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 |
|---|---|---|---|---|
Let $P$ be a regular $n$-gon $A_1A_2\ldots A_n$. Find all positive integers $n$ such that for each permutation $\sigma (1),\sigma (2),\ldots ,\sigma (n)$ there exists $1\le i,j,k\le n$ such that the triangles $A_{i}A_{j}A_{k}$ and $A_{\sigma (i)}A_{\sigma (j)}A_{\sigma (k)}$ are both acute, both right or both obtuse. | n \neq 5 |
Let \( P \) be a regular \( n \)-gon \( A_1A_2\ldots A_n \). We aim to find all positive integers \( n \) such that for each permutation \( \sigma(1), \sigma(2), \ldots, \sigma(n) \), there exists \( 1 \le i, j, k \le n \) such that the triangles \( A_iA_jA_k \) and \( A_{\sigma(i)}A_{\sigma(j)}A_{\sigma(k)} \) are bo... | 7 | [
7,
7,
7,
6,
7,
8,
7,
7
] |
Given positive integers $n$ and $k$, $n > k^2 >4.$ In a $n \times n$ grid, a $k$[i]-group[/i] is a set of $k$ unit squares lying in different rows and different columns.
Determine the maximal possible $N$, such that one can choose $N$ unit squares in the grid and color them, with the following condition holds: in any $... | n(k-1)^2 |
Given positive integers \( n \) and \( k \) such that \( n > k^2 > 4 \), we aim to determine the maximal possible \( N \) such that one can choose \( N \) unit squares in an \( n \times n \) grid and color them, with the condition that in any \( k \)-group from the colored \( N \) unit squares, there are two squares w... | 6.875 | [
7,
7,
7,
7,
7,
6,
7,
7
] |
Find a real number $t$ such that for any set of 120 points $P_1, \ldots P_{120}$ on the boundary of a unit square, there exists a point $Q$ on this boundary with $|P_1Q| + |P_2Q| + \cdots + |P_{120}Q| = t$. | 30(1 + \sqrt{5}) |
We need to find a real number \( t \) such that for any set of 120 points \( P_1, \ldots, P_{120} \) on the boundary of a unit square, there exists a point \( Q \) on this boundary with \( |P_1Q| + |P_2Q| + \cdots + |P_{120}Q| = t \).
Define \(\mathcal{U}\) to be a set of points \( P_1, \ldots, P_{120} \) on the boun... | 7.25 | [
8,
7,
7,
7,
8,
6,
8,
7
] |
Let $ ABP, BCQ, CAR$ be three non-overlapping triangles erected outside of acute triangle $ ABC$. Let $ M$ be the midpoint of segment $ AP$. Given that $ \angle PAB \equal{} \angle CQB \equal{} 45^\circ$, $ \angle ABP \equal{} \angle QBC \equal{} 75^\circ$, $ \angle RAC \equal{} 105^\circ$, and $ RQ^2 \equal{} 6CM^2$, ... | \frac{2}{3} |
Let \( ABP, BCQ, CAR \) be three non-overlapping triangles erected outside of acute triangle \( ABC \). Let \( M \) be the midpoint of segment \( AP \). Given that \( \angle PAB = \angle CQB = 45^\circ \), \( \angle ABP = \angle QBC = 75^\circ \), \( \angle RAC = 105^\circ \), and \( RQ^2 = 6CM^2 \), we aim to compute... | 7.25 | [
6,
7,
7,
8,
8,
7,
7,
8
] |
At a university dinner, there are 2017 mathematicians who each order two distinct entrées, with no two mathematicians ordering the same pair of entrées. The cost of each entrée is equal to the number of mathematicians who ordered it, and the university pays for each mathematician's less expensive entrée (ties broken ar... | 127009 |
To determine the maximum total amount the university could have paid, we can model the problem using graph theory. Consider a graph \( G \) with 2017 edges, where each edge represents a pair of distinct entrées ordered by a mathematician. The cost of each entrée is equal to the number of mathematicians who ordered it,... | 7 | [
7,
7,
7,
6,
7,
8,
7,
7
] |
Let $f:X\rightarrow X$, where $X=\{1,2,\ldots ,100\}$, be a function satisfying:
1) $f(x)\neq x$ for all $x=1,2,\ldots,100$;
2) for any subset $A$ of $X$ such that $|A|=40$, we have $A\cap f(A)\neq\emptyset$.
Find the minimum $k$ such that for any such function $f$, there exist a subset $B$ of $X$, where $|B|=k$, such ... | 69 |
Let \( f: X \rightarrow X \), where \( X = \{1, 2, \ldots, 100\} \), be a function satisfying:
1. \( f(x) \neq x \) for all \( x = 1, 2, \ldots, 100 \);
2. For any subset \( A \) of \( X \) such that \( |A| = 40 \), we have \( A \cap f(A) \neq \emptyset \).
We need to find the minimum \( k \) such that for any such f... | 7.625 | [
8,
7,
8,
8,
7,
7,
8,
8
] |
Consider pairs $(f,g)$ of functions from the set of nonnegative integers to itself such that
[list]
[*]$f(0) \geq f(1) \geq f(2) \geq \dots \geq f(300) \geq 0$
[*]$f(0)+f(1)+f(2)+\dots+f(300) \leq 300$
[*]for any 20 nonnegative integers $n_1, n_2, \dots, n_{20}$, not necessarily distinct, we have $$g(n_1+n_2+\dots+n_{... | 115440 |
Consider pairs \((f, g)\) of functions from the set of nonnegative integers to itself such that:
- \(f(0) \geq f(1) \geq f(2) \geq \dots \geq f(300) \geq 0\),
- \(f(0) + f(1) + f(2) + \dots + f(300) \leq 300\),
- for any 20 nonnegative integers \(n_1, n_2, \dots, n_{20}\), not necessarily distinct, we have \(g(n_1 + n... | 8 | [
8,
8,
8,
8,
8,
8,
8,
8
] |
Find all nonnegative integer solutions $(x,y,z,w)$ of the equation\[2^x\cdot3^y-5^z\cdot7^w=1.\] | (1, 1, 1, 0), (2, 2, 1, 1), (1, 0, 0, 0), (3, 0, 0, 1) |
We are tasked with finding all nonnegative integer solutions \((x, y, z, w)\) to the equation:
\[
2^x \cdot 3^y - 5^z \cdot 7^w = 1.
\]
First, we note that \(x \geq 1\) because if \(x = 0\), the left-hand side would be a fraction, which cannot equal 1.
### Case 1: \(w = 0\)
The equation simplifies to:
\[
2^x \cdot 3... | 6.5 | [
7,
6,
6,
6,
7,
7,
6,
7
] |
Find the largest real number $\lambda$ with the following property: for any positive real numbers $p,q,r,s$ there exists a complex number $z=a+bi$($a,b\in \mathbb{R})$ such that $$ |b|\ge \lambda |a| \quad \text{and} \quad (pz^3+2qz^2+2rz+s) \cdot (qz^3+2pz^2+2sz+r) =0.$$ | \sqrt{3} |
To find the largest real number \(\lambda\) such that for any positive real numbers \(p, q, r, s\), there exists a complex number \(z = a + bi\) (\(a, b \in \mathbb{R}\)) satisfying
\[
|b| \ge \lambda |a|
\]
and
\[
(pz^3 + 2qz^2 + 2rz + s) \cdot (qz^3 + 2pz^2 + 2sz + r) = 0,
\]
we proceed as follows:
The answer is \(... | 7.5 | [
8,
8,
8,
8,
7,
7,
7,
7
] |
Let $S$ be a set, $|S|=35$. A set $F$ of mappings from $S$ to itself is called to be satisfying property $P(k)$, if for any $x,y\in S$, there exist $f_1, \cdots, f_k \in F$ (not necessarily different), such that $f_k(f_{k-1}(\cdots (f_1(x))))=f_k(f_{k-1}(\cdots (f_1(y))))$.
Find the least positive integer $m$, such tha... | 595 |
Let \( S \) be a set with \( |S| = 35 \). A set \( F \) of mappings from \( S \) to itself is said to satisfy property \( P(k) \) if for any \( x, y \in S \), there exist \( f_1, f_2, \ldots, f_k \in F \) (not necessarily different) such that \( f_k(f_{k-1}(\cdots (f_1(x)) \cdots )) = f_k(f_{k-1}(\cdots (f_1(y)) \cdot... | 7.25 | [
7,
7,
8,
7,
7,
8,
7,
7
] |
Let $a_i,b_i,i=1,\cdots,n$ are nonnegitive numbers,and $n\ge 4$,such that $a_1+a_2+\cdots+a_n=b_1+b_2+\cdots+b_n>0$.
Find the maximum of $\frac{\sum_{i=1}^n a_i(a_i+b_i)}{\sum_{i=1}^n b_i(a_i+b_i)}$ | n - 1 |
Let \( a_i, b_i \) for \( i = 1, \ldots, n \) be nonnegative numbers, and let \( n \geq 4 \) such that \( \sum_{i=1}^n a_i = \sum_{i=1}^n b_i > 0 \).
We aim to find the maximum value of the expression:
\[
\frac{\sum_{i=1}^n a_i(a_i + b_i)}{\sum_{i=1}^n b_i(a_i + b_i)}.
\]
We will prove that for \( n \geq 4 \), the m... | 6.75 | [
7,
7,
7,
7,
6,
6,
7,
7
] |
Find in explicit form all ordered pairs of positive integers $(m, n)$ such that $mn-1$ divides $m^2 + n^2$. | (2, 1), (3, 1), (1, 2), (1, 3) |
To find all ordered pairs of positive integers \((m, n)\) such that \(mn-1\) divides \(m^2 + n^2\), we start by considering the condition:
\[
\frac{m^2 + n^2}{mn - 1} = c \quad \text{where} \quad c \in \mathbb{Z}.
