Found problems: 85335
Given a natural number $ n\ge 2 $ and an $ n\times n $ matrix with integer entries, consider the multiplicative monoid
$$ M=\{ M_k=I+kA| k\in \mathbb{Z} \} . $$
[b]a)[/b] Prove that $ M $ is a commutative group if the [url=https://en.wikipedia.org/wiki/Nilpotent_matrix]index[/url] of $ A $ is $ 2. $
[b]b)[/b] Prove that all elements of $ M $ are units if $ M_1,M_2,\ldots M_{2n} $ are all units.
Given is a chessboard 8x8. We have to place $n$ black queens and $n$ white queens, so that no two queens attack. Find the maximal possible $n$.
(Two queens attack each other when they have different colors. The queens of the same color don't attack each other)
A table $2 \times 2010$ is divided to unit cells. Ivan and Peter are playing the following game. Ivan starts, and puts horizontal $2 \times 1$ domino that covers exactly two unit table cells. Then Peter puts vertical $1 \times 2$ domino that covers exactly two unit table cells. Then Ivan puts horizontal domino. Then Peter puts vertical domino, etc. The person who cannot put his domino will lose the game. Find who have winning strategy.
Solve the equation:
\[2^{\lg x}+8=(x-8)^{\frac{1}{\lg 2}}\]
Note: $\lg x=\log_{10}x$.
[i]Daniel Jinga [/i]
On a $ 10 \times 10$ chessboard, some $ 4n$ unit squares are chosen to form a region $ \mathcal{R}.$ This region $ \mathcal{R}$ can be tiled by $ n$ $ 2 \times 2$ squares. This region $ \mathcal{R}$ can also be tiled by a combination of $ n$ pieces of the following types of shapes ([i]see below[/i], with rotations allowed).
Determine the value of $ n.$
Let $p$ be a prime number. Prove that from a $p^2\times p^2$ array of squares, we can select $p^3$ of the squares such that the centers of any four of the selected squares are not the vertices of a rectangle with sides parallel to the edges of the array.
$p$ is a prime. Find the largest integer $d$ such that $p^d$ divides $p^4!$.
Given an equation $$x^3 + ?x^2 + ?x + ? = 0$$ First player substitutes an integer on the place of one of the interrogative marks, than the same do the second with one of the two remained marks, and, finally, the first puts the integer instead of the last mark. Explain how can the first provide the existence of three integer roots in the obtained equation. (The roots may coincide.)
Find all ordered pairs of positive integers $(m,n)$ for which there exists a set $C=\{c_1,\ldots,c_k\}$ ($k\ge1$) of colors and an assignment of colors to each of the $mn$ unit squares of a $m\times n$ grid such that for every color $c_i\in C$ and unit square $S$ of color $c_i$, exactly two direct (non-diagonal) neighbors of $S$ have color $c_i$.
[i]David Yang.[/i]
Real numbers $x_1, x_2, x_3, x_4$ are roots of the fourth degree polynomial $W (x)$ with integer coefficients.
Prove that if $x_3 + x_4$ is a rational number and $x_3x_4$ is a irrational number, then $x_1 + x_2 = x_3 + x_4$.
The number $\overline{13\ldots 3}$, with $k>1$ digits $3$, is a prime. Prove that $6\mid k^{2}-2k+3$.
How many positive integers $n$ satisfy$$\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?$$(Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.)
$\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32$
If $a$ and $b$ are digits, how many are there $4$ digit numbers $\overline{3ab4}$ divisible with $9$ . Which numbers are they ($4$ digit numbers)?
Suppose $a$, $b$, $c$, and $d$ are distinct positive integers such that $a^b$ divides $b^c$, $b^c$ divides $c^d$, and $c^d$ divides $d^a$.
[list](a) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the smallest?
(b) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the largest?[/list]
Let $\mathbb{Z}$ and $\mathbb{Q}$ be the sets of integers and rationals respectively.
a) Does there exist a partition of $\mathbb{Z}$ into three non-empty subsets $A,B,C$ such that the sets $A+B, B+C, C+A$ are disjoint?
b) Does there exist a partition of $\mathbb{Q}$ into three non-empty subsets $A,B,C$ such that the sets $A+B, B+C, C+A$ are disjoint?
