problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | difficulty float64 0.75 2.02k | difficulty_raw listlengths 3 8 |
|---|---|---|---|---|
Find all functions $ f: (0, \infty) \mapsto (0, \infty)$ (so $ f$ is a function from the positive real numbers) such that
\[ \frac {\left( f(w) \right)^2 \plus{} \left( f(x) \right)^2}{f(y^2) \plus{} f(z^2) } \equal{} \frac {w^2 \plus{} x^2}{y^2 \plus{} z^2}
\]
for all positive real numbers $ w,x,y,z,$ satisfying $ wx ... | f(x) = x \text{ or } f(x) = \frac{1}{x} |
To find all functions \( f: (0, \infty) \to (0, \infty) \) satisfying the given functional equation:
\[
\frac {\left( f(w) \right)^2 + \left( f(x) \right)^2}{f(y^2) + f(z^2) } = \frac {w^2 + x^2}{y^2 + z^2}
\]
for all positive real numbers \( w, x, y, z \) such that \( wx = yz \), we proceed as follows:
### Step 1:... | 7.875 | [
8,
8,
7,
7,
9,
8,
8,
8
] |
Find the smallest integer $n$ such that each subset of $\{1,2,\ldots, 2004\}$ with $n$ elements has two distinct elements $a$ and $b$ for which $a^2-b^2$ is a multiple of $2004$. | 1003 |
To solve the problem of finding the smallest integer \( n \) such that each subset of \(\{1, 2, \ldots, 2004\}\) with \( n \) elements has two distinct elements \( a \) and \( b \) for which \( a^2 - b^2 \) is a multiple of \( 2004 \), we start by analyzing the structure of the number \( 2004 \).
Firstly, factorize \... | 5.5 | [
6,
6,
6,
5,
5,
6,
4,
6
] |
Find all positive integers $n$ such that:
\[ \dfrac{n^3+3}{n^2+7} \]
is a positive integer. | 2 \text{ and } 5 | To determine the positive integers \( n \) such that the expression
\[
\dfrac{n^3 + 3}{n^2 + 7}
\]
is a positive integer, we need to analyze when the expression simplifies to a whole number.
### Step 1: Dividing the Polynomials
Consider the division:
\[
\dfrac{n^3 + 3}{n^2 + 7} = q(n) + \dfrac{r(n)}{n^2 + 7},
\]
... | 4.125 | [
5,
4,
4,
4,
4,
4,
4,
4
] |
Let $ f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $ f(f(n) \plus{} f(m)) \equal{} m \plus{} n$ for all positive integers $ n,m.$ Find the possible value for $ f(1988).$ | 1988 |
We start with the functional equation provided:
\[
f(f(n) + f(m)) = m + n
\]
for all positive integers \( n \) and \( m \).
Our goal is to find the possible value for \( f(1988) \).
1. **Substitute special values:**
Let \( n = m = 1 \):
\[
f(f(1) + f(1)) = 2
\]
Let \( n = m = 2 \):
\[
f(f(... | 6 | [
6,
6,
5,
6,
6,
6,
7,
6
] |
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term. | M > 1 |
Consider the sequence \( a_0, a_1, a_2, \ldots \) defined by:
\[
a_0 = M + \frac{1}{2}
\]
and
\[
a_{k+1} = a_k \lfloor a_k \rfloor \quad \text{for} \quad k = 0, 1, 2, \ldots
\]
We are tasked with finding all positive integers \( M \) such that at least one term in the sequence is an integer.
### Analysis of the S... | 6.125 | [
6,
6,
6,
6,
6,
6,
7,
6
] |
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$
[i]Israel[/i] | 8 |
Let \( a, b, c, \) and \( d \) be positive real numbers such that \((a+c)(b+d) = ac + bd\). We are tasked with finding the smallest possible value of
\[
S = \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a}.
\]
To solve this problem, we start by analyzing the condition \((a+c)(b+d) = ac + bd\). Expanding the lef... | 6.625 | [
6,
6,
6,
7,
8,
6,
6,
8
] |
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$... | \mathbb{Z} |
To find all rootiful sets of integers \( S \) that contain all numbers of the form \( 2^a - 2^b \) for positive integers \( a \) and \( b \), we need to analyze the properties of such sets.
### Step 1: Understand the Definition
A set \( S \) is rootiful if, for any positive integer \( n \) and any integers \( a_0, a... | 6.75 | [
8,
7,
6,
6,
7,
7,
7,
6
] |
Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$. | n = 2^k |
We want to determine all positive integers \( n \) for which there exists an integer \( m \) such that \( 2^n - 1 \mid m^2 + 9 \).
To solve this problem, we start by expressing the divisibility condition explicitly:
\[
2^n - 1 \mid m^2 + 9 \quad \Rightarrow \quad m^2 + 9 = k(2^n - 1) \text{ for some integer } k.
\]
... | 6.375 | [
7,
6,
6,
7,
6,
6,
6,
7
] |
An [i]animal[/i] with $n$ [i]cells[/i] is a connected figure consisting of $n$ equal-sized cells[1].
A [i]dinosaur[/i] is an animal with at least $2007$ cells. It is said to be [i]primitive[/i] it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive din... | 4n-3 |
A **dinosaur** is a **polyomino** having at least 2007 cells that is also **primitive**, meaning it cannot be split into smaller dinosaurs. We need to determine the maximum number of cells in a primitive dinosaur.
To tackle this problem, let's consider a primitive dinosaur with \( n \) cells. The goal is to determine... | 6.5 | [
6,
8,
6,
7,
7,
6,
6,
6
] |
Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation\[f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)\]for all real numbers $x$ and $y$.
[i] | f(x) = 2 - x \text{ and } f(x) = x |
To solve the functional equation:
\[
f(x + f(x+y)) + f(xy) = x + f(x+y) + yf(x)
\]
for all \( x, y \in \mathbb{R} \), we start by considering particular values for \( x \) and \( y \) to simplify the equation and gain insight into the form of the function \( f \).
### Step 1: Substitute \( y = 0 \)
Let \( y = 0 \)... | 7.375 | [
7,
7,
8,
8,
7,
7,
7,
8
] |
Find the smallest positive real $\alpha$, such that $$\frac{x+y} {2}\geq \alpha\sqrt{xy}+(1 - \alpha)\sqrt{\frac{x^2+y^2}{2}}$$ for all positive reals $x, y$. | \frac{1}{2} |
To solve the problem of finding the smallest positive real \(\alpha\) such that
\[
\frac{x+y}{2} \geq \alpha\sqrt{xy} + (1 - \alpha)\sqrt{\frac{x^2 + y^2}{2}}
\]
for all positive reals \(x\) and \(y\), we proceed as follows:
### Step 1: Analyze Special Cases
1. **Case \(x = y\):**
If \(x = y\), then both side... | 5.875 | [
5,
6,
6,
6,
6,
6,
6,
6
] |
Let $ABCD$ be a convex quadrilateral with $AB = AD$ and $CB = CD$. The bisector of $\angle BDC$ intersects $BC$ at $L$, and $AL$ intersects $BD$ at $M$, and it is known that $BL = BM$. Determine the value of $2\angle BAD + 3\angle BCD$. | 540^\circ |
Let \(ABCD\) be a convex quadrilateral where \(AB = AD\) and \(CB = CD\). Given that the bisector of \(\angle BDC\) intersects \(BC\) at \(L\), and \(AL\) intersects \(BD\) at \(M\), we are informed that \(BL = BM\). We are to determine the value of \(2\angle BAD + 3\angle BCD\).
First, note the following properties ... | 6.5 | [
7,
6,
7,
7,
6,
6,
7,
6
] |
Given any set $A = \{a_1, a_2, a_3, a_4\}$ of four distinct positive integers, we denote the sum $a_1 +a_2 +a_3 +a_4$ by $s_A$. Let $n_A$ denote the number of pairs $(i, j)$ with $1 \leq i < j \leq 4$ for which $a_i +a_j$ divides $s_A$. Find all sets $A$ of four distinct positive integers which achieve the largest pos... | \{k, 5k, 7k, 11k\} \text{ and } \{k, 11k, 19k, 29k\} |
Let \( A = \{ a_1, a_2, a_3, a_4 \} \) be a set of four distinct positive integers. We define \( s_A = a_1 + a_2 + a_3 + a_4 \) as the sum of these integers. We also define \( n_A \) as the number of pairs \( (i, j) \) with \( 1 \leq i < j \leq 4 \) such that \( a_i + a_j \) divides \( s_A \).