\]
This implies:
\[
m^2 + n^2 = c(mn - 1).
\]
Rewriting, we get:
\[
m^2 - cmn + n^2 + c = 0.
\]
Let \((m... | 6 | [
5,
6,
6,
7,
7,
5,
6,
6
] |
Determine all positive integers $n$, $n\ge2$, such that the following statement is true:
If $(a_1,a_2,...,a_n)$ is a sequence of positive integers with $a_1+a_2+\cdots+a_n=2n-1$, then there is block of (at least two) consecutive terms in the sequence with their (arithmetic) mean being an integer. | 2, 3 |
To determine all positive integers \( n \), \( n \ge 2 \), such that the following statement is true:
If \((a_1, a_2, \ldots, a_n)\) is a sequence of positive integers with \( a_1 + a_2 + \cdots + a_n = 2n - 1 \), then there is a block of (at least two) consecutive terms in the sequence with their (arithmetic) mean be... | 6.125 | [
6,
6,
6,
6,
6,
6,
7,
6
] |
Let $C=\{ z \in \mathbb{C} : |z|=1 \}$ be the unit circle on the complex plane. Let $z_1, z_2, \ldots, z_{240} \in C$ (not necessarily different) be $240$ complex numbers, satisfying the following two conditions:
(1) For any open arc $\Gamma$ of length $\pi$ on $C$, there are at most $200$ of $j ~(1 \le j \le 240)$ suc... | 80 + 40\sqrt{3} |
Let \( C = \{ z \in \mathbb{C} : |z| = 1 \} \) be the unit circle on the complex plane. Let \( z_1, z_2, \ldots, z_{240} \in C \) (not necessarily different) be 240 complex numbers satisfying the following two conditions:
1. For any open arc \(\Gamma\) of length \(\pi\) on \(C\), there are at most 200 of \( j ~(1 \le ... | 7.75 | [
8,
7,
8,
8,
8,
8,
8,
7
] |
Let $n$ be a positive integer. Find, with proof, the least positive integer $d_{n}$ which cannot be expressed in the form \[\sum_{i=1}^{n}(-1)^{a_{i}}2^{b_{i}},\]
where $a_{i}$ and $b_{i}$ are nonnegative integers for each $i.$ | 2 \left( \frac{4^n - 1}{3} \right) + 1 |
Let \( n \) be a positive integer. We aim to find the least positive integer \( d_n \) which cannot be expressed in the form
\[
\sum_{i=1}^{n}(-1)^{a_{i}}2^{b_{i}},
\]
where \( a_i \) and \( b_i \) are nonnegative integers for each \( i \).
We claim that the minimal number that is not \( n \)-good is
\[
d_n = 2 \le... | 6.75 | [
7,
7,
6,
6,
7,
7,
7,
7
] |
Find all positive integers $a$ such that there exists a set $X$ of $6$ integers satisfying the following conditions: for every $k=1,2,\ldots ,36$ there exist $x,y\in X$ such that $ax+y-k$ is divisible by $37$. | 6, 31 |
Find all positive integers \( a \) such that there exists a set \( X \) of \( 6 \) integers satisfying the following conditions: for every \( k = 1, 2, \ldots, 36 \), there exist \( x, y \in X \) such that \( ax + y - k \) is divisible by \( 37 \).
To solve this, we need to find all positive integers \( a \) such tha... | 6.625 | [
7,
6,
6,
7,
6,
7,
7,
7
] |
Let $a_1,a_2,\cdots,a_{41}\in\mathbb{R},$ such that $a_{41}=a_1, \sum_{i=1}^{40}a_i=0,$ and for any $i=1,2,\cdots,40, |a_i-a_{i+1}|\leq 1.$ Determine the greatest possible value of
$(1)a_{10}+a_{20}+a_{30}+a_{40};$
$(2)a_{10}\cdot a_{20}+a_{30}\cdot a_{40}.$ | 10 |
Let \( a_1, a_2, \ldots, a_{41} \in \mathbb{R} \) such that \( a_{41} = a_1 \), \( \sum_{i=1}^{40} a_i = 0 \), and for any \( i = 1, 2, \ldots, 40 \), \( |a_i - a_{i+1}| \leq 1 \). We aim to determine the greatest possible values of:
1. \( a_{10} + a_{20} + a_{30} + a_{40} \)
2. \( a_{10} \cdot a_{20} + a_{30} \cdot a... | 7.625 | [
7,
8,
8,
8,
7,
8,
8,
7
] |
For any $h = 2^{r}$ ($r$ is a non-negative integer), find all $k \in \mathbb{N}$ which satisfy the following condition: There exists an odd natural number $m > 1$ and $n \in \mathbb{N}$, such that $k \mid m^{h} - 1, m \mid n^{\frac{m^{h}-1}{k}} + 1$. | 2^{r+1} |
For any \( h = 2^{r} \) (where \( r \) is a non-negative integer), we need to find all \( k \in \mathbb{N} \) which satisfy the following condition: There exists an odd natural number \( m > 1 \) and \( n \in \mathbb{N} \), such that \( k \mid m^{h} - 1 \) and \( m \mid n^{\frac{m^{h}-1}{k}} + 1 \).
We claim that \( ... | 7.75 | [
8,
8,
8,
7,
8,
8,
7,
8
] |
Find the greatest constant $\lambda$ such that for any doubly stochastic matrix of order 100, we can pick $150$ entries such that if the other $9850$ entries were replaced by $0$, the sum of entries in each row and each column is at least $\lambda$.
Note: A doubly stochastic matrix of order $n$ is a $n\times n$ matrix... | \frac{17}{1900} |
We are given a doubly stochastic matrix of order 100 and need to find the greatest constant \(\lambda\) such that we can select 150 entries in the matrix, and if the other 9850 entries are replaced by 0, the sum of entries in each row and each column is at least \(\lambda\).
To solve this, we construct a bipartite gr... | 7.75 | [
8,
7,
8,
7,
8,
8,
8,
8
] |
Find out the maximum value of the numbers of edges of a solid regular octahedron that we can see from a point out of the regular octahedron.(We define we can see an edge $AB$ of the regular octahedron from point $P$ outside if and only if the intersection of non degenerate triangle $PAB$ and the solid regular octahedro... | 9 |
To determine the maximum number of edges of a regular octahedron that can be seen from a point outside the octahedron, we start by considering the geometric properties of the octahedron and the visibility conditions.
A regular octahedron has 12 edges. The visibility of an edge from an external point depends on whethe... | 6.125 | [
6,
6,
6,
6,
6,
6,
6,
7
] |
Choose positive integers $b_1, b_2, \dotsc$ satisfying
\[1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb\]
and let $r$ denote the largest real number satisfying $\tfrac{b_n}{n^2} \geq r$ for all positive integers $n$. What are the possible values of $r$ across all possible choices of th... | 0 \leq r \leq \frac{1}{2} |
Let \( r \) denote the largest real number satisfying \(\frac{b_n}{n^2} \geq r\) for all positive integers \( n \), where \( b_1, b_2, \dotsc \) are positive integers satisfying
\[
1 = \frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb
\]
We aim to determine the possible values of \( r \).... | 6.875 | [
8,
7,
7,
6,
8,
6,
6,
7
] |
Let $\{ z_n \}_{n \ge 1}$ be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer $k$, $|z_k z_{k+1}|=2^k$. Denote $f_n=|z_1+z_2+\cdots+z_n|,$ for $n=1,2,\cdots$
(1) Find the minimum of $f_{2020}$.
(2) Find the minimum of $f_{2020} \cdot f_{2021}$. | 2 |
Let \(\{ z_n \}_{n \ge 1}\) be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer \(k\), \(|z_k z_{k+1}|=2^k\). Denote \(f_n=|z_1+z_2+\cdots+z_n|,\) for \(n=1,2,\cdots\).
1. To find the minimum of \(f_{2020}\):
Write \(a_k=z_k\) for \(k\) odd and ... | 8.125 | [
8,
8,
8,
8,
9,
8,
8,
8
] |
Find all positive integer pairs $(a,n)$ such that $\frac{(a+1)^n-a^n}{n}$ is an integer. | (a, n) = (a, 1) |
We need to find all positive integer pairs \((a, n)\) such that \(\frac{(a+1)^n - a^n}{n}\) is an integer.
First, observe that for \(\frac{(a+1)^n - a^n}{n}\) to be an integer, \((a+1)^n - a^n\) must be divisible by \(n\).
Consider the smallest prime divisor \(p\) of \(n\). We have:
\[
(a+1)^n \equiv a^n \pmod{p}.
... | 5.875 | [
6,
6,
5,
6,
6,
6,
6,
6
] |
Let $X$ be a set of $100$ elements. Find the smallest possible $n$ satisfying the following condition: Given a sequence of $n$ subsets of $X$, $A_1,A_2,\ldots,A_n$, there exists $1 \leq i < j < k \leq n$ such that
$$A_i \subseteq A_j \subseteq A_k \text{ or } A_i \supseteq A_j \supseteq A_k.$$ | 2 \binom{100}{50} + 2 \binom{100}{49} + 1 |
Let \( X \) be a set of \( 100 \) elements. We aim to find the smallest possible \( n \) such that given a sequence of \( n \) subsets of \( X \), \( A_1, A_2, \ldots, A_n \), there exists \( 1 \leq i < j < k \leq n \) such that
\[ A_i \subseteq A_j \subseteq A_k \text{ or } A_i \supseteq A_j \supseteq A_k. \]
The sm... | 7.375 | [
7,
8,
7,
7,
7,
8,
7,
8
] |
An integer $n>1$ is given . Find the smallest positive number $m$ satisfying the following conditions： for any set $\{a,b\}$ $\subset \{1,2,\cdots,2n-1\}$ ,there are non-negative integers $ x, y$ ( not all zero) such that $2n|ax+by$ and $x+y\leq m.$ | n |
Given an integer \( n > 1 \), we aim to find the smallest positive number \( m \) satisfying the following conditions: for any set \(\{a, b\} \subset \{1, 2, \ldots, 2n-1\}\), there exist non-negative integers \( x \) and \( y \) (not both zero) such that \( 2n \mid ax + by \) and \( x + y \leq m \).
