Found problems: 85335
Let $a_1, a_2, a_3, \ldots$ be a sequence of positive real numbers, and $s$ be a positive integer, such that
\[a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.\]
Prove there exist positive integers $\ell \leq s$ and $N$, such that
\[a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.\]
[i]Proposed by Morteza Saghafiyan, Iran[/i]
At Euclid High School, the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are $11$ students in Mrs. Germain's class, 8 in Mr. Newton, and $9$ in Mrs. Young's class are taking the AMC $8$ this year. How many mathematics students at Euclid High School are taking the contest?
$ \textbf{(A)}\ 26 \qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 29\qquad\textbf{(E)}\ 30 $
Show that there exists a real constant $C>1$ with the following property:
For any positive integer $n$, there are at least $C^n$ positive integers with exactly $n$ decimal digits, which are divisible by the product of their digits. (In particular, these $n$ digits are all non-zero.)
[i]Proposed by Jean-Marie De Koninck and Florian Luca[/i]
Find all natural numbers $N$ (in decimal system) with the following properties:
(i) $N =\overline{aabb}$, where $\overline{aab}$ and $\overline{abb}$ are primes,
(ii) $N = P_1P_2P_3$, where $P_k (k = 1,2,3)$ is a prime consisting of $k$ (decimal) digits.
Prove that for each positive integer $n$ there are at most two pairs $(a, b)$ of positive integers with following two properties:
(i) $a^2 + b = n$,
(ii) $a+b$ is a power of two, i.e. there is an integer $k \ge 0$ such that $a+b = 2^k$.
The number of points with positive rational coordinates selected from the set of points in the xy-plane such that $x+y\leq 5$, is:
$\text{(A)} \ 9 \qquad \text{(B)} \ 10 \qquad \text{(C)} \ 14 \qquad \text{(D)} \ 15 \qquad \text{(E)} \ \text{infinite}$
there are $n$ points on the plane,any two vertex are connected by an edge of red,yellow or green,and any triangle with vertex in the graph contains exactly $2$ colours.prove that $n<13$
Gonçalo writes in a board four of the the following numbers $0, 1, 2, 3, 4$, he can repeat numbers. Nicolas can realize the following operation: change one number of the board, by the remainder(in the division by $5$) of the product of others two numbers of the board. Nicolas wins if all the four numbers are equal, determine if Gonçalo can choose numbers such that Nicolas will never win.
Find all functions $f: R \to R$ that satisfy the condition $(f (x) - f (y)) (u - v) = (f (u) - f (v)) (x -y)$ for arbitrary real $x, y, u, v$ such that $x + y = u + v$.
Let $ a_1,a_2,...,a_6$ be real numbers such that:
$ a_1 \not \equal{} 0, a_1a_6 \plus{} a_3 \plus{} a_4 \equal{} 2a_2a_5 \ \mathrm{and}\ a_1a_3 \ge a_2^2$
Prove that $ a_4a_6\le a_5^2$. When does equality holds?
Consider the set $\mathbb Q^2$ of points in $\mathbb R^2$, both of whose coordinates are rational.
[b](a)[/b] Prove that the union of segments with vertices from $\mathbb Q^2$ is the entire set $\mathbb R^2$.
[b](b)[/b] Is the convex hull of $\mathbb Q^2$ (i.e., the smallest convex set in $\mathbb R^2$ that contains $\mathbb Q^2$) equal to $\mathbb R^2$ ?
For a positive integer number $n$ we denote $d(n)$ as the greatest common divisor of the binomial coefficients $\dbinom{n+1}{n} , \dbinom{n+2}{n} ,..., \dbinom{2n}{n}$.
Find all possible values of $d(n)$
Let x be a real number such that $(\sqrt{6})^x -3^x = 2^{x-2}$. Evaluate $\frac{4^{x+1}}{9^{x-1}}$ .
Each vertex of a cube is to be labeled with an integer $1$ through $8$, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?