Our goal is to find all... | 6.625 | [
7,
6,
7,
7,
7,
6,
6,
7
] |
In a math test, there are easy and hard questions. The easy questions worth 3 points and the hard questions worth D points.\\
If all the questions begin to worth 4 points, the total punctuation of the test increases 16 points.\\
Instead, if we exchange the questions scores, scoring D points for the easy questions and 3... | 20 |
Let \( x \) be the number of hard questions and \( 9x \) be the number of easy questions in the test. Let the total number of questions be \( n = x + 9x = 10x \).
Given:
- Easy questions are worth 3 points each.
- Hard questions are worth \( D \) points each.
**Initial Total Points**
The initial total score of the... | 5.25 | [
5,
5,
4,
5,
5,
6,
6,
6
] |
Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that \[ (a-1)(b-1)(c-1) \] is a divisor of $abc-1.$ | (2, 4, 8) \text{ and } (3, 5, 15) |
We are tasked with finding all integers \( a, b, c \) with \( 1 < a < b < c \) such that
\[
(a-1)(b-1)(c-1)
\]
is a divisor of
\[
abc - 1.
\]
Let's first express \( abc - 1 \) in terms of potential divisors' expressions:
1. We want \((a-1)(b-1)(c-1) \mid abc - 1\), meaning \((a-1)(b-1)(c-1)\) divides \(abc - 1\... | 5.5 | [
6,
5,
6,
5,
6,
5,
5,
6
] |
Find the roots of the equation $(x-a)(x-b)=(x-c)(x-d)$, if you know that $a+d=b+c=2015$ and $a \ne c$ (numbers $a, b, c, d$ are not given). | \frac{2015}{2} |
To find the roots of the equation \((x-a)(x-b)=(x-c)(x-d)\), given that \(a + d = b + c = 2015\) and \(a \neq c\), we will simplify the equation and determine the solutions.
### Step 1: Expand Both Sides
Expanding both sides of the equation, we have:
\[
(x-a)(x-b) = x^2 - (a+b)x + ab
\]
\[
(x-c)(x-d) = x^2 - (c+d)x +... | 4.75 | [
5,
4,
5,
4,
4,
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5,
6
] |
For any integer $n\geq 2$, let $N(n)$ be the maxima number of triples $(a_i, b_i, c_i)$, $i=1, \ldots, N(n)$, consisting of nonnegative integers $a_i$, $b_i$ and $c_i$ such that the following two conditions are satisfied:
[list][*] $a_i+b_i+c_i=n$ for all $i=1, \ldots, N(n)$,
[*] If $i\neq j$ then $a_i\neq a_j$, $b_i\... | \left\lfloor \frac{2n}{3} \right\rfloor + 1 |
To determine \( N(n) \), the maximum number of triples \((a_i, b_i, c_i)\) where each \( a_i, b_i, c_i \) are nonnegative integers satisfying the conditions:
1. \( a_i + b_i + c_i = n \) for all \( i = 1, \ldots, N(n) \),
2. If \( i \neq j \) then \( a_i \neq a_j \), \( b_i \neq b_j \), and \( c_i \neq c_j \),
we pr... | 6.125 | [
6,
6,
6,
6,
6,
7,
6,
6
] |
Determine all positive integers $ n\geq 2$ that satisfy the following condition: for all $ a$ and $ b$ relatively prime to $ n$ we have \[a \equiv b \pmod n\qquad\text{if and only if}\qquad ab\equiv 1 \pmod n.\] | 2, 3, 4, 6, 8, 12, 24 | To determine all positive integers \( n \geq 2 \) that satisfy the given condition, we need to analyze when \( a \equiv b \pmod{n} \) if and only if \( ab \equiv 1 \pmod{n} \) for all \( a \) and \( b \) that are relatively prime to \( n \).
### Step 1: Analyze the given condition
The problem requires:
- \( a \equiv ... | 6.625 | [
6,
7,
7,
7,
6,
6,
7,
7
] |
When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.) | 7 |
To solve the problem, we need to determine the sum of the digits of \( B \), which is derived from processing the large number \( 4444^{4444} \).
### Step 1: Determine the sum of the digits of \( 4444^{4444} \).
The first step is to find \( A \), the sum of the digits of the number \( 4444^{4444} \). Direct computat... | 6 | [
6,
5,
5,
6,
7,
7,
6,
6
] |
Consider $9$ points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of $\,n\,$ such that whenever exactly $\,n\,$ edges are colored, the set of colored edges necessarily co... | 33 |
Consider a configuration where you have 9 points in space, with each pair of points joined by an edge, for a total of \(\binom{9}{2} = 36\) edges. We want to find the smallest \( n \) such that if exactly \( n \) edges are colored (either blue or red), there must exist a monochromatic triangle (a triangle with all edg... | 6.25 | [
6,
6,
8,
6,
6,
6,
6,
6
] |
For a positive integer $n$, let $d(n)$ be the number of all positive divisors of $n$. Find all positive integers $n$ such that $d(n)^3=4n$. | 2, 128, 2000 |
To solve the problem, we need to find all positive integers \( n \) such that the cube of the number of divisors of \( n \), denoted \( d(n)^3 \), is equal to \( 4n \). The equation we need to solve is:
\[
d(n)^3 = 4n.
\]
First, recall that for a number \( n \) with the prime factorization \( n = p_1^{a_1} p_2^{a_2} ... | 5.75 | [
6,
6,
6,
6,
5,
5,
6,
6
] |
An acute triangle $ABC$ is inscribed in a circle of radius 1 with centre $O;$ all the angles of $ABC$ are greater than $45^\circ.$
$B_{1}$ is the foot of perpendicular from $B$ to $CO,$ $B_{2}$ is the foot of perpendicular from $B_{1}$ to $AC.$
Similarly, $C_{1}$ is the foot of perpendicular from $C$ to $BO,$ $C_{2}$... | \frac{1}{2} |
Given an acute triangle \( ABC \) inscribed in a circle with radius 1 and center \( O \), where all angles of \( \triangle ABC \) are greater than \( 45^\circ \), we are tasked with finding the circumradius of triangle \( A_3B_3C_3 \). The construction is defined as follows:
- \( B_1 \): foot of the perpendicular fro... | 7.25 | [
6,
7,
7,
8,
7,
8,
7,
8
] |
Today, Ivan the Confessor prefers continuous functions $f:[0,1]\to\mathbb{R}$ satisfying $f(x)+f(y)\geq |x-y|$ for all pairs $x,y\in [0,1]$. Find the minimum of $\int_0^1 f$ over all preferred functions.
( | \frac{1}{4} |
We are given a continuous function \( f: [0, 1] \to \mathbb{R} \) that satisfies the inequality \( f(x) + f(y) \geq |x-y| \) for all \( x, y \in [0, 1] \). Our goal is to find the minimum value of the integral \(\int_0^1 f(x) \, dx\).
### Step-by-Step Analysis:
1. **Understanding the Inequality:**
The condition ... | 6.625 | [
6,
6,
6,
7,
7,
6,
7,
8
] |
Consider the system \begin{align*}x + y &= z + u,\\2xy & = zu.\end{align*} Find the greatest value of the real constant $m$ such that $m \leq x/y$ for any positive integer solution $(x,y,z,u)$ of the system, with $x \geq y$. | 3 + 2\sqrt{2} |
To solve this problem, we need to analyze the given system of equations:
\[
\begin{align*}
1) \quad & x + y = z + u,\\
2) \quad & 2xy = zu.
\end{align*}
\]
Our goal is to find the greatest value of the real constant \( m \) such that \( m \leq \frac{x}{y} \) for any positive integer solution \((x, y, z, u)\) with \(... | 6.625 | [
6,
7,
7,
7,
6,
7,
6,
7
] |
Let $n>1$ be an integer. For each numbers $(x_1, x_2,\dots, x_n)$ with $x_1^2+x_2^2+x_3^2+\dots +x_n^2=1$, denote
$m=\min\{|x_i-x_j|, 0<i<j<n+1\}$
Find the maximum value of $m$. | {m \leq \sqrt{\frac{12}{n(n-1)(n+1)}}} |
Let \( n > 1 \) be an integer. For any set of numbers \((x_1, x_2, \ldots, x_n)\) such that the condition \( x_1^2 + x_2^2 + x_3^2 + \cdots + x_n^2 = 1 \) holds, we need to determine the maximum possible value of \( m \), where:
\[
m = \min\{|x_i - x_j| \mid 1 \leq i < j \leq n\}.