To determine the... | 6.375 | [
6,
7,
6,
7,
6,
7,
6,
6
] |
Find all positive integers $a,b,c$ and prime $p$ satisfying that
\[ 2^a p^b=(p+2)^c+1.\] | (1, 1, 1, 3) |
We need to find all positive integers \(a, b, c\) and a prime \(p\) that satisfy the equation:
\[
2^a p^b = (p+2)^c + 1.
\]
First, we note that \(p\) cannot be 2 because the left-hand side would be even, while the right-hand side would be odd.
### Case 1: \(a > 1\)
Consider the equation modulo 4:
\[
(p+2)^c + 1 \equ... | 6.875 | [
7,
7,
6,
7,
7,
7,
7,
7
] |
Find the smallest positive number $\lambda$, such that for any $12$ points on the plane $P_1,P_2,\ldots,P_{12}$(can overlap), if the distance between any two of them does not exceed $1$, then $\sum_{1\le i<j\le 12} |P_iP_j|^2\le \lambda$. | 48 |
We are tasked with finding the smallest positive number \(\lambda\) such that for any 12 points on the plane \(P_1, P_2, \ldots, P_{12}\) (which can overlap), if the distance between any two of them does not exceed 1, then \(\sum_{1 \le i < j \le 12} |P_iP_j|^2 \le \lambda\).
Let \(O\) be an arbitrary point, and let ... | 6.625 | [
7,
7,
7,
6,
7,
6,
6,
7
] |
There are $10$ birds on the ground. For any $5$ of them, there are at least $4$ birds on a circle. Determine the least possible number of birds on the circle with the most birds. | 9 |
Given that there are 10 birds on the ground and for any 5 of them, there are at least 4 birds on a circle, we need to determine the least possible number of birds on the circle with the most birds.
To solve this, consider the following steps:
1. **Initial Assumption**: Let \( n \) be the number of birds on the circl... | 4.5 | [
4,
4,
4,
5,
4,
5,
5,
5
] |
$x$, $y$ and $z$ are positive reals such that $x+y+z=xyz$. Find the minimum value of:
\[ x^7(yz-1)+y^7(zx-1)+z^7(xy-1) \] | 162\sqrt{3} |
Given that \( x \), \( y \), and \( z \) are positive reals such that \( x + y + z = xyz \), we aim to find the minimum value of:
\[
x^7(yz-1) + y^7(zx-1) + z^7(xy-1).
\]
First, we use the given condition \( x + y + z = xyz \). By the AM-GM inequality, we have:
\[
xyz = x + y + z \geq 3\sqrt[3]{xyz},
\]
which implies... | 6.75 | [
7,
7,
7,
7,
7,
6,
7,
6
] |
For each positive integer $ n$, let $ c(n)$ be the largest real number such that
\[ c(n) \le \left| \frac {f(a) \minus{} f(b)}{a \minus{} b}\right|\]
for all triples $ (f, a, b)$ such that
--$ f$ is a polynomial of degree $ n$ taking integers to integers, and
--$ a, b$ are integers with $ f(a) \neq f(b)$.
Find... | \frac{1}{L_n} |
For each positive integer \( n \), let \( c(n) \) be the largest real number such that
\[
c(n) \le \left| \frac{f(a) - f(b)}{a - b} \right|
\]
for all triples \( (f, a, b) \) such that:
- \( f \) is a polynomial of degree \( n \) taking integers to integers, and
- \( a, b \) are integers with \( f(a) \neq f(b) \).
To... | 8.125 | [
9,
8,
8,
8,
8,
8,
8,
8
] |
An integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition.
[quote]For example, 4 can be partitioned in five distinct ways:
4
3 + 1
2 + 2
2 + 1 + 1
1 + 1 + 1 + 1[/quote]
The number of partitions of n is given by t... | 1, 3, 5 |
We need to determine all positive integers \( n \) such that
\[
p(n) + p(n+4) = p(n+2) + p(n+3),
\]
where \( p(n) \) denotes the partition function, which counts the number of ways \( n \) can be partitioned into positive integers.
To solve this, we consider the equivalent equation by setting \( N = n + 4 \):
\[
p(N... | 6.5 | [
6,
6,
6,
7,
7,
7,
6,
7
] |
$101$ people, sitting at a round table in any order, had $1,2,... , 101$ cards, respectively.
A transfer is someone give one card to one of the two people adjacent to him.
Find the smallest positive integer $k$ such that there always can through no more than $ k $ times transfer, each person hold c... | 42925 |
Given 101 people sitting at a round table, each holding a unique card numbered from 1 to 101, we need to determine the smallest positive integer \( k \) such that through no more than \( k \) transfers, each person can hold the same number of cards, regardless of the initial sitting order.
To find the smallest \( k \... | 7.75 | [
8,
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8,
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9,
6
] |
In a sports league, each team uses a set of at most $t$ signature colors. A set $S$ of teams is[i] color-identifiable[/i] if one can assign each team in $S$ one of their signature colors, such that no team in $S$ is assigned any signature color of a different team in $S$.
For all positive integers $n$ and $t$, determi... | \lceil \frac{n}{t} \rceil |
In a sports league, each team uses a set of at most \( t \) signature colors. A set \( S \) of teams is color-identifiable if one can assign each team in \( S \) one of their signature colors, such that no team in \( S \) is assigned any signature color of a different team in \( S \).
For all positive integers \( n \... | 6.625 | [
7,
7,
7,
7,
7,
7,
6,
5
] |
Let $G$ be a simple graph with 100 vertices such that for each vertice $u$, there exists a vertice $v \in N \left ( u \right )$ and $ N \left ( u \right ) \cap N \left ( v \right ) = \o $. Try to find the maximal possible number of edges in $G$. The $ N \left ( . \right )$ refers to the neighborhood. | 3822 |
Let \( G \) be a simple graph with 100 vertices such that for each vertex \( u \), there exists a vertex \( v \in N(u) \) and \( N(u) \cap N(v) = \emptyset \). We aim to find the maximal possible number of edges in \( G \).
We claim that the maximal number of edges is \( \boxed{3822} \).
To prove this, we consider t... | 7.75 | [
7,
8,
7,
8,
8,
8,
8,
8
] |
A social club has $2k+1$ members, each of whom is fluent in the same $k$ languages. Any pair of members always talk to each other in only one language. Suppose that there were no three members such that they use only one language among them. Let $A$ be the number of three-member subsets such that the three distinct ... | \binom{2k+1}{3} - k(2k+1) |
Let \( A \) be the number of three-member subsets such that the three distinct pairs among them use different languages. We aim to find the maximum possible value of \( A \).
Given that the social club has \( 2k+1 \) members, each fluent in \( k \) languages, and that no three members use only one language among them... | 6.875 | [
7,
7,
7,
7,
6,
7,
7,
7
] |
Find all positive integer $ m$ if there exists prime number $ p$ such that $ n^m\minus{}m$ can not be divided by $ p$ for any integer $ n$. | m \neq 1 |
We are asked to find all positive integers \( m \) such that there exists a prime number \( p \) for which \( n^m - m \) is not divisible by \( p \) for any integer \( n \).
We claim that the answer is all \( m \neq 1 \).
First, consider \( m = 1 \). In this case, the expression becomes \( n - 1 \), which can clearl... | 6.75 | [
7,
7,
6,
7,
7,
6,
7,
7
] |
Determine the greatest real number $ C $, such that for every positive integer $ n\ge 2 $, there exists $ x_1, x_2,..., x_n \in [-1,1]$, so that
$$\prod_{1\le i<j\le n}(x_i-x_j) \ge C^{\frac{n(n-1)}{2}}$$. | \frac{1}{2} |
To determine the greatest real number \( C \) such that for every positive integer \( n \geq 2 \), there exist \( x_1, x_2, \ldots, x_n \in [-1, 1] \) satisfying
\[
\prod_{1 \le i < j \le n} (x_i - x_j) \geq C^{\frac{n(n-1)}{2}},
\]
we consider the example where \( x_i = \cos\left(\frac{i\pi}{n}\right) \) for \( i = 1... | 7.5 | [
7,
8,
7,
8,
7,
7,
8,
8
] |
Find all the pairs of prime numbers $ (p,q)$ such that $ pq|5^p\plus{}5^q.$ | (2, 3), (2, 5), (3, 2), (5, 2), (5, 5), (5, 313), (313, 5) |
To find all pairs of prime numbers \((p, q)\) such that \(pq \mid 5^p + 5^q\), we consider several cases:
1. **Case 1: Either \(p\) or \(q\) is 2.**
- Assume \(p = 2\). Then \(2q \mid 5^2 + 5^q\).
- This simplifies to \(2q \mid 25 + 5^q\).
- Since \(25 \equiv 1 \pmod{2}\), we have \(5^q + 1 \equiv 0 \pmod{q}... | 7 | [
7,
7,
7,
7,
7,
7,
7,
7
] |
Suppose $a_i, b_i, c_i, i=1,2,\cdots ,n$, are $3n$ real numbers in the interval $\left [ 0,1 \right ].$ Define $$S=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k<1 \right \}, \; \; T=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k>2 \right \}.$$ Now we know that $\left | S \right |\ge 2018,\, \left | T \right |\ge 201... | 18 |
Suppose \( a_i, b_i, c_i \) for \( i = 1, 2, \ldots, n \) are \( 3n \) real numbers in the interval \([0, 1]\). Define the sets
\[
S = \{ (i, j, k) \mid a_i + b_j + c_k < 1 \}
\]
and
\[
T = \{ (i, j, k) \mid a_i + b_j + c_k > 2 \}.
