that $P T$ is tangent to $C_1$. Find the locus of $P$. |
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Unique Math Problems (AOPS subset)
This dataset contains 81,901 unique problem statements extracted from the AOPS subset of rakeshb4r/Nemotron-Math-v2.
Dataset Structure
problem_statement(string): The text of the math problem.
Source
Original source: Nemotron-Math-v2
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