\]
Our goal is to find the maximum ... | 6.25 | [
7,
6,
6,
6,
7,
6,
6,
6
] |
Let $n \ge 3$ be an integer. What is the largest possible number of interior angles greater than $180^\circ$ in an $n$-gon in the plane, given that the $n$-gon does not intersect itself and all its sides have the same length? | 0 |
Let \( n \ge 3 \) be an integer, and consider an \( n \)-gon in the plane with equal side lengths. We are asked to find the largest possible number of interior angles greater than \( 180^\circ \), given that the \( n \)-gon does not intersect itself.
To solve this, we will use the following geometric principles:
1. ... | 4 | [
4,
4,
4,
4,
4,
4,
4,
4
] |
Let $\mathbb{Z}$ be the set of all integers. Find all pairs of integers $(a,b)$ for which there exist functions $f \colon \mathbb{Z}\rightarrow \mathbb{Z}$ and $g \colon \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying
\[ f(g(x))=x+a \quad\text{and}\quad g(f(x))=x+b \]
for all integers $x$. | $\left | a \right | = \left | b \right |$ |
We are tasked with finding all pairs of integers \((a, b)\) such that there exist functions \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) and \( g: \mathbb{Z} \rightarrow \mathbb{Z} \) satisfying the conditions:
\[ f(g(x)) = x + a \quad \text{and} \quad g(f(x)) = x + b \]
for all integers \( x \).
To solve this proble... | 7.25 | [
7,
8,
8,
7,
7,
7,
7,
7
] |
Determine all the pairs $ (p , n )$ of a prime number $ p$ and a positive integer $ n$ for which $ \frac{ n^p + 1 }{p^n + 1} $ is an integer. | $(p,n)=(p,p),(2,4)$ |
To solve the problem of finding all pairs \((p, n)\) of a prime number \(p\) and a positive integer \(n\) for which \(\frac{n^p + 1}{p^n + 1}\) is an integer, we start by analyzing the expression:
\[
\frac{n^p + 1}{p^n + 1}.
\]
**Step 1: Initial observation**
We need to determine when this ratio is an integer. Clea... | 6.375 | [
7,
6,
7,
6,
6,
7,
6,
6
] |
For a given positive integer $ k$ denote the square of the sum of its digits by $ f_1(k)$ and let $ f_{n\plus{}1}(k) \equal{} f_1(f_n(k)).$ Determine the value of $ f_{1991}(2^{1990}).$ | 256 | Let \( k \) be a positive integer, and define the function \( f_1(k) \) as the square of the sum of the digits of \( k \). We are also given a recursive function \( f_{n+1}(k) = f_1(f_n(k)) \). We need to find the value of \( f_{1991}(2^{1990}) \).
### Step-by-Step Solution:
1. **Calculate the Sum of Digits of \( 2^{... | 6.625 | [
7,
7,
7,
6,
7,
6,
7,
6
] |
Find all pairs of natural numbers $ (a, b)$ such that $ 7^a \minus{} 3^b$ divides $ a^4 \plus{} b^2$.
[i]Author: Stephan Wagner, Austria[/i] | (2, 4) |
To find all pairs of natural numbers \((a, b)\) such that \(7^a - 3^b\) divides \(a^4 + b^2\), we proceed as follows:
1. Let \(d = 7^a - 3^b\). We need to ensure \(d \mid a^4 + b^2\). This implies that \(a^4 + b^2 = k \cdot (7^a - 3^b)\) for some integer \(k\).
2. We know that for any potential solution, \(7^a > 3^b... | 6.75 | [
6,
7,
7,
7,
7,
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6,
7
] |
Let $P$ be a regular $2006$-gon. A diagonal is called [i]good[/i] if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$. The sides of $P$ are also called [i]good[/i].
Suppose $P$ has been dissected into triangles by $2003$ diagonals, no two of which have a common poi... | 1003 |
Let \( P \) be a regular 2006-gon. We are tasked with finding the maximum number of isosceles triangles that can be formed by dissecting \( P \) using 2003 diagonals such that each triangle has two good sides, where a side is called good if it divides the boundary of \( P \) into two parts, each having an odd number o... | 7.125 | [
7,
6,
7,
8,
7,
7,
8,
7
] |
Let $n>1$ be a positive integer. Ana and Bob play a game with other $n$ people. The group of $n$ people form a circle, and Bob will put either a black hat or a white one on each person's head. Each person can see all the hats except for his own one. They will guess the color of his own hat individually.
Before Bob dis... | \left\lfloor \frac{n-1}{2} \right\rfloor |
Given a group of \( n \) people forming a circle, Ana and Bob play a strategy-based game where Bob assigns each person either a black hat or a white hat. The challenge is that each person can see every other hat except their own. The goal is for Ana to devise a strategy to maximize the number of correct guesses about ... | 6.375 | [
7,
7,
6,
6,
7,
6,
6,
6
] |
Evaluate
\[\left \lfloor \ \prod_{n=1}^{1992} \frac{3n+2}{3n+1} \ \right \rfloor\] | 12 |
Given the problem, we want to evaluate:
\[
\left\lfloor \prod_{n=1}^{1992} \frac{3n+2}{3n+1} \right\rfloor
\]
To solve this, we will analyze the product:
\[
P = \prod_{n=1}^{1992} \frac{3n+2}{3n+1}
\]
### Step 1: Simplify the Expression
Write the product as follows:
\[
P = \frac{5}{4} \cdot \frac{8}{7} \cdot \fr... | 5.75 | [
6,
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6,
6,
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5,
6
] |
Find all ordered pairs of positive integers$ (x, y)$ such that:$$x^3+y^3=x^2+42xy+y^2.$$ | $(7, 1), (1, 7), (22,22)$ | To find all ordered pairs of positive integers \((x, y)\) such that \(x^3 + y^3 = x^2 + 42xy + y^2\), we start by rewriting the given equation as follows:
\[
x^3 + y^3 - x^2 - y^2 - 42xy = 0
\]
We rearrange and factor the left-hand side:
\[
(x^3 - x^2) + (y^3 - y^2) = 42xy
\]
This equation can be simplified by fact... | 5.75 | [
6,
5,
6,
6,
5,
7,
6,
5
] |
Find all positive integers $a$ and $b$ such that
\[ {a^2+b\over b^2-a}\quad\mbox{and}\quad{b^2+a\over a^2-b} \]
are both integers. | (2,2)(3,3)(1,2)(2,1)(2,3)(3,2) |
We are tasked with finding all positive integer pairs \((a, b)\) such that both expressions
\[
\frac{a^2 + b}{b^2 - a}
\]
and
\[
\frac{b^2 + a}{a^2 - b}
\]
are integers.
### Analysis and Approach
To solve this problem, we'll start by analyzing the conditions under which each expression is an integer:
1. **Firs... | 6.125 | [
6,
6,
6,
6,
6,
6,
7,
6
] |
A positive integer $n$ is $inverosimil$ if there exists $n$ integers not necessarily distinct such that the sum and the product of this integers are equal to $n$. How many positive integers less than or equal to $2022$ are $inverosimils$? | 1010 |
We are tasked with determining how many positive integers \( n \leq 2022 \) are $inversosimil$, which means \( n \) can be expressed with \( n \) integers such that both the sum and the product of these integers equal \( n \).
To solve this problem, let's first consider the sequence of integers that can satisfy the c... | 5.625 | [
5,
5,
5,
6,
4,
7,
7,
6
] |
Determine all functions $f : \mathbb{R}^2 \to\mathbb {R}$ for which \[f(A)+f(B)+f(C)+f(D)=0,\]whenever $A,B,C,D$ are the vertices of a square with side-length one. | $f(x)=0$ | To determine all functions \( f : \mathbb{R}^2 \to \mathbb{R} \) for which
\[
f(A) + f(B) + f(C) + f(D) = 0,
\]
whenever \( A, B, C, D \) are the vertices of a square with side-length one, consider the following steps:
1. **Translation Invariance**: The property holds for any square, particularly for squares center... | 7.5 | [
7,
7,
8,
7,
8,
8,
7,
8
] |
Among a group of 120 people, some pairs are friends. A [i]weak quartet[/i] is a set of four people containing exactly one pair of friends. What is the maximum possible number of weak quartets ? | 4769280 |
Given a group of 120 people, where some pairs are friends, we need to determine the maximum possible number of weak quartets. A weak quartet is defined as a set of four people containing exactly one pair of friends.