\]
We are given that \( |S| \geq 2018 \) and \( |T| \geq 2018 \). We aim to find the m... | 6.5 | [
7,
7,
7,
6,
6,
5,
7,
7
] |
Let $\triangle ABC$ be an equilateral triangle of side length 1. Let $D,E,F$ be points on $BC,AC,AB$ respectively, such that $\frac{DE}{20} = \frac{EF}{22} = \frac{FD}{38}$. Let $X,Y,Z$ be on lines $BC,CA,AB$ respectively, such that $XY\perp DE, YZ\perp EF, ZX\perp FD$. Find all possible values of $\frac{1}{[DEF]} + \f... | \frac{97 \sqrt{2} + 40 \sqrt{3}}{15} |
Let \(\triangle ABC\) be an equilateral triangle of side length 1. Let \(D, E, F\) be points on \(BC, AC, AB\) respectively, such that \(\frac{DE}{20} = \frac{EF}{22} = \frac{FD}{38}\). Let \(X, Y, Z\) be on lines \(BC, CA, AB\) respectively, such that \(XY \perp DE\), \(YZ \perp EF\), \(ZX \perp FD\). We aim to find ... | 6.875 | [
7,
7,
7,
7,
6,
7,
7,
7
] |
Given positive integer $n$ and $r$ pairwise distinct primes $p_1,p_2,\cdots,p_r.$ Initially, there are $(n+1)^r$ numbers written on the blackboard: $p_1^{i_1}p_2^{i_2}\cdots p_r^{i_r} (0 \le i_1,i_2,\cdots,i_r \le n).$
Alice and Bob play a game by making a move by turns, with Alice going first. In Alice's round, she e... | M^{\lfloor \frac{n}{2} \rfloor} |
Given positive integer \( n \) and \( r \) pairwise distinct primes \( p_1, p_2, \cdots, p_r \). Initially, there are \( (n+1)^r \) numbers written on the blackboard: \( p_1^{i_1} p_2^{i_2} \cdots p_r^{i_r} \) where \( 0 \le i_1, i_2, \cdots, i_r \le n \).
Alice and Bob play a game by making a move by turns, with Ali... | 7.375 | [
8,
7,
8,
7,
7,
8,
7,
7
] |
Given integer $n\geq 2$. Find the minimum value of $\lambda {}$, satisfy that for any real numbers $a_1$, $a_2$, $\cdots$, ${a_n}$ and ${b}$,
$$\lambda\sum\limits_{i=1}^n\sqrt{|a_i-b|}+\sqrt{n\left|\sum\limits_{i=1}^na_i\right|}\geqslant\sum\limits_{i=1}^n\sqrt{|a_i|}.$$ | \frac{n-1 + \sqrt{n-1}}{\sqrt{n}} |
Given an integer \( n \geq 2 \), we aim to find the minimum value of \( \lambda \) such that for any real numbers \( a_1, a_2, \ldots, a_n \) and \( b \), the following inequality holds:
\[
\lambda \sum_{i=1}^n \sqrt{|a_i - b|} + \sqrt{n \left| \sum_{i=1}^n a_i \right|} \geq \sum_{i=1}^n \sqrt{|a_i|}.
\]
To determine... | 7 | [
8,
7,
7,
6,
7,
7,
7,
7
] |
Let $k$ be a positive real. $A$ and $B$ play the following game: at the start, there are $80$ zeroes arrange around a circle. Each turn, $A$ increases some of these $80$ numbers, such that the total sum added is $1$. Next, $B$ selects ten consecutive numbers with the largest sum, and reduces them all to $0$. $A$ then w... | 1 + 1 + \frac{1}{2} + \ldots + \frac{1}{7} |
Let \( k \) be a positive real number. \( A \) and \( B \) play the following game: at the start, there are 80 zeroes arranged around a circle. Each turn, \( A \) increases some of these 80 numbers such that the total sum added is 1. Next, \( B \) selects ten consecutive numbers with the largest sum and reduces them a... | 7.375 | [
8,
8,
8,
8,
6,
7,
7,
7
] |
$ S$ is a non-empty subset of the set $ \{ 1, 2, \cdots, 108 \}$, satisfying:
(1) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c \in S$, such that $ \gcd(a,c)\equal{}\gcd(b,c)\equal{}1$.
(2) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c' \in S$, $ c' \neq a$, $ c' ... | 79 |
Let \( S \) be a non-empty subset of the set \( \{ 1, 2, \ldots, 108 \} \) satisfying the following conditions:
1. For any two numbers \( a, b \in S \) (not necessarily distinct), there exists \( c \in S \) such that \( \gcd(a, c) = \gcd(b, c) = 1 \).
2. For any two numbers \( a, b \in S \) (not necessarily distinct)... | 7.125 | [
7,
7,
7,
8,
7,
7,
7,
7
] |
Let $m>1$ be an integer. Find the smallest positive integer $n$, such that for any integers $a_1,a_2,\ldots ,a_n; b_1,b_2,\ldots ,b_n$ there exists integers $x_1,x_2,\ldots ,x_n$ satisfying the following two conditions:
i) There exists $i\in \{1,2,\ldots ,n\}$ such that $x_i$ and $m$ are coprime
ii) $\sum^n_{i=1} a_... | 2\omega(m) + 1 |
Let \( m > 1 \) be an integer. We are tasked with finding the smallest positive integer \( n \) such that for any integers \( a_1, a_2, \ldots, a_n \) and \( b_1, b_2, \ldots, b_n \), there exist integers \( x_1, x_2, \ldots, x_n \) satisfying the following two conditions:
1. There exists \( i \in \{1, 2, \ldots, n\}... | 6.875 | [
7,
6,
7,
7,
7,
8,
6,
7
] |
Fix positive integers $k,n$. A candy vending machine has many different colours of candy, where there are $2n$ candies of each colour. A couple of kids each buys from the vending machine $2$ candies of different colours. Given that for any $k+1$ kids there are two kids who have at least one colour of candy in common, f... | n(3k) |
Fix positive integers \( k \) and \( n \). Consider a candy vending machine that has many different colors of candy, with \( 2n \) candies of each color. A couple of kids each buys from the vending machine 2 candies of different colors. We are to find the maximum number of kids such that for any \( k+1 \) kids, there ... | 6.875 | [
7,
7,
7,
7,
6,
7,
7,
7
] |
Find the smallest positive integer $ K$ such that every $ K$-element subset of $ \{1,2,...,50 \}$ contains two distinct elements $ a,b$ such that $ a\plus{}b$ divides $ ab$. | 26 |
To find the smallest positive integer \( K \) such that every \( K \)-element subset of \( \{1, 2, \ldots, 50\} \) contains two distinct elements \( a \) and \( b \) such that \( a + b \) divides \( ab \), we need to analyze the properties of the set and the divisibility condition.
Consider the set \( \{1, 2, \ldots,... | 5.25 | [
6,
5,
5,
5,
5,
6,
5,
5
] |
Find the largest positive integer $m$ which makes it possible to color several cells of a $70\times 70$ table red such that [list] [*] There are no two red cells satisfying: the two rows in which they are have the same number of red cells, while the two columns in which they are also have the same number of red cells; ... | 32 |
To find the largest positive integer \( m \) that allows coloring several cells of a \( 70 \times 70 \) table red such that:
1. There are no two red cells satisfying: the two rows in which they are have the same number of red cells, while the two columns in which they are also have the same number of red cells.
2. The... | 6.25 | [
6,
7,
6,
6,
6,
6,
7,
6
] |
Convex quadrilateral $ ABCD$ is inscribed in a circle, $ \angle{A}\equal{}60^o$, $ BC\equal{}CD\equal{}1$, rays $ AB$ and $ DC$ intersect at point $ E$, rays $ BC$ and $ AD$ intersect each other at point $ F$. It is given that the perimeters of triangle $ BCE$ and triangle $ CDF$ are both integers. Find the perimeter o... | \frac{38}{7} |
Given a convex quadrilateral \(ABCD\) inscribed in a circle with \(\angle A = 60^\circ\), \(BC = CD = 1\), and the intersections of rays \(AB\) and \(DC\) at point \(E\), and rays \(BC\) and \(AD\) at point \(F\), we aim to find the perimeter of quadrilateral \(ABCD\) given that the perimeters of triangles \(BCE\) and... | 6.25 | [
6,
6,
6,
7,
6,
7,
6,
6
] |
Find all positive real numbers $\lambda$ such that for all integers $n\geq 2$ and all positive real numbers $a_1,a_2,\cdots,a_n$ with $a_1+a_2+\cdots+a_n=n$, the following inequality holds:
$\sum_{i=1}^n\frac{1}{a_i}-\lambda\prod_{i=1}^{n}\frac{1}{a_i}\leq n-\lambda$. | \lambda \geq e |
Find all positive real numbers \(\lambda\) such that for all integers \(n \geq 2\) and all positive real numbers \(a_1, a_2, \ldots, a_n\) with \(a_1 + a_2 + \cdots + a_n = n\), the following inequality holds:
\[
\sum_{i=1}^n \frac{1}{a_i} - \lambda \prod_{i=1}^{n} \frac{1}{a_i} \leq n - \lambda.