Here $X+Y$ denotes the set $\{ x+y : x \in X, y \in Y \}$, for $X,Y \subseteq \mathbb{Z}$ and for $X,Y \subseteq \mathbb{Q}$.
An ant crawls along the edges of a cube turning only at its vertices. It has visited one of the vertices $25$ times. Is it possible that it has visited each of the other $7$ vertices exactly $20$ times?
(S Tokarev)
If $8/19$ of the product of largest two elements of a positive integer set is not greater than the sum of other elements, what is the minimum possible value of the largest number in the set?
$
\textbf{(A)}\ 8
\qquad\textbf{(B)}\ 12
\qquad\textbf{(C)}\ 13
\qquad\textbf{(D)}\ 19
\qquad\textbf{(E)}\ 20
$
A 3 × 3 grid of blocks is labeled from 1 through 9. Cindy paints each block orange or
lime with equal probability and gives the grid to her friend Sophia.
Sophia then plays with the grid of blocks. She can take the top row of blocks and move
it to the bottom, as shown.
1 2 3
4 5 6
7 8 9
4 5 6
7 8 9
1 2 3
Grid A Grid A0
She can also take the leftmost column of blocks and move it to the right end, as shown.
1 2 3
4 5 6
7 8 9
2 3 1
5 6 4
8 9 7
Grid B Grid B0
Sophia calls the grid of blocks citrus if it is impossible for her to use a sequence of the
moves described above to obtain another grid with the same coloring but a different numbering
scheme. For example, Grid B is citrus, but Grid A is not citrus because moving the
top row of blocks to the bottom results in a grid with a different numbering but the same
coloring as Grid A.
What is the probability that Sophia receives a citrus grid of blocks?
In the set of real numbers solve the system of equations
$x^4+y^2+4=5yz$
$y^4+z^2+4=5zx$
$z^4+x^2+4=5xy$
A circle of radius $5$ is inscribed in a rectangle as shown. The ratio of the the length of the rectangle to its width is $2\ :\ 1$. What is the area of the rectangle?
[asy]
draw((0,0)--(0,10)--(20,10)--(20,0)--cycle);
draw(circle((10,5),5));
[/asy]
$ \textbf{(A)}\ 50\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 125\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 200 $
Let $ Q $ be a point, $ H,O $ be the orthocenter and circumcenter, respectively, of a triangle $ ABC, $ and $ D,E,F, $ be the symmetric points of $ Q $ with respect to $ A,B,C, $ respectively. Also, $ M,N,P $ are the middle of the segments $ AE,BF,CD, $ and $ G,G',G'' $ are the centroids of $ ABC,MNP,DEF, $ respectively. Prove the following propositions:
[b]a)[/b] $ \frac{1}{2}\overrightarrow{OG} =\frac{1}{3}\overrightarrow{OG'}=\frac{1}{4}\overrightarrow{OG''} $
[b]b)[/b] $ Q=O\implies \overrightarrow{OG'} =\overrightarrow{G'H} $
[b]c)[/b] $ Q=H\implies G'=O $
[i]Cătălin Zîrnă[/i]
Let \( p \geq 7 \) be a prime number and \( m \in \mathbb{N} \). Prove that
\[\left| p^m - (p - 2)! \right| > p^2.\]
[i]Proposed by Miloš Milićev[/i]
Let $ n $ $(n\geq2)$ be an integer. Find the greatest possible value of the expression $$E=\frac{a_1}{1+a_1^2}+\frac{a_2}{1+a_2^2}+\ldots+\frac{a_n}{1+a_n^2}$$ if the positive real numbers $a_1,a_2,\ldots,a_n$ satisfy $a_1+a_2+\ldots+a_n=\frac{n}{2}.$ What are the values of $a_1,a_2,\ldots,a_n$ when the greatest value is achieved?
[u]Round 9[/u]
[b]p25. [/b]Define a sequence $\{a_n\}_{n \ge 1}$ of positive real numbers by $a_1 = 2$ and $a^2_n -2a_n +5 =4a_{n-1}$ for $n \ge 2$. Suppose $k$ is a positive real number such that $a_n <k$ for all positive integers $n$. Find the minimum possible value of $k$.