To solve this, we need to analyze the structure of weak quartets:
1. **Count the total number of qua... | 5.875 | [
5,
6,
6,
6,
6,
6,
6,
6
] |
Let $n > 0$ be an integer. We are given a balance and $n$ weights of weight $2^0, 2^1, \cdots, 2^{n-1}$. We are to place each of the $n$ weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed ... | (2n-1)!! |
Consider an integer \( n > 0 \) and a balance with \( n \) weights of weights \( 2^0, 2^1, \ldots, 2^{n-1} \). Our task is to place each of these weights on the balance, one by one, so that the right pan is never heavier than the left pan. We aim to determine the number of ways to achieve this.
### Understanding the ... | 6.125 | [
6,
6,
6,
6,
6,
6,
6,
7
] |
Let $a$ and $b$ be positive real numbers such that $3a^2 + 2b^2 = 3a + 2b$. Find the minimum value of $A =\sqrt{\frac{a}{b(3a+2)}} + \sqrt{\frac{b}{a(2b+3)}} $ | \frac{2}{\sqrt{5}} |
Let \( a \) and \( b \) be positive real numbers such that:
\[
3a^2 + 2b^2 = 3a + 2b.
\]
We aim to find the minimum value of:
\[
A = \sqrt{\frac{a}{b(3a+2)}} + \sqrt{\frac{b}{a(2b+3)}}.
\]
First, observe the given equality:
\[
3a^2 + 2b^2 = 3a + 2b.
\]
Rearrange the terms:
\[
3a^2 - 3a + 2b^2 - 2b = 0.
\]
Rewr... | 6 | [
6,
6,
6,
6,
6,
6,
6,
6
] |
Let $n\ge 2$ be a given integer. Find the greatest value of $N$, for which the following is true: there are infinitely many ways to find $N$ consecutive integers such that none of them has a divisor greater than $1$ that is a perfect $n^{\mathrm{th}}$ power. | 2^n - 1 |
Let \( n \geq 2 \) be a given integer. We are tasked with finding the greatest value of \( N \) such that there are infinitely many ways to select \( N \) consecutive integers where none of them has a divisor greater than 1 that is a perfect \( n^{\text{th}} \) power.
To solve this, consider the properties of divisor... | 6.25 | [
6,
6,
6,
6,
6,
7,
7,
6
] |
Let $R^+$ be the set of positive real numbers. Determine all functions $f:R^+$ $\rightarrow$ $R^+$ such that for all positive real numbers $x$ and $y:$
$$f(x+f(xy))+y=f(x)f(y)+1$$
[i]Ukraine | f(x) = x + 1 |
Let \( R^+ \) be the set of positive real numbers. We need to determine all functions \( f: R^+ \rightarrow R^+ \) such that for all positive real numbers \( x \) and \( y \), the following equation holds:
\[
f(x + f(xy)) + y = f(x)f(y) + 1
\]
### Step-by-Step Solution:
1. **Assumption and Simplification:**
Let'... | 7 | [
7,
6,
6,
7,
6,
8,
8,
8
] |
Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition
\[ f(m \plus{} n) \geq f(m) \plus{} f(f(n)) \minus{} 1
\]
for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$
[i]Author: Nikolai Nikolov, Bulgaria[/i] | 1, 2, \ldots, 2008 |
To solve the problem, we need to determine all possible values of \( f(2007) \) for functions \( f: \mathbb{N} \to \mathbb{N} \) that satisfy the given functional inequality:
\[
f(m + n) \geq f(m) + f(f(n)) - 1
\]
for all \( m, n \in \mathbb{N} \).
Firstly, let's consider the functional inequality with the specific... | 6.25 | [
6,
6,
6,
6,
6,
7,
7,
6
] |
The diagonals of parallelogram $ABCD$ intersect at $E$ . The bisectors of $\angle DAE$ and $\angle EBC$ intersect at $F$. Assume $ECFD$ is a parellelogram . Determine the ratio $AB:AD$. | {\sqrt{3}} |
Let's denote the lengths \( AB = x \) and \( AD = y \). We are tasked with finding the ratio \( \frac{x}{y} \).
Since \( ABCD \) is a parallelogram, its diagonals \( AC \) and \( BD \) bisect each other at \( E \). Therefore, \( AE = EC \) and \( BE = ED \).
Given that \( ECFD \) is a parallelogram, \( EC \parallel ... | 5.5 | [
5,
6,
5,
4,
6,
6,
6,
6
] |
Let $X_r=x^r+y^r+z^r$ with $x,y,z$ real. It is known that if $S_1=0$, \[(*)\quad\frac{S_{m+n}}{m+n}=\frac{S_m}{m}\frac{S_n}{n}\] for $(m,n)=(2,3),(3,2),(2,5)$, or $(5,2)$. Determine [i]all[/i] other pairs of integers $(m,n)$ if any, so that $(*)$ holds for all real numbers $x,y,z$ such that $x+y+z=0$. | (2, 3), (3, 2), (2, 5), (5, 2) |
Let's start by understanding the problem statement correctly. We have a sequence defined by
\[ S_r = x^r + y^r + z^r \]
where \( x, y, \) and \( z \) are real numbers. We are informed that if \( S_1 = x + y + z = 0 \), then the following relationship holds:
\[
(*)\quad \frac{S_{m+n}}{m+n} = \frac{S_m}{m} \cdot \frac{... | 6.5 | [
4,
7,
7,
7,
7,
8,
6,
6
] |
Let $\mathbf{Z}$ denote the set of all integers. Find all real numbers $c > 0$ such that there exists a labeling of the lattice points $ ( x, y ) \in \mathbf{Z}^2$ with positive integers for which:
[list]
[*] only finitely many distinct labels occur, and
[*] for each label $i$, the distance between any two points labe... | c < \sqrt{2} |
To solve this problem, we need to determine all real numbers \( c > 0 \) such that there exists a labeling of the lattice points \( (x, y) \in \mathbf{Z}^2 \) with positive integers while satisfying the given conditions:
- Only finitely many distinct labels occur.
- For each label \( i \), the distance between any two... | 7 | [
7,
7,
7,
7,
7,
7,
7,
7
] |
$P, A, B, C,$ and $D$ are five distinct points in space such that $\angle APB = \angle BPC = \angle CPD = \angle DPA = \theta$, where $\theta$ is a given acute angle. Determine the greatest and least values of $\angle APC + \angle BPD$. | 0^\circ \text{ and } 360^\circ |
Consider five distinct points \( P, A, B, C, \) and \( D \) in space where the angles formed at \( P \) satisfy \( \angle APB = \angle BPC = \angle CPD = \angle DPA = \theta \). We are tasked with finding the greatest and least possible values of the sum of angles \( \angle APC + \angle BPD \).
### Analyzing the Geom... | 7.375 | [
7,
7,
7,
8,
6,
8,
8,
8
] |
Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$,
\[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \] | 120 |
To find the least possible value of \( f(1998) \), where \( f: \mathbb{N} \to \mathbb{N} \) satisfies the functional equation
\[
f\left( n^{2}f(m)\right) = m\left( f(n)\right) ^{2}
\]
for all \( m, n \in \mathbb{N} \), we begin by analyzing the given equation.
Firstly, let's examine the case when \( m = 1 \):
\[
... | 6.75 | [
6,
8,
7,
7,
7,
6,
6,
7
] |
Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\]
[i] | (9, 3), (6, 3), (9, 5), (54, 5) |
We are tasked with finding all pairs \((m, n)\) of nonnegative integers that satisfy the equation:
\[
m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).
\]
To solve this equation, we rearrange terms to express it in a form that can be factored:
\[
m^2 - m(2^{n+1} - 1) + 2 \cdot 3^n = 0.
\]
This is a quadratic equation i... | 5.375 | [
6,
5,
6,
5,
6,
5,
5,
5
] |
Find all prime numbers $ p,q,r$, such that $ \frac{p}{q}\minus{}\frac{4}{r\plus{}1}\equal{}1$ | (7, 3, 2), (5, 3, 5), (3, 2, 7) |
We are tasked with finding all prime numbers \( p, q, r \) that satisfy the equation:
\[
\frac{p}{q} - \frac{4}{r+1} = 1.
\]
First, we rearrange the equation to find a common denominator:
\[
\frac{p}{q} - \frac{4}{r+1} = 1 \implies \frac{p(r+1) - 4q}{q(r+1)} = 1.
\]
This simplifies to:
\[
p(r+1) - 4q = q(r+1).