\]
To find the value... | 7.5 | [
8,
7,
8,
8,
7,
8,
6,
8
] |
Find all functions $f:\mathbb {Z}\to\mathbb Z$, satisfy that for any integer ${a}$, ${b}$, ${c}$,
$$2f(a^2+b^2+c^2)-2f(ab+bc+ca)=f(a-b)^2+f(b-c)^2+f(c-a)^2$$ | f(x) = 0 \text{ or } f(x) = x |
We are given the functional equation for \( f: \mathbb{Z} \to \mathbb{Z} \):
\[
2f(a^2 + b^2 + c^2) - 2f(ab + bc + ca) = f(a - b)^2 + f(b - c)^2 + f(c - a)^2
\]
for any integers \( a, b, \) and \( c \).
To find all such functions \( f \), we proceed as follows:
### Step 1: Initial Analysis
Let \( P(a, b, c) \) denot... | 7.75 | [
8,
6,
9,
7,
8,
8,
8,
8
] |
Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying the following conditions:
(1) $f(1)=1$;
(2) $\forall n\in \mathbb{N}$, $3f(n) f(2n+1) =f(2n) ( 1+3f(n) )$;
(3) $\forall n\in \mathbb{N}$, $f(2n) < 6 f(n)$.
Find all solutions of equation $f(k) +f(l)=293$, where $k<l$.
($\mathbb{N}$ denotes the set of a... | (5, 47), (7, 45), (13, 39), (15, 37) |
Let \( f: \mathbb{N} \rightarrow \mathbb{N} \) be a function satisfying the following conditions:
1. \( f(1) = 1 \);
2. For all \( n \in \mathbb{N} \), \( 3f(n) f(2n+1) = f(2n) (1 + 3f(n)) \);
3. For all \( n \in \mathbb{N} \), \( f(2n) < 6 f(n) \).
We need to find all solutions of the equation \( f(k) + f(l) = 293 \... | 7.375 | [
7,
8,
8,
8,
6,
7,
8,
7
] |
Find all natural numbers $n (n \geq 2)$ such that there exists reals $a_1, a_2, \dots, a_n$ which satisfy \[ \{ |a_i - a_j| \mid 1\leq i<j \leq n\} = \left\{1,2,\dots,\frac{n(n-1)}{2}\right\}. \]
Let $A=\{1,2,3,4,5,6\}, B=\{7,8,9,\dots,n\}$. $A_i(i=1,2,\dots,20)$ contains eight numbers, three of which are chosen fro... | 2, 3, 4 |
We need to find all natural numbers \( n \) (where \( n \geq 2 \)) such that there exist real numbers \( a_1, a_2, \dots, a_n \) which satisfy the condition:
\[ \{ |a_i - a_j| \mid 1 \leq i < j \leq n \} = \left\{ 1, 2, \dots, \frac{n(n-1)}{2} \right\}. \]
We claim that only \( n = 2, 3, 4 \) work. We can construct ... | 6.125 | [
6,
6,
7,
6,
6,
6,
6,
6
] |
Find all triples $ (x,y,z)$ of real numbers that satisfy the system of equations
\[ \begin{cases}x^3 \equal{} 3x\minus{}12y\plus{}50, \\ y^3 \equal{} 12y\plus{}3z\minus{}2, \\ z^3 \equal{} 27z \plus{} 27x. \end{cases}\]
[i]Razvan Gelca.[/i] | (2, 4, 6) |
We are given the system of equations:
\[
\begin{cases}
x^3 = 3x - 12y + 50, \\
y^3 = 12y + 3z - 2, \\
z^3 = 27z + 27x.
\end{cases}
\]
To find all triples \((x, y, z)\) of real numbers that satisfy these equations, we analyze the behavior of the functions involved.
1. **Case \(x > 2\):**
- For \(x > 2\), the funct... | 6.5 | [
6,
7,
6,
7,
6,
6,
7,
7
] |
There are arbitrary 7 points in the plane. Circles are drawn through every 4 possible concyclic points. Find the maximum number of circles that can be drawn. | 7 |
Given 7 arbitrary points in the plane, we need to determine the maximum number of circles that can be drawn through every 4 possible concyclic points.
To solve this, we consider the combinatorial aspect of selecting 4 points out of 7. The number of ways to choose 4 points from 7 is given by the binomial coefficient:
... | 6 | [
6,
7,
5,
6,
6,
6,
6,
6
] |
Number $a$ is such that $\forall a_1, a_2, a_3, a_4 \in \mathbb{R}$, there are integers $k_1, k_2, k_3, k_4$ such that $\sum_{1 \leq i < j \leq 4} ((a_i - k_i) - (a_j - k_j))^2 \leq a$. Find the minimum of $a$. | 1.25 |
Let \( a \) be such that for all \( a_1, a_2, a_3, a_4 \in \mathbb{R} \), there exist integers \( k_1, k_2, k_3, k_4 \) such that
\[
\sum_{1 \leq i < j \leq 4} ((a_i - k_i) - (a_j - k_j))^2 \leq a.
\]
We aim to find the minimum value of \( a \).
Consider the numbers \( a_i = \frac{i}{4} \) for \( i = 1, 2, 3, 4 \). L... | 6.5 | [
6,
6,
6,
7,
7,
7,
7,
6
] |
For a positive integer $M$, if there exist integers $a$, $b$, $c$ and $d$ so that:
\[ M \leq a < b \leq c < d \leq M+49, \qquad ad=bc \]
then we call $M$ a GOOD number, if not then $M$ is BAD. Please find the greatest GOOD number and the smallest BAD number. | 576 |
For a positive integer \( M \), we need to determine if it is a GOOD or BAD number based on the existence of integers \( a, b, c, \) and \( d \) such that:
\[ M \leq a < b \leq c < d \leq M + 49, \qquad ad = bc. \]
We aim to find the greatest GOOD number and the smallest BAD number.
### Greatest GOOD Number
**Lemma... | 6.375 | [
7,
7,
6,
6,
6,
6,
6,
7
] |
Let $n$ be a positive integer. Initially, a $2n \times 2n$ grid has $k$ black cells and the rest white cells. The following two operations are allowed :
(1) If a $2\times 2$ square has exactly three black cells, the fourth is changed to a black cell;
(2) If there are exactly two black cells in a $2 \times 2$ square, t... | n^2 + n + 1 |
Let \( n \) be a positive integer. Initially, a \( 2n \times 2n \) grid has \( k \) black cells and the rest white cells. The following two operations are allowed:
1. If a \( 2 \times 2 \) square has exactly three black cells, the fourth is changed to a black cell.
2. If there are exactly two black cells in a \( 2 \ti... | 6.875 | [
6,
7,
7,
7,
7,
7,
7,
7
] |
Find the smallest prime number $p$ that cannot be represented in the form $|3^{a} - 2^{b}|$, where $a$ and $b$ are non-negative integers. | 41 |
We need to find the smallest prime number \( p \) that cannot be represented in the form \( |3^a - 2^b| \), where \( a \) and \( b \) are non-negative integers.
First, we verify that all primes less than 41 can be expressed in the form \( |3^a - 2^b| \):
- For \( p = 2 \): \( 2 = |3^0 - 2^1| \)
- For \( p = 3 \): \(... | 6.5 | [
6,
8,
7,
6,
6,
6,
6,
7
] |
A positive integer $n$ is known as an [i]interesting[/i] number if $n$ satisfies
\[{\ \{\frac{n}{10^k}} \} > \frac{n}{10^{10}} \]
for all $k=1,2,\ldots 9$.
Find the number of interesting numbers. | 999989991 |
A positive integer \( n \) is known as an interesting number if \( n \) satisfies
\[
\left\{ \frac{n}{10^k} \right\} > \frac{n}{10^{10}}
\]
for all \( k = 1, 2, \ldots, 9 \), where \( \{ x \} \) denotes the fractional part of \( x \).
To determine the number of interesting numbers, we can use a computational approach... | 6.375 | [
6,
6,
7,
6,
5,
7,
7,
7
] |
Determine all functions $f: \mathbb{Q} \to \mathbb{Q}$ such that
$$f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2}$$
for all $x,y \in \mathbb{Q}$. | f(x) = x^2 + \frac{1}{2} |
Let \( f: \mathbb{Q} \to \mathbb{Q} \) be a function such that
\[
f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2}
\]
for all \( x, y \in \mathbb{Q} \).
First, we denote the given functional equation as \( P(x, y) \):
\[
P(x, y): f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2}.
\]
By considering \( P(... | 8.25 | [
8,
9,
8,
8,
8,
9,
8,
8
] |
Given a square $ABCD$ whose side length is $1$, $P$ and $Q$ are points on the sides $AB$ and $AD$. If the perimeter of $APQ$ is $2$ find the angle $PCQ$. | 45^\circ |
Given a square \(ABCD\) with side length \(1\), points \(P\) and \(Q\) are on sides \(AB\) and \(AD\) respectively. We are to find the angle \( \angle PCQ \) given that the perimeter of \( \triangle APQ \) is \(2\).
Let \( AP = x \) and \( AQ = y \). Then, \( PB = 1 - x \) and \( QD = 1 - y \). We need to find \( \ta... | 4 | [
4,
4,
4,
3,
4,
5,
4,
4
] |
Find the smallest positive number $\lambda $ , such that for any complex numbers ${z_1},{z_2},{z_3}\in\{z\in C\big| |z|<1\}$ ,if $z_1+z_2+z_3=0$, then $$\left|z_1z_2 +z_2z_3+z_3z_1\right|^2+\left|z_1z_2z_3\right|^2 <\lambda .$$ | 1 |
We aim to find the smallest positive number \(\lambda\) such that for any complex numbers \(z_1, z_2, z_3 \in \{z \in \mathbb{C} \mid |z| < 1\}\) with \(z_1 + z_2 + z_3 = 0\), the following inequality holds:
\[
\left|z_1z_2 + z_2z_3 + z_3z_1\right|^2 + \left|z_1z_2z_3\right|^2 < \lambda.
