Let $m$ and $n$ be positive integers. A people planted two kind of different trees on a plot tabular grid size $ m \times n $ (each square plant one tree.) A plant called [i]inpressive[/i] if two conditions following conditions are met simultaneously: i) The number of trees in each of kind is equal; ii) In each row the number of tree of each kind is diffrenent not less than a half of number of cells on that row and In each colum the number of tree of each kind is diffrenent not less than a half of number of cells on that colum. a) Find an inpressive plant when $m=n=2016$; b) Prove that if there at least a inpressive plant then $4|m$ and $4|n$.

Find the largest and smallest value of the function $$y=\sqrt{7+5\cos x}-\cos x.$$

In a plane a set of $n\geq 3$ points is given. Each pair of points is connected by a segment. Let $d$ be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length $d$. Prove that the number of diameters of the given set is at most $n$.

Given a trihedral angle Sxyz. A plane (P) not through S cuts Sx,Sy,Sz respectively at A,B,C. On the plane (P), outside triangle ABC, construct triangles DAB,EBC,FCA which are confruent to the triangles SAB,SBC,SCA respectively. Let (T) be the sphere lying inside Sxyz, but not inside the tetrahedron SABC, toucheing the planes containing the faces of SABC. Prove that (T) touches the plane (P) at the circumcenter of triangle DEF.

For the real number $a$ find the number of solutions $(x, y, z)$ of a system of the equations: $\left\{\begin{array}{lll} x+y^2+z^2=a \\ x^2+y+z^2=a \\ x^2+y^2+z=a\end{array}\right.$

$AB$ is the diameter of circle $O$, $CD$ is a non-diameter chord that is perpendicular to $AB$. Let $E$ be the midpoint of $OC$, connect $AE$ and extend it to meet the circle at point $P$. Let $DP$ and $BC$ meet at $F$. Prove that $F$ is the midpoint of $BC$.

The arithmetic mean of the nine numbers in the set $ \{9,99,999,9999,...,999999999\}$ is a $ 9$-digit number $ M$, all of whose digits are distinct. The number $ M$ does not contain the digit $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8$

Determine all pairs of $(m, n)$ such that is possible to tile the table $ m \times n$ with figure ”corner” as in figure with condition that in that tilling does not exist rectangle (except $m \times n$) regularly covered with figures.

Find $ \lim_{a\rightarrow \plus{} \infty} \frac {\int_0^a \sin ^ 4 x\ dx}{a}$.

The points $A, B, C, D$ are selected on the circle as followed so that $AB = BC = CD$. Bisectors of $\angle ABD$ and $\angle ACD$ intersect at point $E$. Find $\angle ABC$, if it is known that $AE \parallel CD$.

Let $n\ge 1$ be an odd integer. Alice and Bob play the following game, taking alternating turns, with Alice playing first. The playing area consists of $n$ spaces, arranged in a line. Initially all spaces are empty. At each turn, a player either • places a stone in an empty space, or • removes a stone from a nonempty space $s,$ places a stone in the nearest empty space to the left of $s$ (if such a space exists), and places a stone in the nearest empty space to the right of $s$ (if such a space exists). Furthermore, a move is permitted only if the resulting position has not occurred previously in the game. A player loses if he or she is unable to move. Assuming that both players play optimally throughout the game, what moves may Alice make on her first turn?

Compute the sum of all positive integers $n$ for which there exists a real number $x$ satisfying $$\left(x +\frac{n}{x} \right)^n= 2^{20}.$$

Determine, with proof, whether a square can be dissected into finitely many (not necessarily congruent) triangles, each of which has interior angles $30^\circ, 75^\circ,$ and $75^\circ.$

For real numbers $(x, y, z)$ satisfying the following equations, find all possible values of $x+y+z$ $x^2y+y^2z+z^2x=-1$ $xy^2+yz^2+zx^2=5$ $xyz=-2$

Prove that if all four vertices of a parallelogram are lattice points and there are some other lattice points in or on the parallelogram, then its area exceeds $1$.

Determine if there exists a subset $E$ of $Z \times Z$ with the properties: (i) $E$ is closed under addition, (ii) $E$ contains $(0,0),$ (iii) For every $(a,b) \ne (0,0), E$ contains exactly one of $(a,b)$ and $-(a,b)$. Remark: We define $(a,b)+(a',b') = (a+a',b+b')$ and $-(a,b) = (-a,-b)$.

Marisa has two identical cubical dice labeled with the numbers $\{1, 2, 3, 4, 5, 6\}$. However, the two dice are not fair, meaning that they can land on each face with different probability. Marisa rolls the two dice and calculates their sum. Given that the sum is $2$ with probability $0.04$, and $12$ with probability $0.01$, the maximum possible probability of the sum being $7$ is $p$. Compute $\lfloor 100p \rfloor$.