\]... | 5 | [
5,
5,
5,
5,
5,
5,
5,
5
] |
Find all odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$.
($d(n)$ is the number of divisors of the number n including $1$ and $n$ ). | 9 |
We are tasked with finding all odd natural numbers \( n \) such that \( d(n) \), the number of divisors of \( n \), is the largest divisor of \( n \) different from \( n \) itself. Recall that \( d(n) \) counts all divisors of \( n \), including \( 1 \) and \( n \).
To solve the problem, let's begin by analyzing the ... | 5.625 | [
4,
6,
6,
5,
6,
6,
6,
6
] |
What is the smallest positive integer $t$ such that there exist integers $x_1,x_2,\ldots,x_t$ with \[x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?\] | 4 |
To determine the smallest positive integer \( t \) such that there exist integers \( x_1, x_2, \ldots, x_t \) satisfying
\[
x_1^3 + x_2^3 + \cdots + x_t^3 = 2002^{2002},
\]
we will apply Fermat's Last Theorem and results regarding sums of cubes.
### Step 1: Understanding the Sum of Cubes
The problem requires expres... | 6.125 | [
6,
7,
6,
6,
6,
6,
6,
6
] |
Find all integers $n \geq 3$ such that the following property holds: if we list the divisors of $n!$ in increasing order as $1 = d_1 < d_2 < \dots < d_k = n!$, then we have
\[ d_2 - d_1 \leq d_3 - d_2 \leq \dots \leq d_k - d_{k-1}. \] | 3 \text{ and } 4 |
Consider the property that for integers \( n \geq 3 \), the divisors of \( n! \), listed in increasing order as \( 1 = d_1 < d_2 < \dots < d_k = n! \), satisfy:
\[
d_2 - d_1 \leq d_3 - d_2 \leq \dots \leq d_k - d_{k-1}.
\]
To solve this problem, we analyze the differences \( d_{i+1} - d_i \) for the sequence of divi... | 6.125 | [
6,
6,
6,
6,
7,
6,
6,
6
] |
For any integer $n \ge2$, we define $ A_n$ to be the number of positive integers $ m$ with the following property: the distance from $n$ to the nearest multiple of $m$ is equal to the distance from $n^3$ to the nearest multiple of $ m$. Find all integers $n \ge 2 $ for which $ A_n$ is odd. (Note: The distance between t... | $\boxed{n=(2k)^2}$ |
Let \( n \) be an integer such that \( n \ge 2 \). We need to find the integers \( n \) for which the number of positive integers \( m \), denoted by \( A_n \), is odd. The integers \( m \) have the property that the distance from \( n \) to the nearest multiple of \( m \) is equal to the distance from \( n^3 \) to th... | 6.25 | [
7,
6,
6,
6,
6,
6,
7,
6
] |
Determine all the functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[ f(x^2 + f(y)) = f(f(x)) + f(y^2) + 2f(xy) \]
for all real numbers $x$ and $y$. | $f(x) = x^2 \text{ and } f(x) = 0$ |
To determine all functions \( f : \mathbb{R} \to \mathbb{R} \) satisfying the functional equation:
\[
f(x^2 + f(y)) = f(f(x)) + f(y^2) + 2f(xy)
\]
for all real numbers \( x \) and \( y \), we will go through the following steps:
### Step 1: Substitution and Initial Analysis
First, consider substituting special val... | 7.375 | [
7,
7,
8,
7,
7,
8,
8,
7
] |
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for each $x \in \mathbb{R}^+$, there is exactly one $y \in \mathbb{R}^+$ satisfying $$xf(y)+yf(x) \leq 2$$ | f(x) = \frac{1}{x} |
To solve the given functional equation problem, we must find all functions \( f: \mathbb{R}^+ \to \mathbb{R}^+ \) such that for each \( x \in \mathbb{R}^+ \), there is exactly one \( y \in \mathbb{R}^+ \) satisfying
\[
xf(y) + yf(x) \leq 2.
\]
### Step 1: Analyze the Condition
Given the condition \( xf(y) + yf(x) \... | 8.125 | [
8,
7,
9,
8,
8,
8,
8,
9
] |
We say a triple of real numbers $ (a_1,a_2,a_3)$ is [b]better[/b] than another triple $ (b_1,b_2,b_3)$ when exactly two out of the three following inequalities hold: $ a_1 > b_1$, $ a_2 > b_2$, $ a_3 > b_3$. We call a triple of real numbers [b]special[/b] when they are nonnegative and their sum is $ 1$.
For which natu... | n\geq4 |
To solve this problem, we need to determine for which natural numbers \( n \) there exists a set \( S \) of special triples, with \( |S| = n \), such that any special triple is bettered by at least one element of \( S \).
### Understanding the Definitions
A **special triple** \((a_1, a_2, a_3)\) is defined as a trip... | 6.75 | [
7,
7,
7,
7,
7,
7,
6,
6
] |
Petya bought one cake, two cupcakes and three bagels, Apya bought three cakes and a bagel, and Kolya bought six cupcakes. They all paid the same amount of money for purchases. Lena bought two cakes and two bagels. And how many cupcakes could be bought for the same amount spent to her? | $\frac{13}{4}$ |
To solve this problem, we need to determine how many cupcakes can be purchased for the same amount that Lena spent, given the prices of each pastry type.
Let's denote the prices:
- The price of one cake as \( c \).
- The price of one cupcake as \( p \).
- The price of one bagel as \( b \).
According to the problem,... | 4.125 | [
4,
4,
4,
4,
4,
4,
4,
5
] |
The International Mathematical Olympiad is being organized in Japan, where a folklore belief is that the number $4$ brings bad luck. The opening ceremony takes place at the Grand Theatre where each row has the capacity of $55$ seats. What is the maximum number of contestants that can be seated in a single row with the ... | 30 |
To address the problem, we need to determine the maximum number of contestants that can be seated in a single row of 55 seats under the restriction that no two contestants are seated 4 seats apart.
Let's denote the seats in the row as positions \(1, 2, 3, \ldots, 55\). The condition that no two contestants are 4 seat... | 4.875 | [
5,
5,
5,
5,
5,
5,
4,
5
] |
Two players play alternately on a $ 5 \times 5$ board. The first player always enters a $ 1$ into an empty square and the second player always enters a $ 0$ into an empty square. When the board is full, the sum of the numbers in each of the nine $ 3 \times 3$ squares is calculated and the first player's score is the la... | 6 |
To determine the largest score the first player can achieve, we must analyze how the scores are calculated and devise a strategy for maximizing the score in any \(3 \times 3\) square.
The board is a \(5 \times 5\) grid, so we have several overlapping \(3 \times 3\) squares to consider. When full, there are exactly ni... | 6.125 | [
6,
6,
6,
6,
6,
6,
6,
7
] |
Peter has $2022$ pieces of magnetic railroad cars, which are of two types: some have the front with north and the rear with south magnetic polarity, and some have the rear with north and the rear with south magnetic polarity (on these railroad cars the front and the rear can be distinguished). Peter wants to decide whe... | 2021 |
Peter has 2022 pieces of magnetic railroad cars, which are of two types:
- Type 1: The front with north polarity and the rear with south polarity.
- Type 2: The rear with north polarity and the front with south polarity.
To determine whether there is the same number of both types of cars, Peter can try to fit two ca... | 5.25 | [
5,
6,
5,
5,
4,
6,
6,
5
] |
For every $a \in \mathbb N$ denote by $M(a)$ the number of elements of the set
\[ \{ b \in \mathbb N | a + b \text{ is a divisor of } ab \}.\]
Find $\max_{a\leq 1983} M(a).$ | 121 |
To solve the problem, we need to analyze the set \( S(a) = \{ b \in \mathbb{N} \mid a + b \text{ is a divisor of } ab \} \) for a given \( a \) in the natural numbers, and we need to find the maximum number of elements \( M(a) \) in this set for \( a \leq 1983 \).
### Step 1: Understand the Condition
For \( a + b \m... | 6.5 | [
6,
6,
6,
7,
8,
6,
6,
7
] |
Each of given $100$ numbers was increased by $1$. Then each number was increased by $1$ once more. Given that the fi rst time the sum of the squares of the numbers was not changed find how this sum was changed the second time. | 200 |
Let the original 100 numbers be \( a_1, a_2, \ldots, a_{100} \). Initially, the sum of their squares is
\[
S = \sum_{i=1}^{100} a_i^2.
