Given a poistive integer $ m, $ determine the smallest integer $ n\ge 2 $ such that for any coloring of the $ n^2 $ unit squares of a $ n\times n $ square with $ m $ colors, there are, at least, two unit squares $ (i,j),(k,l) $ that share the same color, where $ 1\le i,j,k,l\le n,i\neq j,k\neq l. $ [i]American Mathematical Monthly[/i]

Let $ABC$ be a triangle inscribed in the circle $K$ and consider a point $M$ on the arc $BC$ that do not contain $A$. The tangents from $M$ to the incircle of $ABC$ intersect the circle $K$ at the points $N$ and $P$. Prove that if $\angle BAC = \angle NMP$, then triangles $ABC$ and $MNP$ are congruent. Valentin Vornicu [hide= about Romania JBMO TST 2004 in aops]I found the Romania JBMO TST 2004 links [url=https://artofproblemsolving.com/community/c6h5462p17656]here [/url] but they were inactive. So I am asking for solution for the only geo I couldn't find using search. The problems were found [url=https://artofproblemsolving.com/community/c6h5135p16284]here[/url].[/hide]

There is a $19\times19$ board. Is it possible to mark some $1\times 1$ squares so that each of $10\times 10$ squares contain different number of marked squares?

As evidence that the correct answer does not mean the correctness of the proof, the teacher cited next example. Let's take the fraction $\frac{19}{95}$. After crossing out $9$ in the numerator and denominator (“reduction” by $9$), we get $\frac{1}{5}$ which is the correct answer. In the same way, a fraction $\frac{1999}{9995}$ can be “reduced” by three nines (cross out $999$ in the numerator and denominator). Is it possible that as a result of such a “reduction” we also get the correct answer, equal to $\frac13$ ? (We consider fractions of the form $\frac{1a}{a3}$. Here, with the letter $a$ we denote several numbers that follow in the same order in the numerator after $1$, and in the denominator before $3$. “Reduce” by $a$.)

Let $a,\ b>0$ be real numbers, $n\geq 2$ be integers. Evaluate $I_n=\int_{-\infty}^{\infty} \frac{exp(ia(x-ib))}{(x-ib)^n}dx.$

Let $ f: \mathbb R \rightarrow \mathbb R$ be a function such that $ f(tx_1\plus{}(1\minus{}t)x_2)\leq tf(x_1)\plus{}(1\minus{}t)f(x_2)$ for all $ x_1 , x_2 \in \mathbb R$ and $ t\in (0,1)$. Show that $ \sum_{k\equal{}1}^{2003}f(a_{k\plus{}1})a_k \geq \sum_{k\equal{}1}^{2003}f(a_k)a_{k\plus{}1}$ for all real numbers $ a_1,a_2,...,a_{2004}$ such that $ a_1\geq a_2\geq ... \geq a_{2003}$ and $ a_{2004}\equal{}a_1$

It is known that the graph of a quadratic trinomial $y = x^2 + px + q$ touches the graph of a straight line $y = 2x + p$. Prove that all such quadratic trinomials have the same minimum value. Find this smallest value.

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

Let $ABC$ be an acute triangle with $AB \neq AC$ . Also let $M$ be the midpoint of the side $BC$ , $H$ the orthocenter of the triangle $ABC$ , $O_1$ the midpoint of the segment $AH$ and $O_2$ the center of the circumscribed circle of the triangle $BCH$ . Prove that $O_1AMO_2$ is a parallelogram .

Let $m$ be a positive integer, and let $a_0, a_1,\ldots,a_m$ be a sequence of reals such that $a_0=37$, $a_1=72$, $a_m=0$, and \[a_{k+1}=a_{k-1}-\frac{3}{a_k}\] for $k=1,2, \dots, m-1$. Find $m$.

Given a triangle $ABC$ in which the angle $B$ is equal to $60^\circ$. A circle inscribed in a triangle with a center $I$ touches the side $AC$ at point $K$. A line passing through the points of touching of this circle with the other sides of the triangle intersects the its circumcircle at points $M$ and $N$. Prove that the ray $KI$ divides the arc $MN$ in half.

Show that there exists the maximum value of the function $f(x,\ y)=(3xy+1)e^{-(x^2+y^2)}$ on $\mathbb{R}^2$, then find the value.