\]
First, we show that \(\lam... | 6.75 | [
7,
7,
7,
7,
6,
7,
6,
7
] |
Define the sequences $(a_n),(b_n)$ by
\begin{align*}
& a_n, b_n > 0, \forall n\in\mathbb{N_+} \\
& a_{n+1} = a_n - \frac{1}{1+\sum_{i=1}^n\frac{1}{a_i}} \\
& b_{n+1} = b_n + \frac{1}{1+\sum_{i=1}^n\frac{1}{b_i}}
\end{align*}
1) If $a_{100}b_{100} = a_{101}b_{101}$, find the value of $a_1-b_1$;
2) If $a_{100} = b_{99}... | 199 |
Define the sequences \( (a_n) \) and \( (b_n) \) by
\[
\begin{align*}
& a_n, b_n > 0, \forall n \in \mathbb{N_+}, \\
& a_{n+1} = a_n - \frac{1}{1 + \sum_{i=1}^n \frac{1}{a_i}}, \\
& b_{n+1} = b_n + \frac{1}{1 + \sum_{i=1}^n \frac{1}{b_i}}.
\end{align*}
\]
1. If \( a_{100} b_{100} = a_{101} b_{101} \), find the value... | 7 | [
7,
7,
7,
7,
7,
7,
7,
7
] |
Let $a=2001$. Consider the set $A$ of all pairs of integers $(m,n)$ with $n\neq0$ such that
(i) $m<2a$;
(ii) $2n|(2am-m^2+n^2)$;
(iii) $n^2-m^2+2mn\leq2a(n-m)$.
For $(m, n)\in A$, let \[f(m,n)=\frac{2am-m^2-mn}{n}.\]
Determine the maximum and minimum values of $f$. | 2 \text{ and } 3750 |
Let \( a = 2001 \). Consider the set \( A \) of all pairs of integers \((m, n)\) with \( n \neq 0 \) such that:
1. \( m < 2a \),
2. \( 2n \mid (2am - m^2 + n^2) \),
3. \( n^2 - m^2 + 2mn \leq 2a(n - m) \).
For \((m, n) \in A\), let
\[ f(m, n) = \frac{2am - m^2 - mn}{n}. \]
We need to determine the maximum and minimum... | 7.125 | [
7,
8,
7,
7,
7,
7,
7,
7
] |
Given a positive integer $n$, find all $n$-tuples of real number $(x_1,x_2,\ldots,x_n)$ such that
\[ f(x_1,x_2,\cdots,x_n)=\sum_{k_1=0}^{2} \sum_{k_2=0}^{2} \cdots \sum_{k_n=0}^{2} \big| k_1x_1+k_2x_2+\cdots+k_nx_n-1 \big| \]
attains its minimum. | \left( \frac{1}{n+1}, \frac{1}{n+1}, \ldots, \frac{1}{n+1} \right) |
Given a positive integer \( n \), we aim to find all \( n \)-tuples of real numbers \( (x_1, x_2, \ldots, x_n) \) such that
\[
f(x_1, x_2, \cdots, x_n) = \sum_{k_1=0}^{2} \sum_{k_2=0}^{2} \cdots \sum_{k_n=0}^{2} \left| k_1 x_1 + k_2 x_2 + \cdots + k_n x_n - 1 \right|
\]
attains its minimum.
To solve this, we first cl... | 6.75 | [
7,
7,
7,
7,
7,
6,
7,
6
] |
For a given integer $n\ge 2$, let $a_0,a_1,\ldots ,a_n$ be integers satisfying $0=a_0<a_1<\ldots <a_n=2n-1$. Find the smallest possible number of elements in the set $\{ a_i+a_j \mid 0\le i \le j \le n \}$. | 3n |
For a given integer \( n \ge 2 \), let \( a_0, a_1, \ldots, a_n \) be integers satisfying \( 0 = a_0 < a_1 < \ldots < a_n = 2n-1 \). We aim to find the smallest possible number of elements in the set \( \{ a_i + a_j \mid 0 \le i \le j \le n \} \).
First, we prove that the set \( \{ a_i + a_j \mid 1 \le i \le j \le n-... | 6.125 | [
6,
7,
6,
6,
6,
6,
6,
6
] |
Find all functions $f: \mathbb R \to \mathbb R$ such that for any $x,y \in \mathbb R$, the multiset $\{(f(xf(y)+1),f(yf(x)-1)\}$ is identical to the multiset $\{xf(f(y))+1,yf(f(x))-1\}$.
[i]Note:[/i] The multiset $\{a,b\}$ is identical to the multiset $\{c,d\}$ if and only if $a=c,b=d$ or $a=d,b=c$. | f(x) \equiv x \text{ or } f(x) \equiv -x |
Let \( f: \mathbb{R} \to \mathbb{R} \) be a function such that for any \( x, y \in \mathbb{R} \), the multiset \( \{ f(xf(y) + 1), f(yf(x) - 1) \} \) is identical to the multiset \( \{ xf(f(y)) + 1, yf(f(x)) - 1 \} \).
We aim to find all such functions \( f \).
Let \( P(x, y) \) denote the assertion that \( \{ f(xf(... | 8 | [
8,
8,
8,
7,
8,
8,
9,
8
] |
What is the smallest integer $n$ , greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?
$\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \cdots, a_n$ is defined to be \[\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}\] | \(\boxed{337}\) | Let's first obtain an algebraic expression for the root mean square of the first $n$ integers, which we denote $I_n$ . By repeatedly using the identity $(x+1)^3 = x^3 + 3x^2 + 3x + 1$ , we can write \[1^3 + 3\cdot 1^2 + 3 \cdot 1 + 1 = 2^3,\] \[1^3 + 3 \cdot(1^2 + 2^2) + 3 \cdot (1 + 2) + 1 + 1 = 3^3,\] and \[1^3 + 3... | 5.375 | [
6,
5,
5,
4,
5,
6,
6,
6
] |
For a positive integer $n$, and a non empty subset $A$ of $\{1,2,...,2n\}$, call $A$ good if the set $\{u\pm v|u,v\in A\}$ does not contain the set $\{1,2,...,n\}$. Find the smallest real number $c$, such that for any positive integer $n$, and any good subset $A$ of $\{1,2,...,2n\}$, $|A|\leq cn$. | \frac{6}{5} |
For a positive integer \( n \), and a non-empty subset \( A \) of \(\{1, 2, \ldots, 2n\}\), we call \( A \) good if the set \(\{u \pm v \mid u, v \in A\}\) does not contain the set \(\{1, 2, \ldots, n\}\). We aim to find the smallest real number \( c \) such that for any positive integer \( n \), and any good subset \... | 7.125 | [
7,
7,
7,
7,
7,
7,
8,
7
] |
Find all primes $p$ such that there exist positive integers $x,y$ that satisfy $x(y^2-p)+y(x^2-p)=5p$ . | \[
\boxed{2, 3, 7}
\] | Rearrange the original equation to get \begin{align*} 5p &= xy^2 + x^2 y - xp - yp \\ &= xy(x+y) -p(x+y) \\ &= (xy-p)(x+y) \end{align*} Since $5p$ , $xy-p$ , and $x+y$ are all integers, $xy-p$ and $x+y$ must be a factor of $5p$ . Now there are four cases to consider.
Case 1: and
Since $x$ and $y$ are positive integers... | 6.375 | [
6,
6,
6,
7,
7,
7,
6,
6
] |
Let $n \geq 2$ be a natural. Define
$$X = \{ (a_1,a_2,\cdots,a_n) | a_k \in \{0,1,2,\cdots,k\}, k = 1,2,\cdots,n \}$$.
For any two elements $s = (s_1,s_2,\cdots,s_n) \in X, t = (t_1,t_2,\cdots,t_n) \in X$, define
$$s \vee t = (\max \{s_1,t_1\},\max \{s_2,t_2\}, \cdots , \max \{s_n,t_n\} )$$
$$s \wedge t = (\min \{s_1... | (n + 1)! - (n - 1)! |
Let \( n \geq 2 \) be a natural number. Define
\[
X = \{ (a_1, a_2, \cdots, a_n) \mid a_k \in \{0, 1, 2, \cdots, k\}, k = 1, 2, \cdots, n \}.
\]
For any two elements \( s = (s_1, s_2, \cdots, s_n) \in X \) and \( t = (t_1, t_2, \cdots, t_n) \in X \), define
\[
s \vee t = (\max \{s_1, t_1\}, \max \{s_2, t_2\}, \cdots... | 7.125 | [
8,
7,
7,
7,
6,
7,
8,
7
] |
For each integer $n\ge 2$ , determine, with proof, which of the two positive real numbers $a$ and $b$ satisfying \[a^n=a+1,\qquad b^{2n}=b+3a\] is larger. | \[ a > b \] | Solution 1
Square and rearrange the first equation and also rearrange the second. \begin{align} a^{2n}-a&=a^2+a+1\\ b^{2n}-b&=3a \end{align} It is trivial that \begin{align*} (a-1)^2 > 0 \tag{3} \end{align*} since $a-1$ clearly cannot equal $0$ (Otherwise $a^n=1\neq 1+1$ ). Thus \begin{align*} a^2+a+1&>3a \tag{4}\\ a^... | 6.25 | [
7,
6,
7,
6,
6,
6,
6,
6
] |
Assume $n$ is a positive integer. Considers sequences $a_0, a_1, \ldots, a_n$ for which $a_i \in \{1, 2, \ldots , n\}$ for all $i$ and $a_n = a_0$.