\]
When each number is increased by 1 for the first time, the new numbers are \( a_1 + 1, a_2 + 1, \ldots, a_{100} + 1 \). The new sum of squares is:
\[
S_1 = \sum_{i=1}^{100} (a_i... | 5.5 | [
6,
6,
6,
5,
4,
5,
6,
6
] |
Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the
sequence $1$, $2$, $\dots$ , $n$ satisfying
$$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. | F_{n+1} |
Consider the problem of counting the number of permutations of the sequence \(1, 2, \ldots, n\) that satisfy the inequality:
\[
a_1 \le 2a_2 \le 3a_3 \le \cdots \le na_n.
\]
To solve this, we relate the problem to a known sequence, specifically, the Fibonacci numbers. This can be approached using a combinatorial arg... | 6.25 | [
6,
7,
6,
7,
6,
6,
6,
6
] |
In how many ways can we fill the cells of a $4\times4$ grid such that each cell contains exactly one positive integer and the product of the numbers in each row and each column is $2020$? | 576 |
To solve the problem, we need to fill the cells of a \(4 \times 4\) grid such that each cell contains exactly one positive integer, and the product of the numbers in each row and each column is 2020. We must determine the number of ways to achieve this configuration.
First, observe that the prime factorization of 202... | 7 | [
7,
7,
7,
6,
7,
7,
8,
7
] |
The integer numbers from $1$ to $2002$ are written in a blackboard in increasing order $1,2,\ldots, 2001,2002$. After that, somebody erases the numbers in the $ (3k+1)-th$ places i.e. $(1,4,7,\dots)$. After that, the same person erases the numbers in the $(3k+1)-th$ positions of the new list (in this case, $2,5,9,\ldot... | 2,6,10 |
Let's analyze the process of elimination step by step, starting from the list of integers from 1 to 2002:
### Step 1:
Initially, the numbers \(1, 2, \ldots, 2002\) are written on the board. In this first step, numbers at positions \(1, 4, 7, \ldots\) (i.e., \( (3k+1) \)-th positions for \(k = 0, 1, 2, \ldots\)) are e... | 5.75 | [
6,
5,
6,
6,
6,
6,
6,
5
] |
Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values ... | \sqrt{2} |
Given an acute triangle \( ABC \), let \( M \) be the midpoint of \( AC \). A circle \( \omega \) that passes through points \( B \) and \( M \) intersects side \( AB \) at point \( P \) and side \( BC \) at point \( Q \). Point \( T \) is such that \( BPTQ \) forms a parallelogram, and it is given that \( T \) lies o... | 6.75 | [
7,
8,
6,
6,
7,
7,
6,
7
] |
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ that satisfy the conditions
\[f(1+xy)-f(x+y)=f(x)f(y) \quad \text{for all } x,y \in \mathbb{R},\]
and $f(-1) \neq 0$. | f(x) = x - 1 |
To find all functions \( f:\mathbb{R} \rightarrow \mathbb{R} \) satisfying the functional equation
\[
f(1+xy) - f(x+y) = f(x)f(y)
\]
for all \( x, y \in \mathbb{R} \), and also given that \( f(-1) \neq 0 \), we proceed as follows:
### Step 1: Investigate possible solutions
Assume a potential solution of the form \... | 7.25 | [
8,
6,
7,
8,
7,
7,
8,
7
] |
Let $S(n)$ be the sum of the digits of the positive integer $n$. Find all $n$ such that $S(n)(S(n)-1)=n-1$. | $n=\{1,13,43,91,157\}$ |
Let \( S(n) \) be the sum of the digits of the positive integer \( n \). We want to find all \( n \) such that:
\[
S(n)(S(n) - 1) = n - 1.
\]
Rearranging the equation gives:
\[
S(n)^2 - S(n) = n - 1 \quad \Rightarrow \quad S(n)^2 - S(n) - n + 1 = 0.
\]
This can be rewritten as:
\[
S(n)^2 - S(n) = n - 1.
\]
Denot... | 5.5 | [
5,
5,
5,
6,
6,
6,
5,
6
] |
We consider dissections of regular $n$-gons into $n - 2$ triangles by $n - 3$ diagonals which do not intersect inside the $n$-gon. A [i]bicoloured triangulation[/i] is such a dissection of an $n$-gon in which each triangle is coloured black or white and any two triangles which share an edge have different colours. We c... | 3\mid n |
To solve the problem, we need to determine which positive integers \( n \ge 4 \) allow a regular \( n \)-gon to be dissected into a bicoloured triangulation under the condition that, for each vertex \( A \), the number of black triangles having \( A \) as a vertex is greater than the number of white triangles having \... | 7.125 | [
7,
7,
7,
7,
8,
7,
8,
6
] |
Find the smallest value that the expression takes $x^4 + y^4 - x^2y - xy^2$, for positive numbers $x$ and $y$ satisfying $x + y \le 1$. | -\frac{1}{8} |
We wish to find the minimum value of the expression \( x^4 + y^4 - x^2y - xy^2 \) subject to the constraint \( x + y \leq 1 \) where \( x \) and \( y \) are positive real numbers.
First, consider using the Lagrange multipliers method to incorporate the constraint \( x + y = c \leq 1 \). We define the Lagrangian funct... | 5.875 | [
5,
6,
6,
6,
6,
6,
6,
6
] |
Let $\mathbb{R}$ denote the set of the reals. Find all $f : \mathbb{R} \to \mathbb{R}$ such that
$$ f(x)f(y) = xf(f(y-x)) + xf(2x) + f(x^2) $$
for all real $x, y$. | f(x)=0,f(x)=3x \text{ }\forall x |
To solve the functional equation for \( f : \mathbb{R} \to \mathbb{R} \),
\[
f(x)f(y) = xf(f(y-x)) + xf(2x) + f(x^2),
\]
for all real \( x, y \), we proceed as follows:
1. **Substitute \( y = 0 \):**
Considering \( y = 0 \), the equation becomes:
\[
f(x)f(0) = xf(f(-x)) + xf(2x) + f(x^2).
\]
Notice... | 7.25 | [
8,
8,
7,
7,
7,
8,
7,
6
] |
Find all triples $(x; y; p)$ of two non-negative integers $x, y$ and a prime number p such that $ p^x-y^p=1 $ | (0, 0, 2), (1, 1, 2), (2, 2, 3) |
The problem requires us to find all triples \((x, y, p)\) consisting of two non-negative integers \(x\) and \(y\), and a prime number \(p\), such that:
\[
p^x - y^p = 1
\]
To solve this problem, we'll analyze it case by case, beginning with small values for \(x\) and considering the nature of \(y^p\) and \(p^x\).
#... | 7.25 | [
7,
8,
7,
8,
7,
8,
6,
7
] |
Let $\alpha$ be a real number. Determine all polynomials $P$ with real coefficients such that $$P(2x+\alpha)\leq (x^{20}+x^{19})P(x)$$ holds for all real numbers $x$. | P(x)\equiv 0 |
Let \(\alpha\) be a real number. We need to determine all polynomials \(P(x)\) with real coefficients satisfying:
\[
P(2x + \alpha) \leq (x^{20} + x^{19})P(x)
\]
for all real numbers \(x\).
### Step-by-Step Solution
1. **Analyzing the inequality:**
The inequality \(P(2x + \alpha) \leq (x^{20} + x^{19})P(x)\) i... | 7.125 | [
7,
8,
7,
7,
7,
7,
7,
7
] |
A positive integer with 3 digits $\overline{ABC}$ is $Lusophon$ if $\overline{ABC}+\overline{CBA}$ is a perfect square. Find all $Lusophon$ numbers. | 110,143,242,341,440,164,263,362,461,560,198,297,396,495,594,693,792,891,990 | To find all three-digit Lusophon numbers \( \overline{ABC} \), we first need to establish the conditions under which a number meets the Lusophon criteria. A number is defined as Lusophon if the sum of the number and its digit reversal is a perfect square. Therefore, we need to consider the number \(\overline{ABC}\) and... | 5.125 | [
6,
5,
5,
5,
5,
5,
5,
5
] |
For an integer $x \geq 1$, let $p(x)$ be the least prime that does not divide $x$, and define $q(x)$ to be the product of all primes less than $p(x)$. In particular, $p(1) = 2.$ For $x$ having $p(x) = 2$, define $q(x) = 1$. Consider the sequence $x_0, x_1, x_2, \ldots$ defined by $x_0 = 1$ and \[ x_{n+1} = \frac{x_n p(... | 142 |
We are given a sequence \( x_0, x_1, x_2, \ldots \) defined by \( x_0 = 1 \) and the recursive formula
\[
x_{n+1} = \frac{x_n p(x_n)}{q(x_n)}
\]
where \( p(x) \) is the least prime that does not divide \( x \) and \( q(x) \) is the product of all primes less than \( p(x) \). If \( p(x) = 2 \), then \( q(x) = 1 \).