The $ 8\times 18$ rectangle $ ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $ y$? [asy] unitsize(2mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=4; draw((0,4)--(18,4)--(18,-4)--(0,-4)--cycle); draw((6,4)--(6,0)--(12,0)--(12,-4)); label("$D$",(0,4),NW); label("$C$",(18,4),NE); label("$B$",(18,-4),SE); label("$A$",(0,-4),SW); label("$y$",(9,1)); [/asy]$ \textbf{(A) } 6\qquad \textbf{(B) } 7\qquad \textbf{(C) } 8\qquad \textbf{(D) } 9\qquad \textbf{(E) } 10$

Determine all positive integers $d,$ such that there exists an integer $k\geq 3,$ such that One can arrange the numbers $d,2d,\ldots,kd$ in a row, such that the sum of every two consecutive of them is a perfect square.

Let $AB$ be a segment of length $2$. The locus of points $P$ such that the $P$-median of triangle $ABP$ and its reflection over the $P$-angle bisector of triangle $ABP$ are perpendicular determines some region $R$. Find the area of $R$.

Let $ \omega $ be the incircle of the triangle $ABC$ and with centre $I$. Let $\Gamma $ be the circumcircle of the triangle $AIB$. Circles $ \omega $ and $ \Gamma $ intersect at the point $X$ and $Y$. Let $Z$ be the intersection of the common tangents of the circles $\omega$ and $\Gamma$. Show that the circumcircle of the triangle $XYZ$ is tangent to the circumcircle of the triangle $ABC$.

Prove that there are infinitely many positive integers $m$, such that $n^4+m$ is not prime for any positive integer $n$.

A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are $ 6.2$ cm, $ 8.3$ cm, and $ 9.5$ cm. The area of the square is \[ \textbf{(A)}\ 24 \text{ cm}^2 \qquad \textbf{(B)}\ 36 \text{ cm}^2 \qquad \textbf{(C)}\ 48 \text{ cm}^2 \qquad \textbf{(D)}\ 64 \text{ cm}^2 \qquad \textbf{(E)}\ 144 \text{ cm}^2 \]

Let $ x,y,z$ be nonnegative real numbers such that $ x \plus{} y \plus{} z \equal{} 2$. Prove that $ x^{2}y^{2} \plus{} y^{2}z^{2} \plus{} z^{2}x^{2} \plus{} xyz\leq 1$. When does the equality occur?

Find all quadruples of positive integers $(k,a,b,c)$ such that $2^k=a!+b!+c!$ and $a\geq b\geq c$.

$n$ is a given positive integer, such that it’s possible to weigh out the mass of any product weighing $1,2,3,\cdots ,n\text{g}$ with a counter balance and $k$ counterweights, whose weights are positive integers. [b](a)[/b] Find $f(n)$: the minumum value of $k$. [b](b)[/b] Find all possible number of $n,$ such that the mass of $f(n)$ counterweights is uniquely determined.

Let $[mn]$ be a diameter of the circle $C$ and $[AB]$ a chord with given length on this circle. $[AB]$ neither coincides nor is perpendicular to $[MN]$. Let $C,D$ be the orthogonal projections of $A$ and $B$ on $[MN]$ and $P$ the midpoint of $[AB]$. Prove that $\angle CPD$ does not depend on the chord $[AB]$.

Let $n\geq 2$ be a positive integer and \begin{align*} P(x) = c_0 X^n + c_1 X^{n-1} + \ldots + c_{n-1} X +c_n \end{align*} be a polynomial with integer coefficients, such that $\mid c_n \mid$ is a prime number and \begin{align*} |c_0| + |c_1| + \ldots + |c_{n-1}| < |c_n| \end{align*} Prove that the polynomial $P(X)$ is irreducible in the $\mathbb{Z}[x]$

Let $\alpha$ and $\beta$ be nonnegative integers. Suppose the number of strictly increasing sequences of integers $a_0,a_1,\dots,a_{2014}$ satisfying $0 \leq a_m \leq 3m$ is $2^\alpha (2\beta + 1)$. Find $\alpha$. [i]Proposed by Lewis Chen[/i]

Let $ABC$ be an acute triangle. Let $D$ be a point on side $AB$ and $E$ be a point on side $AC$ such that lines $BC$ and $DE$ are parallel. Let $X$ be an interior point of $BCED$. Suppose rays $DX$ and $EX$ meet side $BC$ at points $P$ and $Q$, respectively, such that both $P$ and $Q$ lie between $B$ and $C$. Suppose that the circumcircles of triangles $BQX$ and $CPX$ intersect at a point $Y \neq X$. Prove that the points $A, X$, and $Y$ are collinear.