(a) Suppose $n$ is odd. Find the number of such sequences if $a_i - a_{i-1} \not \equiv i \pmod{n}$ for all $i = 1, 2, \ldots, n$.
(b) Suppose $n$ is an odd prime. F... | (n-1)(n-2)^{n-1} - \frac{2^{n-1} - 1}{n} - 1 |
Let \( n \) be a positive integer. Consider sequences \( a_0, a_1, \ldots, a_n \) for which \( a_i \in \{1, 2, \ldots , n\} \) for all \( i \) and \( a_n = a_0 \).
### Part (a)
Suppose \( n \) is odd. We need to find the number of such sequences if \( a_i - a_{i-1} \not\equiv i \pmod{n} \) for all \( i = 1, 2, \ldots... | 6.75 | [
7,
7,
7,
7,
7,
7,
6,
6
] |
In convex quadrilateral $ ABCD$, $ AB\equal{}a$, $ BC\equal{}b$, $ CD\equal{}c$, $ DA\equal{}d$, $ AC\equal{}e$, $ BD\equal{}f$. If $ \max \{a,b,c,d,e,f \}\equal{}1$, then find the maximum value of $ abcd$. | 2 - \sqrt{3} |
Given a convex quadrilateral \(ABCD\) with side lengths \(AB = a\), \(BC = b\), \(CD = c\), \(DA = d\), and diagonals \(AC = e\), \(BD = f\), where \(\max \{a, b, c, d, e, f\} = 1\), we aim to find the maximum value of \(abcd\).
We claim that the maximum value of \(abcd\) is \(2 - \sqrt{3}\).
To show that this value... | 7 | [
7,
7,
8,
8,
6,
7,
6,
7
] |
16 students took part in a competition. All problems were multiple choice style. Each problem had four choices. It was said that any two students had at most one answer in common, find the maximum number of problems. | 5 |
Let 16 students take part in a competition where each problem is multiple choice with four choices. We are to find the maximum number of problems such that any two students have at most one answer in common.
Let \( T \) denote the number of triples \((S_i, S_j, Q_k)\) such that students \( S_i \) and \( S_j \) answer... | 6.125 | [
6,
7,
6,
6,
6,
6,
7,
5
] |
For positive integer $k>1$, let $f(k)$ be the number of ways of factoring $k$ into product of positive integers greater than $1$ (The order of factors are not countered, for example $f(12)=4$, as $12$ can be factored in these $4$ ways: $12,2\cdot 6,3\cdot 4, 2\cdot 2\cdot 3$.
Prove: If $n$ is a positive integer greater... | \frac{n}{p} |
For a positive integer \( k > 1 \), let \( f(k) \) represent the number of ways to factor \( k \) into a product of positive integers greater than 1. For example, \( f(12) = 4 \) because 12 can be factored in these 4 ways: \( 12 \), \( 2 \cdot 6 \), \( 3 \cdot 4 \), and \( 2 \cdot 2 \cdot 3 \).
We aim to prove that i... | 6.875 | [
7,
7,
7,
7,
6,
7,
7,
7
] |
Three distinct vertices are chosen at random from the vertices of a given regular polygon of $(2n+1)$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen random points? | \[
\boxed{\frac{n+1}{4n-2}}
\] | There are $\binom{2n+1}{3}$ ways how to pick the three vertices. We will now count the ways where the interior does NOT contain the center. These are obviously exactly the ways where all three picked vertices lie among some $n+1$ consecutive vertices of the polygon.
We will count these as follows: We will go clockwise ... | 6.625 | [
6,
6,
6,
7,
7,
7,
7,
7
] |
Given positive integers $k, m, n$ such that $1 \leq k \leq m \leq n$. Evaluate
\[\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}.\] | 0 |
Given positive integers \( k, m, n \) such that \( 1 \leq k \leq m \leq n \), we aim to evaluate the sum
\[
\sum_{i=0}^n \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}.
\]
To solve this, we employ a calculus-based approach. We start by expressing the sum in terms of an integral:
\[
\sum_{i=0}^n \frac{(-1)... | 7.625 | [
8,
8,
7,
7,
7,
8,
8,
8
] |
Determine all integers $k$ such that there exists infinitely many positive integers $n$ [b]not[/b] satisfying
\[n+k |\binom{2n}{n}\] | k \neq 1 |
Determine all integers \( k \) such that there exist infinitely many positive integers \( n \) not satisfying
\[
n + k \mid \binom{2n}{n}.
\]
We claim that all integers \( k \neq 1 \) satisfy the desired property.
First, recall that \(\frac{1}{n + 1} \binom{2n}{n}\) is the \( n \)-th Catalan number. Since the Catal... | 7 | [
7,
7,
7,
7,
7,
7,
7,
7
] |
Given two integers $ m,n$ satisfying $ 4 < m < n.$ Let $ A_{1}A_{2}\cdots A_{2n \plus{} 1}$ be a regular $ 2n\plus{}1$ polygon. Denote by $ P$ the set of its vertices. Find the number of convex $ m$ polygon whose vertices belongs to $ P$ and exactly has two acute angles. | (2n + 1) \left[ \binom{n}{m - 1} + \binom{n + 1}{m - 1} \right] |
Given two integers \( m \) and \( n \) satisfying \( 4 < m < n \), let \( A_1A_2\cdots A_{2n+1} \) be a regular \( 2n+1 \) polygon. Denote by \( P \) the set of its vertices. We aim to find the number of convex \( m \)-gons whose vertices belong to \( P \) and have exactly two acute angles.
Notice that if a regular \... | 7.375 | [
7,
8,
7,
7,
7,
8,
8,
7
] |
Find all positive integers $x,y$ satisfying the equation \[9(x^2+y^2+1) + 2(3xy+2) = 2005 .\] | \[
\boxed{(7, 11), (11, 7)}
\] | Solution 1
We can re-write the equation as:
$(3x)^2 + y^2 + 2(3x)(y) + 8y^2 + 9 + 4 = 2005$
or $(3x + y)^2 = 4(498 - 2y^2)$
The above equation tells us that $(498 - 2y^2)$ is a perfect square.
Since $498 - 2y^2 \ge 0$ . this implies that $y \le 15$
Also, taking $mod 3$ on both sides we see that $y$ cannot be a multi... | 5.5 | [
6,
6,
4,
5,
6,
6,
6,
5
] |
Let $n \geq 3$ be an odd number and suppose that each square in a $n \times n$ chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex and two squares $a,b$ are considered connected if there exists a sequence of squares $c_1,\ldots,c_k$ wi... | \left(\frac{n+1}{2}\right)^2 + 1 |
Let \( n \geq 3 \) be an odd number and suppose that each square in an \( n \times n \) chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex. Two squares \( a \) and \( b \) are considered connected if there exists a sequence of square... | 6.625 | [
7,
6,
6,
7,
6,
7,
7,
7
] |
A $5 \times 5$ table is called regular if each of its cells contains one of four pairwise distinct real numbers, such that each of them occurs exactly once in every $2 \times 2$ subtable.The sum of all numbers of a regular table is called the total sum of the table. With any four numbers, one constructs all possible re... | \boxed{60} | Solution 1 (Official solution)
We will prove that the maximum number of total sums is $60$ .
The proof is based on the following claim:
In a regular table either each row contains exactly two of the numbers or each column contains exactly two of the numbers.
Proof of the Claim:
Let R be a row containing at least three... | 6.875 | [
6,
7,
7,
7,
7,
8,
6,
7
] |
Let $S_r=x^r+y^r+z^r$ with $x,y,z$ real. It is known that if $S_1=0$ ,
$(*)$ $\frac{S_{m+n}}{m+n}=\frac{S_m}{m}\frac{S_n}{n}$
for $(m,n)=(2,3),(3,2),(2,5)$ , or $(5,2)$ . Determine all other pairs of integers $(m,n)$ if any, so that $(*)$ holds for all real numbers $x,y,z$ such that $x+y+z=0$ . | \((m, n) = (5, 2), (2, 5), (3, 2), (2, 3)\) | Claim Both $m,n$ can not be even.
Proof $x+y+z=0$ , $\implies x=-(y+z)$ .
Since $\frac{S_{m+n}}{m+n} = \frac{S_m S_n}{mn}$ ,
by equating cofficient of $y^{m+n}$ on LHS and RHS ,get
$\frac{2}{m+n}=\frac{4}{mn}$ .
$\implies \frac{m}{2} + \frac {n}{2} = \frac{m\cdot n}{2\cdot2}$ .
So we have, $\frac{m}{2} \biggm{|} \frac... | 7.375 | [
8,
7,
8,
7,
7,
7,
7,
8
] |
A computer screen shows a $98 \times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minim... | \[ 98 \] | Answer: $98$ .
There are $4\cdot97$ adjacent pairs of squares in the border and each pair has one black and one white square. Each move can fix at most $4$ pairs, so we need at least $97$ moves. However, we start with two corners one color and two another, so at least one rectangle must include a corner square. But suc... | 6 | [
6,
6,
7,
6,
5,
6,
6,
6
] |
Find the maximum possible number of three term arithmetic progressions in a monotone sequence of $n$ distinct reals. | \[
f(n) = \left\lfloor \frac{(n-1)^2}{2} \right\rfloor
\] | Consider the first few cases for $n$ with the entire $n$ numbers forming an arithmetic sequence \[(1, 2, 3, \ldots, n)\] If $n = 3$ , there will be one ascending triplet (123). Let's only consider the ascending order for now.