... | 6 | [
6,
6,
5,
7,
6,
6,
6,
6
] |
Find all triples of positive integers $(x,y,z)$ that satisfy the equation
$$2(x+y+z+2xyz)^2=(2xy+2yz+2zx+1)^2+2023.$$ | (2, 3, 3) |
To solve the given equation for triples \((x, y, z)\) of positive integers:
\[
2(x + y + z + 2xyz)^2 = (2xy + 2yz + 2zx + 1)^2 + 2023,
\]
we start by analyzing the structure of the equation. The equation can be seen as comparing the square of two polynomials with an additional constant term of 2023. Let's explore po... | 7 | [
6,
7,
7,
7,
8,
7,
7,
7
] |
The writer Arthur has $n \ge1$ co-authors who write books with him. Each book has a list of authors including Arthur himself. No two books have the same set of authors. At a party with all his co-author, each co-author writes on a note how many books they remember having written with Arthur. Inspecting the numbers on t... | $n\le6$ |
To solve the problem, we need to determine the values of \( n \) for which it is possible that each co-author accurately remembers the number of books written with Arthur, and these numbers correspond to the first \( n \) Fibonacci numbers.
The Fibonacci sequence is defined by:
\[
F_1 = 1, \quad F_2 = 1, \quad F_{k+... | 5.875 | [
6,
6,
6,
6,
6,
6,
5,
6
] |
Knowing that the system
\[x + y + z = 3,\]\[x^3 + y^3 + z^3 = 15,\]\[x^4 + y^4 + z^4 = 35,\]
has a real solution $x, y, z$ for which $x^2 + y^2 + z^2 < 10$, find the value of $x^5 + y^5 + z^5$ for that solution. | 83 |
To solve for \( x^5 + y^5 + z^5 \) given the system of equations:
1. \( x + y + z = 3 \)
2. \( x^3 + y^3 + z^3 = 15 \)
3. \( x^4 + y^4 + z^4 = 35 \)
and the condition:
\[ x^2 + y^2 + z^2 < 10, \]
we will utilize symmetric polynomials and Newton's identities.
### Step 1: Establish Variables and Polynomials
Let:
-... | 6.375 | [
6,
7,
6,
7,
7,
6,
6,
6
] |
Let $ ABC$ be an isosceles triangle with $ AB\equal{}AC$ and $ \angle A\equal{}20^\circ$. On the side $ AC$ consider point $ D$ such that $ AD\equal{}BC$. Find $ \angle BDC$. | $30^\circ$ |
Let triangle \( ABC \) be an isosceles triangle with \( AB = AC \) and \( \angle A = 20^\circ \). We are given a point \( D \) on side \( AC \) such that \( AD = BC \). Our task is to find \( \angle BDC \).
#### Step-by-Step Solution:
1. **Base Angle Calculation:**
Since \( ABC \) is an isosceles triangle with \(... | 5.625 | [
6,
5,
6,
5,
6,
6,
5,
6
] |
Determine which integers $n > 1$ have the property that there exists an infinite sequence $a_1, a_2, a_3, \ldots$ of nonzero integers such that the equality \[a_k+2a_{2k}+\ldots+na_{nk}=0\]holds for every positive integer $k$. | n > 2 | Consider the problem to determine which integers \( n > 1 \) have the property that there exists an infinite sequence \( a_1, a_2, a_3, \ldots \) of nonzero integers satisfying the equality:
\[
a_k + 2a_{2k} + \ldots + na_{nk} = 0
\]
for every positive integer \( k \).
### Step-by-Step Solution:
1. **Express the Co... | 6 | [
6,
6,
6,
6,
6,
6,
6,
6
] |
Find all positive integers $n$ for which there exist positive integers $x_1, x_2, \dots, x_n$ such that
$$ \frac{1}{x_1^2}+\frac{2}{x_2^2}+\frac{2^2}{x_3^2}+\cdots +\frac{2^{n-1}}{x_n^2}=1.$$ | n\ge 3 \text{ and } n=1 |
We need to determine all positive integers \( n \) such that there exist positive integers \( x_1, x_2, \ldots, x_n \) satisfying the equation:
\[
\frac{1}{x_1^2} + \frac{2}{x_2^2} + \frac{2^2}{x_3^2} + \cdots + \frac{2^{n-1}}{x_n^2} = 1.
\]
### Case \( n = 1 \)
For \( n = 1 \), the equation simplifies to:
\[
\frac... | 5.875 | [
6,
5,
6,
6,
6,
6,
6,
6
] |
A broken line consists of $31$ segments. It has no self intersections, and its start and end points are distinct. All segments are extended to become straight lines. Find the least possible number of straight lines. | 16 |
Let us consider a broken line made up of 31 segments with no self-intersections, where the start and end points are distinct. Each segment of the broken line can be extended indefinitely to form a straight line. The problem asks us to find the least possible number of distinct straight lines that can be created from t... | 6.125 | [
6,
6,
6,
6,
7,
6,
6,
6
] |
Let $A$ be a given set with $n$ elements. Let $k<n$ be a given positive integer. Find the maximum value of $m$ for which it is possible to choose sets $B_i$ and $C_i$ for $i=1,2,\ldots,m$ satisfying the following conditions:
[list=1]
[*]$B_i\subset A,$ $|B_i|=k,$
[*]$C_i\subset B_i$ (there is no additional condition fo... | {2^k} |
Let \( A \) be a set with \( n \) elements, and let \( k < n \) be a given positive integer. We need to find the maximum value of \( m \) such that it is possible to choose sets \( B_i \) and \( C_i \) for \( i = 1, 2, \ldots, m \) satisfying the following conditions:
1. \( B_i \subset A \), with \(|B_i| = k\).
2. \(... | 6.125 | [
6,
6,
7,
8,
6,
6,
5,
5
] |
A given finite number of lines in the plane, no two of which are parallel and no three of which are concurrent, divide the plane into finite and infinite regions. In each finite region we write $1$ or $-1$. In one operation, we can choose any triangle made of three of the lines (which may be cut by other lines in the c... | \text{No} |
Consider a finite number of lines in the plane, none of which are parallel, and no three of which are concurrent. These lines divide the plane into several regions—both finite and infinite.
### Problem Analysis
In this scenario, we assign the number \( 1 \) or \( -1 \) to each of the finite regions. The operation al... | 8.125 | [
8,
8,
7,
9,
8,
8,
8,
9
] |
A subset of a student group is called an [i]ideal company[/i] if
1) in this subset, all girls are liked by all young men,
2) no one can be added to this subset without violating condition $1$.
In a certain group, $9$ female students and $15$ students study. Warden of the group made a list of all kinds of ideal companie... | 512 |
To solve this problem, we need to understand the concept of an "ideal company" as defined by the question. An ideal company is a subset of the student group where all female students in the subset are liked by all male students within the same subset, and no additional student can be added to this group without breaki... | 5.625 | [
5,
6,
5,
6,
6,
6,
6,
5
] |
Let $ a_1 \equal{} 11^{11}, \, a_2 \equal{} 12^{12}, \, a_3 \equal{} 13^{13}$, and $ a_n \equal{} |a_{n \minus{} 1} \minus{} a_{n \minus{} 2}| \plus{} |a_{n \minus{} 2} \minus{} a_{n \minus{} 3}|, n \geq 4.$ Determine $ a_{14^{14}}$. | 1 |
To determine \( a_{14^{14}} \), we need to evaluate the recursive relationship given by \( a_n = |a_{n-1} - a_{n-2}| + |a_{n-2} - a_{n-3}| \) starting from the initial terms \( a_1 = 11^{11} \), \( a_2 = 12^{12} \), and \( a_3 = 13^{13} \).
### Step-by-step Calculation:
1. **Base Cases:**
Given:
\[
a_1 = ... | 5.5 | [
6,
6,
6,
5,
4,
6,
6,
5
] |
Find all pairs $(a,\, b)$ of positive integers such that $2a-1$ and $2b+1$ are coprime and $a+b$ divides $4ab+1.$ | (a, a+1) |
We need to find all pairs \((a, b)\) of positive integers such that:
1. \(2a-1\) and \(2b+1\) are coprime,
2. \(a+b\) divides \(4ab+1\).