$\textbf{(A) } 1\qquad\textbf{(B) } 3\qquad\textbf{(C) }6 \qquad\textbf{(D) }12 \qquad\textbf{(E) }24$
Prove that for every positive integer $n$ there is an $n$-digit number $z$ with none of its digits $0$ and such that $z$ is divisible by its sum of digits.
Let $m \ge n$ be positive integers. Frieder is given $mn$ posters of Linus with different integer dimensions of $k \times l$ with $1 \ge k \ge m$ and $1 \ge l \ge n$. He must put them all up one by one on his bedroom wall without rotating them. Every time he puts up a poster, he can either put it on an empty spot on the wall or on a spot where it entirely covers a single visible poster and does not overlap any other visible poster. Determine the minimal area of the wall that will be covered by posters.
Fill in each cell of the grid with a positive digit so that the following conditions hold:
1. each row and column contains ve distinct digits;
2. for any cage containing multiple cells of a row, the label on the cage is the GCD of the sum of the digits in the cage and the sum of the digits in the whole row, and
3. for any cage containing multiple cells of a column, the label on the cage is the GCD of the sum of the digits in the cage and the sum of the digits in the whole column.
You do not need to prove that your answer is the only one possible; you merely need to fi nd an answer that satis fies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justi cation acceptable.)
[asy]
unitsize(48);
int[][] a = {
{0, 0, 0, 0, 0},
{0, 0, 0, 0, 0},
{0, 0, 0, 0, 0},
{0, 0, 0, 0, 0},
{0, 0, 0, 0, 0}};
for (int i = 0; i < 5; ++i) {
for (int j = 0; j < 5; ++j) {
draw((i, -j)--(i+1, -j)--(i+1, -j-1)--(i, -j-1)--cycle);
if (a[j][i] > 0 && a[j][i] < 999) label(string(a[j][i]), (i+0.5, -j-0.5), fontsize(24pt));
}
}
real ep=0.1;
real s=3;
pen lw=linewidth(.12mm);
real x=0.9;
real y=1.2;
draw((0+s*ep,0-ep)--(2-ep,0-ep)--(2-ep,-1+ep)--(0+ep,-1+ep)--(0+ep,0-s*ep),dashed+lw);
draw((2+s*ep,0-ep)--(4-ep,0-ep)--(4-ep,-1+ep)--(2+ep,-1+ep)--(2+ep,0-s*ep),dashed+lw);
draw((1+s*ep,-1-ep)--(3-ep,-1-ep)--(3-ep,-2+ep)--(1+ep,-2+ep)--(1+ep,-1-s*ep),dashed+lw);
draw((2+s*ep,-3-ep)--(4-ep,-3-ep)--(4-ep,-4+ep)--(2+ep,-4+ep)--(2+ep,-3-s*ep),dashed+lw);
draw((1+s*ep,-4-ep)--(3-ep,-4-ep)--(3-ep,-5+ep)--(1+ep,-5+ep)--(1+ep,-4-s*ep),dashed+lw);
draw((3+s*ep,-4-ep)--(5-ep,-4-ep)--(5-ep,-5+ep)--(3+ep,-5+ep)--(3+ep,-4-s*ep),dashed+lw);
label(scale(x)*"5", (0+ep,0-y*ep));
label(scale(x)*"7", (2+ep,0-y*ep));
label(scale(x)*"10", (1+ep,-1-y*ep));
label(scale(x)*"5", (2+ep,-3-y*ep));
label(scale(x)*"2", (1+ep, -4-y*ep));
label(scale(x)*"13", (3+ep, -4-y*ep));
draw((4+s*ep,0-ep)--(5-ep,0-ep)--(5-ep,-2+ep)--(4+ep,-2+ep)--(4+ep,0-s*ep),dashed+lw);
draw((0+s*ep,-1-ep)--(1-ep,-1-ep)--(1-ep,-3+ep)--(0+ep,-3+ep)--(0+ep,-1-s*ep),dashed+lw);
draw((3+s*ep,-1-ep)--(4-ep,-1-ep)--(4-ep,-3+ep)--(3+ep,-3+ep)--(3+ep,-1-s*ep),dashed+lw);
draw((0+s*ep,-3-ep)--(1-ep,-3-ep)--(1-ep,-5+ep)--(0+ep,-5+ep)--(0+ep,-3-s*ep),dashed+lw);