If $n = 4$ , the first 3 numbers give 1 triplet, the addition of the 4 gives one more, for 2 i... | 6.5 | [
7,
6,
7,
6,
7,
6,
6,
7
] |
Let $ABCDE$ be a convex pentagon such that $AB=AE=CD=1$ , $\angle ABC=\angle DEA=90^\circ$ and $BC+DE=1$ . Compute the area of the pentagon. | \[ 1 \] | Solution 1
Let $BC = a, ED = 1 - a$
Let $\angle DAC = X$
Applying cosine rule to $\triangle DAC$ we get:
$\cos X = \frac{AC ^ {2} + AD ^ {2} - DC ^ {2}}{ 2 \cdot AC \cdot AD }$
Substituting $AC^{2} = 1^{2} + a^{2}, AD ^ {2} = 1^{2} + (1-a)^{2}, DC = 1$ we get:
$\cos^{2} X = \frac{(1 - a - a ^ {2}) ^ {2}}{(1 + a^{2})... | 5.25 | [
4,
6,
6,
5,
5,
5,
6,
5
] |
Let $A,B,C,D$ denote four points in space such that at most one of the distances $AB,AC,AD,BC,BD,CD$ is greater than $1$ . Determine the maximum value of the sum of the six distances. | \[ 5 + \sqrt{3} \] | Suppose that $AB$ is the length that is more than $1$ . Let spheres with radius $1$ around $A$ and $B$ be $S_A$ and $S_B$ . $C$ and $D$ must be in the intersection of these spheres, and they must be on the circle created by the intersection to maximize the distance. We have $AC + BC + AD + BD = 4$ .
In fact, $CD$ must ... | 7 | [
7,
7,
7,
7,
7,
7,
7,
7
] |
On a given circle, six points $A$ , $B$ , $C$ , $D$ , $E$ , and $F$ are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangles $ABC$ and $DEF$ are disjoint, i.e., have no common points. | \[
\frac{3}{10}
\] | First we give the circle an orientation (e.g., letting the circle be the unit circle in polar coordinates). Then, for any set of six points chosen on the circle, there are exactly $6!$ ways to label them one through six. Also, this does not affect the probability we wish to calculate. This will, however, make calculat... | 6.5 | [
6,
6,
7,
7,
6,
7,
6,
7
] |
Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1. | -1 - pq \text{ and } 1 + pq |
Given distinct prime numbers \( p \) and \( q \) and a natural number \( n \geq 3 \), we aim to find all \( a \in \mathbb{Z} \) such that the polynomial \( f(x) = x^n + ax^{n-1} + pq \) can be factored into two integral polynomials of degree at least 1.
To solve this, we use the following reasoning:
1. **Lemma (Eise... | 7.5 | [
7,
7,
7,
8,
8,
8,
8,
7
] |
Let $ABC$ be an isosceles triangle with $AC=BC$ , let $M$ be the midpoint of its side $AC$ , and let $Z$ be the line through $C$ perpendicular to $AB$ . The circle through the points $B$ , $C$ , and $M$ intersects the line $Z$ at the points $C$ and $Q$ . Find the radius of the circumcircle of the triangle $ABC$ in term... | \[ R = \frac{2}{3}m \] | Let length of side $CB = x$ and length of $QM = a$ . We shall first prove that $QM = QB$ .
Let $O$ be the circumcenter of $\triangle ACB$ which must lie on line $Z$ as $Z$ is a perpendicular bisector of isosceles $\triangle ACB$ .
So, we have $\angle ACO = \angle BCO = \angle C/2$ .
Now $MQBC$ is a cyclic quadrilateral... | 6.125 | [
6,
6,
5,
6,
7,
6,
7,
6
] |
In the polynomial $x^4 - 18x^3 + kx^2 + 200x - 1984 = 0$ , the product of $2$ of its roots is $- 32$ . Find $k$ . | \[ k = 86 \] | Using Vieta's formulas, we have:
\begin{align*}a+b+c+d &= 18,\\ ab+ac+ad+bc+bd+cd &= k,\\ abc+abd+acd+bcd &=-200,\\ abcd &=-1984.\\ \end{align*}
From the last of these equations, we see that $cd = \frac{abcd}{ab} = \frac{-1984}{-32} = 62$ . Thus, the second equation becomes $-32+ac+ad+bc+bd+62=k$ , and so $ac+ad+bc+b... | 6 | [
6,
6,
6,
6,
6,
6,
6,
6
] |
Let $n \geq 5$ be an integer. Find the largest integer $k$ (as a function of $n$ ) such that there exists a convex $n$ -gon $A_{1}A_{2}\dots A_{n}$ for which exactly $k$ of the quadrilaterals $A_{i}A_{i+1}A_{i+2}A_{i+3}$ have an inscribed circle. (Here $A_{n+j} = A_{j}$ .) | \[
k = \left\lfloor \frac{n}{2} \right\rfloor
\] | Lemma: If quadrilaterals $A_iA_{i+1}A_{i+2}A_{i+3}$ and $A_{i+2}A_{i+3}A_{i+4}A_{i+5}$ in an equiangular $n$ -gon are tangential, and $A_iA_{i+3}$ is the longest side quadrilateral $A_iA_{i+1}A_{i+2}A_{i+3}$ for all $i$ , then quadrilateral $A_{i+1}A_{i+2}A_{i+3}A_{i+4}$ is not tangential.
Proof:
[asy] import geometry;... | 6.75 | [
6,
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7,
7,
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7,
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] |
Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$ , and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$ , evaluate the angle $NMB$ . | \[
\boxed{\frac{\pi}{4}}
\] | [asy] size(15cm,0); draw((0,0)--(0,2)--(4,2)--(4,-3)--(0,0)); draw((-1,2)--(9,2)); draw((0,0)--(2,2)); draw((2,2)--(1,1)); draw((0,0)--(4,2)); draw((0,2)--(1,1)); draw(circle((0,1),1)); draw(circle((4,-3),5)); dot((0,0)); dot((0,2)); dot((2,2)); dot((4,2)); dot((4,-3)); dot((1,1)); dot((0,1)); label("A",(0,0),NW); labe... | 6.125 | [
6,
6,
6,
6,
7,
6,
6,
6
] |
Let $a_1,a_2,a_3,\cdots$ be a non-decreasing sequence of positive integers. For $m\ge1$ , define $b_m=\min\{n: a_n \ge m\}$ , that is, $b_m$ is the minimum value of $n$ such that $a_n\ge m$ . If $a_{19}=85$ , determine the maximum value of $a_1+a_2+\cdots+a_{19}+b_1+b_2+\cdots+b_{85}$ . | \boxed{1700} | We create an array of dots like so: the array shall go out infinitely to the right and downwards, and at the top of the $i$ th column we fill the first $a_i$ cells with one dot each. Then the $19$ th row shall have 85 dots. Now consider the first 19 columns of this array, and consider the first 85 rows. In row $j$ , we... | 6.125 | [
6,
6,
6,
6,
6,
7,
6,
6
] |
Two positive integers $p,q \in \mathbf{Z}^{+}$ are given. There is a blackboard with $n$ positive integers written on it. A operation is to choose two same number $a,a$ written on the blackboard, and replace them with $a+p,a+q$. Determine the smallest $n$ so that such operation can go on infinitely. | \frac{p+q}{\gcd(p,q)} |
Given two positive integers \( p \) and \( q \), we are to determine the smallest number \( n \) such that the operation of choosing two identical numbers \( a, a \) on the blackboard and replacing them with \( a+p \) and \( a+q \) can go on infinitely.
To solve this, we first note that we can assume \(\gcd(p, q) = 1... | 6.625 | [
7,
5,
7,
7,
7,
6,
7,
7
] |
Let $\angle XOY = \frac{\pi}{2}$; $P$ is a point inside $\angle XOY$ and we have $OP = 1; \angle XOP = \frac{\pi}{6}.$ A line passes $P$ intersects the Rays $OX$ and $OY$ at $M$ and $N$. Find the maximum value of $OM + ON - MN.$ | 2 |
Given that \(\angle XOY = \frac{\pi}{2}\), \(P\) is a point inside \(\angle XOY\) with \(OP = 1\) and \(\angle XOP = \frac{\pi}{6}\). We need to find the maximum value of \(OM + ON - MN\) where a line passing through \(P\) intersects the rays \(OX\) and \(OY\) at \(M\) and \(N\), respectively.
To solve this problem, ... | 6 | [
6,
6,
6,
6,
6,
6,
6,
6
] |
Find all ordered triples of primes $(p, q, r)$ such that \[ p \mid q^r + 1, \quad q \mid r^p + 1, \quad r \mid p^q + 1. \] [i]Reid Barton[/i] | (2, 3, 5), (2, 5, 3), (3, 2, 5), (3, 5, 2), (5, 2, 3), (5, 3, 2) |
We are tasked with finding all ordered triples of primes \((p, q, r)\) such that
\[ p \mid q^r + 1, \quad q \mid r^p + 1, \quad r \mid p^q + 1. \]
Assume \( p = \min(p, q, r) \) and \( p \neq 2 \). Note the following conditions:
\[
\begin{align*}
\text{ord}_p(q) &\mid 2r \implies \text{ord}_p(q) = 2 \text{ or } 2r, \... | 8.375 | [
8,
8,
8,
8,
9,
9,
9,
8
] |
Problem
Steve is piling $m\geq 1$ indistinguishable stones on the squares of an $n\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform stone moves, defined as follows. Consider any four grid squares, which are corners of a recta... | \[
\binom{n+m-1}{m}^{2}
\] | Let the number of stones in row $i$ be $r_i$ and let the number of stones in column $i$ be $c_i$ . Since there are $m$ stones, we must have $\sum_{i=1}^n r_i=\sum_{i=1}^n c_i=m$
Lemma 1: If any $2$ pilings are equivalent, then $r_i$ and $c_i$ are the same in both pilings $\forall i$ .
Proof: We suppose the contrary. N... | 7.125 | [
8,
7,
7,
7,
7,
7,
7,
7
] |
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