### Step 1: Analyze the Conditions
**Condition 1:**
The integers \(2a-1\) and \(2b+1\) are coprime, meaning their greatest common divisor (GCD) is 1. Therefore:
\[
\gcd(2a-1, 2b+... | 6.375 | [
7,
6,
7,
6,
6,
6,
6,
7
] |
There are 60 empty boxes $B_1,\ldots,B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game.
In the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps:
(a) Bob... | 960 | To solve this problem, we need to find the smallest integer \( n \) such that Alice can always prevent Bob from winning regardless of how the game progresses. The setup is as follows:
1. Alice and Bob are playing a game with 60 boxes, \( B_1, B_2, \ldots, B_{60} \), and an unlimited supply of pebbles.
2. In the first ... | 6.75 | [
7,
6,
6,
7,
7,
7,
7,
7
] |
Consider the function $ f: \mathbb{N}_0\to\mathbb{N}_0$, where $ \mathbb{N}_0$ is the set of all non-negative
integers, defined by the following conditions :
$ (i)$ $ f(0) \equal{} 0$; $ (ii)$ $ f(2n) \equal{} 2f(n)$ and $ (iii)$ $ f(2n \plus{} 1) \equal{} n \plus{} 2f(n)$ for all $ n\geq 0$.
$ (a)$ Determine the... | 2^k - 1 |
### Part (a)
We have the function \( f: \mathbb{N}_0 \rightarrow \mathbb{N}_0 \) defined by:
- \( f(0) = 0 \)
- \( f(2n) = 2f(n) \)
- \( f(2n + 1) = n + 2f(n) \)
We need to determine the sets:
- \( L = \{ n \mid f(n) < f(n + 1) \} \)
- \( E = \{ n \mid f(n) = f(n + 1) \} \)
- \( G = \{ n \mid f(n) > f(n + 1) \} \)
... | 6.625 | [
6,
7,
6,
8,
6,
7,
6,
7
] |
Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for any two real numbers $x,y$ holds
$$f(xf(y)+2y)=f(xy)+xf(y)+f(f(y)).$$ | f(x) = 2x \text{ and } f(x) = 0 |
To solve the functional equation problem, we need to identify all functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the given functional equation for any real numbers \( x, y \):
\[
f(xf(y) + 2y) = f(xy) + xf(y) + f(f(y)).
\]
We will explore potential solutions by substituting specific values for \( x \) and... | 7.875 | [
8,
8,
7,
8,
8,
8,
8,
8
] |
Let $a, b,c$ and $d$ be real numbers such that $a + b + c + d = 2$ and $ab + bc + cd + da + ac + bd = 0$.
Find the minimum value and the maximum value of the product $abcd$. | 0\frac{1}{16} |
Given the conditions:
\[ a + b + c + d = 2 \]
\[ ab + bc + cd + da + ac + bd = 0, \]
we are required to find the minimum and maximum values of the product \( abcd \).
### Step 1: Consider the Polynomial Approach
We associate the real numbers \( a, b, c, \) and \( d \) with the roots of a polynomial \( P(x) \).
The... | 5.875 | [
5,
6,
6,
6,
6,
6,
6,
6
] |
$2014$ points are placed on a circumference. On each of the segments with end points on two of the $2014$ points is written a non-negative real number. For any convex polygon with vertices on some of the $2014$ points, the sum of the numbers written on their sides is less or equal than $1$. Find the maximum possible va... | 507024.5 |
Given the problem, we are tasked with finding the maximum possible sum of numbers written on segments between 2014 points uniformly placed on a circumference, under the condition that for any convex polygon formed using these points as vertices, the sum of the numbers on its sides must not exceed 1.
Consider the foll... | 7.75 | [
8,
8,
8,
7,
7,
8,
8,
8
] |
Let $ n \geq 2$ be a positive integer and $ \lambda$ a positive real number. Initially there are $ n$ fleas on a horizontal line, not all at the same point. We define a move as choosing two fleas at some points $ A$ and $ B$, with $ A$ to the left of $ B$, and letting the flea from $ A$ jump over the flea from $ B$ to ... | \lambda \ge \frac{1}{n-1} |
Let \( n \geq 2 \) be a positive integer and \( \lambda \) a positive real number. There are \( n \) fleas on a horizontal line, and we need to find the values of \( \lambda \) for which, given any point \( M \) and any initial positions of the fleas, there is a sequence of moves that can place all fleas to the right ... | 6.75 | [
7,
7,
7,
7,
7,
6,
6,
7
] |
Calculate $\displaystyle \sum_{n=1}^\infty \ln \left(1+\frac{1}{n}\right) \ln\left( 1+\frac{1}{2n}\right)\ln\left( 1+\frac{1}{2n+1}\right)$. | \[\frac{\ln^3(2)}{3}\] |
The problem requires evaluating the infinite series:
\[
\sum_{n=1}^\infty \ln \left(1+\frac{1}{n}\right) \ln\left( 1+\frac{1}{2n}\right)\ln\left( 1+\frac{1}{2n+1}\right).
\]
Firstly, observe the behavior of the logarithmic terms for large \( n \). Using the approximation \(\ln(1+x) \approx x\) for small \( x \), we ... | 7.25 | [
6,
7,
8,
7,
8,
7,
7,
8
] |
The polynomial $Q(x)=x^3-21x+35$ has three different real roots. Find real numbers $a$ and $b$ such that the polynomial $x^2+ax+b$ cyclically permutes the roots of $Q$, that is, if $r$, $s$ and $t$ are the roots of $Q$ (in some order) then $P(r)=s$, $P(s)=t$ and $P(t)=r$. | a = 2, b = -14 |
Given the polynomial \( Q(x) = x^3 - 21x + 35 \), which has three different real roots, we need to find real numbers \( a \) and \( b \), such that the polynomial \( P(x) = x^2 + ax + b \) cyclically permutes the roots of \( Q \). Let the roots of \( Q \) be \( r, s, \) and \( t \). The cyclic permutation property req... | 6.25 | [
7,
6,
6,
6,
6,
6,
6,
7
] |
For a finite graph $G$, let $f(G)$ be the number of triangles and $g(G)$ the number of tetrahedra formed by edges of $G$. Find the least constant $c$ such that \[g(G)^3\le c\cdot f(G)^4\] for every graph $G$.
[i] | \frac{3}{32} |
Let \( G \) be a finite graph. We denote by \( f(G) \) the number of triangles and by \( g(G) \) the number of tetrahedra in \( G \). We seek to establish the smallest constant \( c \) such that
\[
g(G)^3 \le c \cdot f(G)^4
\]
for every graph \( G \).
### Step 1: Understanding the Problem
A triangle in a graph con... | 6.875 | [
8,
7,
6,
6,
7,
7,
8,
6
] |
Determine the least real number $M$ such that the inequality \[|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \leq M(a^{2}+b^{2}+c^{2})^{2}\] holds for all real numbers $a$, $b$ and $c$. | M=\frac 9{16\sqrt 2} |
To find the least real number \( M \) such that the inequality
\[
|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \leq M(a^{2}+b^{2}+c^{2})^{2}
\]
holds for all real numbers \( a, b, \) and \( c \), we proceed as follows:
### Step 1: Expression Expansion
First, expand the left-hand side of the equation:
\[
ab(... | 7.25 | [
7,
8,
8,
7,
6,
6,
8,
8
] |
Three players $A,B$ and $C$ play a game with three cards and on each of these $3$ cards it is written a positive integer, all $3$ numbers are different. A game consists of shuffling the cards, giving each player a card and each player is attributed a number of points equal to the number written on the card and then the... | C |
We are given that players \( A \), \( B \), and \( C \) each receive one card per game, and the points received correspond to the numbers written on their respective cards. After several games, the total points are as follows: \( A \) has 20 points, \( B \) has 10 points, and \( C \) has 9 points. In the last game, \(... | 5.625 | [
6,
5,
6,
5,
5,
6,
6,
6
] |
Find all the integers $x, y$ and $z$ greater than or equal to $0$ such that $2^x + 9 \cdot 7^y = z^3$ | (0, 1, 4) |
We are tasked with finding all non-negative integer solutions \((x, y, z)\) to the equation:
\[
2^x + 9 \cdot 7^y = z^3
\]
Given that our solution must satisfy integer constraints and each variable is greater than or equal to zero, we will systematically explore potential solutions.
### Step 1: Analyze small values... | 5.125 | [
5,
6,
5,
5,
6,
5,
4,
5
] |
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