draw((1+s*ep,-2-ep)--(2-ep,-2-ep)--(2-ep,-4+ep)--(1+ep,-4+ep)--(1+ep,-2-s*ep),dashed+lw);
draw((4+s*ep,-2-ep)--(5-ep,-2-ep)--(5-ep,-4+ep)--(4+ep,-4+ep)--(4+ep,-2-s*ep),dashed+lw);
label(scale(x)*"10", (4+ep,0-y*ep));
label(scale(x)*"3", (0+ep,-1-y*ep));
label(scale(x)*"8", (3+ep,-1-y*ep));
label(scale(x)*"16", (1+ep,-2-y*ep));
label(scale(x)*"6", (4+ep,-2-y*ep));
label(scale(x)*"11", (0+ep,-3-y*ep));
[/asy]
Let $A$ be a finite set of non-negative integers. Determine all functions $f:\mathbb{Z}_{\ge 0} \to A$ such that \[f(|x-y|)=|f(x)-f(y)|\] for each $x,y\in\mathbb Z_{\ge 0}$.
[i]Andrei Bâra[/i]
Let $x_1, \ldots , x_{100}$ be nonnegative real numbers such that $x_i + x_{i+1} + x_{i+2} \leq 1$ for all $i = 1, \ldots , 100$ (we put $x_{101 } = x_1, x_{102} = x_2).$ Find the maximal possible value of the sum $S = \sum^{100}_{i=1} x_i x_{i+2}.$
[i]Proposed by Sergei Berlov, Ilya Bogdanov, Russia[/i]
On competition which has $16$ teams, it is played $55$ games. Prove that among them exists $3$ teams such that they have not played any matches between themselves.
Define $M_n = \{ 1, 2, \ldots , n \} $ for all positive integers $n$. A collection of $3$-element subsets
of $M_n$ is said to be fine if for any colouring of elements of $M_n$ in two colours there is a subset of the
collection all three elements of which are of the same colour. For each $n \geq 5$ find the minimal
possible number of the $3$-element subsets of a fine collection
In the parallelogram $CMNP$ extend the bisectors of angles $MCN$ and $PCN$ and intersect with extensions of sides PN and $MN$ at points $A$ and $B$, respectively. Prove that the bisector of the original angle $C$ of the the parallelogram is perpendicular to $AB$.
[img]https://cdn.artofproblemsolving.com/attachments/f/3/fde8ef133758e06b1faf8bdd815056173f9233.png[/img]
Two turtles, the leader and the slave, are crawling along the plane from point $A$ to point $B$. They crawl in turn: first the leader crawls some distance, then the slave crawls some distance in a straight line towards the leading one. Then the leader crawls somewhere again, after which the slave crawls towards the leader, etc. Finally, they both crawl to $B$. Prove that the slave turtle crawled no more than the leading one.
Circles $\Gamma_1$ and $\Gamma_2$ meet at points $X$ and $Y$. A circle $S_1$ touches internally $\Gamma_1$ at $A$ and $\Gamma_2$ externally at $B$. A circle $S_2$ touches $\Gamma_2$ internally at $C$ and $\Gamma_1$ externally at $D$. Prove that the points $A, B, C, D$ are either collinear or concyclic.
(A. Voidelevich)
Al[u][b]l[/b][/u] edges and all diagonals of regular hexagon $A_1A_2A_3A_4A_5A_6$ are colored blue or red such that each triangle $A_jA_kA_m, 1 \leq j < k < m\leq 6$ has at least one red edge. Let $R_k$ be the number of red segments $A_kA_j, (j \neq k)$. Prove the inequality
\[\sum_{k=1}^6 (2R_k-7)^2 \leq 54.\]