Bisector of one angle of triangle $ABC$ is equal to the bisector of its external angle at the same vertex (see figure). Find the difference between the other two angles of the triangle. [img]https://cdn.artofproblemsolving.com/attachments/c/3/d2efeb65544c45a15acccab8db05c8314eb5f2.png[/img]

A word of length $n$ that consists only of the digits $0$ and $1$, is called a bit-string of length $n$. (For example, $000$ and $01101$ are bit-strings of length 3 and 5.) Consider the sequence $s(1), s(2), ...$ of bit-strings of length $n > 1$ which is obtained as follows : (1) $s(1)$ is the bit-string $00...01$, consisting of $n - 1$ zeros and a $1$ ; (2) $s(k+1)$ is obtained as follows : (a) Remove the digit on the left of $s(k)$. This gives a bit-string $t$ of length $n - 1$. (b) Examine whether the bit-string $t1$ (length $n$, adding a $1$ after $t$) is already in $\{s(1), s(2), ..., s(k)\}$. If this is the not case, then $s(k+1) = t1$. If this is the case then $s(k+1) = t0$. For example, if $n = 3$ we get : $s(1) = 001 \rightarrow s(2) = 011 \rightarrow s(3) = 111 \rightarrow s(4) = 110 \rightarrow s(5) = 101$ $\rightarrow s(6) = 010 \rightarrow s(7) = 100 \rightarrow s(8) = 000 \rightarrow s(9) = 001 \rightarrow ...$ Suppose $N = 2^n$. Prove that the bit-strings $s(1), s(2), ..., s(N)$ of length $n$ are all different.

For what value(s) of $k$ does the pair of equations $y=x^2$ and $y=3x+k$ have two identical solutions? $\textbf{(A)}\ \dfrac{4}{9} \qquad \textbf{(B)}\ -\dfrac{4}{9} \qquad \textbf{(C)}\ \dfrac{9}{4} \qquad \textbf{(D)}\ -\dfrac{9}{4} \qquad \textbf{(E)}\ \pm\dfrac{9}{4}$

We call a matrix $\textsl{binary matrix}$ if all its entries equal to $0$ or $1$. A binary matrix is $\textsl{Good}$ if it simultaneously satisfies the following two conditions: (1) All the entries above the main diagonal (from left to right), not including the main diagonal, are equal. (2) All the entries below the main diagonal (from left to right), not including the main diagonal, are equal. Given positive integer $m$, prove that there exists a positive integer $M$, such that for any positive integer $n>M$ and a given $n \times n$ binary matrix $A_n$, we can select integers $1 \leq i_1 <i_2< \cdots < i_{n-m} \leq n$ and delete the $i_i$-th, $i_2$-th,$\cdots$, $i_{n-m}$-th rows and $i_i$-th, $i_2$-th,$\cdots$, $i_{n-m}$-th columns of $A_n$, then the resulting binary matrix $B_m$ is $\textsl{Good}$.

Let $A$ be a countable set, some of its countable subsets are selected such that; the intersection of any two selected subsets has at most one element. Find the smallest $k$ for which one can ensure that we can color elements of $A$ with $k$ colors such that each selected subsets exactly contain one element of one of the colors and an infinite number of elements of each of the other colors.

How many pairs $(a, b)$ for integers $a, b \ge 2$ which exist the sequence $x_1, x_2, . . . , x_{1000}$ which satisfy conditions as below? 1.Terms $x_1, x_2, . . . , x_{1000}$ are sorting of $1, 2, . . . , 1000$. 2.For each integers $1 \le i < 1000$, the sequence forms $x_{i+1} = x_i + a$ or $x_{i+1} = x_i - b$.

Let $P(x)$ be a quadratic polynomial with real coefficients. Suppose that $P(1) = 20, P(-1) = 22,$ and $P(P(0)) = 400.$ Compute the largest possible value of $P(10).$

Let $n$ be a positive integer, and let $A$ be a subset of $\{ 1,\cdots ,n\}$. An $A$-partition of $n$ into $k$ parts is a representation of n as a sum $n = a_1 + \cdots + a_k$, where the parts $a_1 , \cdots , a_k $ belong to $A$ and are not necessarily distinct. The number of different parts in such a partition is the number of (distinct) elements in the set $\{ a_1 , a_2 , \cdots , a_k \} $. We say that an $A$-partition of $n$ into $k$ parts is optimal if there is no $A$-partition of $n$ into $r$ parts with $r<k$. Prove that any optimal $A$-partition of $n$ contains at most $\sqrt[3]{6n}$ different parts.