[b]p26.[/b] Let $\vartriangle ABC$ be a triangle with $AB = 13$, $BC = 14$, and $C A = 15$. Suppose the incenter of $\vartriangle ABC$ is $I$ and the incircle is tangent to $BC$ and $AB$ at $D$ and $E$, respectively. Line $\ell$ passes through the midpoints of $BD$ and $BE$ and point $X$ is on $\ell$ such that $AX \parallel BC$. Find $X I$ .
[b]p27.[/b] Let $x, y, z$ be positive real numbers such that $x y + yz +zx = 20$ and $x^2yz +x y^2z +x yz^2 = 100$. Additionally, let $s = \max (x y, yz,xz)$ and $m = \min(x, y, z)$. If $s$ is maximal, find $m$.
[u]Round 10[/u]
[b]p28.[/b] Let $\omega_1$ be a circle with center $O$ and radius $1$ that is internally tangent to a circle $\omega_2$ with radius $2$ at $T$ . Let $R$ be a point on $\omega_1$ and let $N$ be the projection of $R$ onto line $TO$. Suppose that $O$ lies on segment $NT$ and $\frac{RN}{NO} = \frac4 3$ . Additionally, let $S$ be a point on $\omega_2$ such that $T,R,S$ are collinear. Tangents are drawn from $S$ to $\omega_1$ and touch $\omega_1$ at $P$ and $Q$. The tangent to $\omega_1$ at $R$ intersects $PQ$ at $Z$. Find the area of triangle $\vartriangle ZRS$.
[b]p29.[/b] Let $m$ and $n$ be positive integers such that $k =\frac{ m^2+n^2}{mn-1}$ is also a positive integer. Find the sum of all possible values of $k$.
[b]p30.[/b] Let $f_k (x) = k \cdot \ min (x,1-x)$. Find the maximum value of $k \le 2$ for which the equation $f_k ( f_k ( f_k (x))) = x$ has fewer than $8$ solutions for $x$ with $0 \le x \le 1$.
[u]Round 11[/u]
In the following problems, $A$ is the answer to Problem $31$, $B$ is the answer to Problem $32$, and $C$ is the answer to Problem $33$. For this set, you should find the values of $A$,$B$, and $C$ and submit them as answers to problems $31$, $32$, and $33$, respectively. Although these answers depend on each other, each problem will be scored separately.
[b]p31.[/b] Find $$A \cdot B \cdot C + \dfrac{1}{B+ \dfrac{1}{C +\dfrac{1}{B+\dfrac{1}{...}}}}$$
[b]p32.[/b] Let $D = 7 \cdot B \cdot C$. An ant begins at the bottom of a unit circle. Every turn, the ant moves a distance of $r$ units clockwise along the circle, where $r$ is picked uniformly at random from the interval $\left[ \frac{\pi}{2D} , \frac{\pi}{D} \right]$. Then, the entire unit circle is rotated $\frac{\pi}{4}$ radians counterclockwise. The ant wins the game if it doesn’t get crushed between the circle and the $x$-axis for the first two turns. Find the probability that the ant wins the game.
[b]p33.[/b] Let $m$ and $n$ be the two-digit numbers consisting of the products of the digits and the sum of the digits of the integer $2016 \cdot B$, respectively. Find $\frac{n^2}{m^2 - mn}$.
[u]Round 12[/u]
[b]p34.[/b] There are five regular platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. For each of these solids, define its adjacency angle to be the dihedral angle formed between two adjacent faces. Estimate the sum of the adjacency angles of all five solids, in degrees. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, \lfloor 15 -\frac12 |A-E| \rfloor \right).$
[b]p35.[/b] Estimate the value of $$\log_{10} \left(\prod_{k|2016} k!\right), $$ where the product is taken over all positive divisors $k$ of $2016$. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, \lceil 15 \cdot \min \left(\frac{E}{A}, 2- \frac{E}{A}\right) \rceil \right).$
[b]p36.[/b] Estimate the value of $\sqrt{2016}^{\sqrt[4]{2016}}$. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, \lceil 15 \cdot \min \left(\frac{\ln E}{\ln A}, 2- \frac{\ln E}{\ln A}\right) \rceil \right).$
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3158461p28714996]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3158474p28715078]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Prove that the sum of the squares of two consecutive positive integers cannot be equal to a sum of the fourth powers of two consecutive positive integers.