Find the smallest positive integer $ u$ such that there exists only one positive integer $ a$ and satisfies the inequality \[ 20u < 19a < 21u \ \text{?} \]

$\boxed{\text{A1}}$Let $a,b,c$ be positive reals numbers such that $a+b+c=1$.Prove that $2(a^2+b^2+c^2)\ge \frac{1}{9}+15abc$

A square of side length $5$ is inscribed in a square of side length $7$. If we construct a grid of $1\times1$ squares for both squares, as shown to the right, then we find that the two grids have $8$ lattice points in common. If we do the same construction by inscribing a square of side length $1489$ in a square of side length $2009$, and construct a grid of $1\times1$ squares in each large square, then how many lattice points will the two grids of $1\times1$ squares have in common? [asy] import graph; size(6cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.3,xmax=11.88,ymin=-4.69,ymax=8.77; pair H_2=(0,3), I_2=(3,7), J_2=(7,4), K_2=(4,0), L_2=(3.01,1.99), M_2=(2.01,4), N_2=(4.01,5.01), O_2=(5.01,3); draw((0,0)--(0,7)); draw((0,7)--(7,7)); draw((7,7)--(7,0)); draw((7,0)--(0,0)); draw((0,6)--(7,6)); draw((0,5)--(7,5)); draw(J_2--(0,4)); draw(H_2--(7,3)); draw((0,2)--(7,2)); draw((0,1)--(7,1)); draw((1,0)--(1,7)); draw((2,7)--(2,0)); draw((3,0)--I_2); draw(K_2--(4,7)); draw((5,0)--(5,7)); draw((6,7)--(6,0)); draw(H_2--I_2); draw(I_2--J_2); draw(J_2--K_2); draw(K_2--H_2); draw(H_2--I_2); draw(I_2--J_2); draw((2.41,6.21)--(6.4,3.2)); draw((5.8,2.4)--(1.81,5.41)); draw((1.2,4.61)--(5.2,1.6)); draw((4.6,0.8)--(0.6,3.8)); draw((3.8,6.4)--(0.8,2.4)); draw((1.61,1.79)--(4.6,5.8)); draw((5.4,5.2)--(2.41,1.19)); draw((3.21,0.59)--(6.2,4.6)); draw((0,7)--(7,7),linewidth(1.2)); draw((7,7)--(7,0),linewidth(1.2)); draw((0,0)--(7,0),linewidth(1.2)); draw((0,7)--(0,0),linewidth(1.2)); dot(H_2,linewidth(4pt)+ds); dot(I_2,linewidth(4pt)+ds); dot(J_2,linewidth(4pt)+ds); dot(K_2,linewidth(4pt)+ds); dot(L_2,linewidth(4pt)+ds); dot(M_2,linewidth(4pt)+ds); dot(N_2,linewidth(4pt)+ds); dot(O_2,linewidth(4pt)+ds); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy]

Choose a point $A$ inside a circle of radius $R$. Construct a pair of perpendicular lines through $A$. Then rotate these lines through the same angle $V$ about $A$. The figure formed inside the circle, as the lines move from their initial to their final position, is in the form of a cross with its centre at $A$. Find the area of this cross. (Problem from Latvia)

The set of solutions of the equation $\log_{10}\left( a^2-15a\right)=2$ consists of $ \textbf{(A)}\ \text{two integers } \qquad\textbf{(B)}\ \text{one integer and one fraction}\qquad$ $\textbf{(C)}\ \text{two irrational numbers }\qquad\textbf{(D)}\ \text{two non-real numbers} \qquad\textbf{(E)}\ \text{no numbers, that is, the empty set} $

Let $ABC$ be a triangle and $D$ be the foot of the altitude from $C$ to $AB$. If $|CH|=|HD|$ where $H$ is the orthocenter, what is $\tan \widehat {A} \cdot \tan \widehat{B}$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ \sqrt 2 \qquad\textbf{(C)}\ 3/2 \qquad\textbf{(D)}\ \sqrt 3 \qquad\textbf{(E)}\ \text{None of the preceding} $

Find the maximum value of: $E(a,b)=\frac{a+b}{(4a^2+3)(4b^2+3)}$ For $a,b$ real numbers.

Fill in the grid with the numbers 1 to 6 so that each number appears exactly once in each row and column. A horizontal gray line marks any cell when it is the middle cell of the three consecutive cells with the largest sum in that row. Similarly, a vertical gray line marks any cell when it is the middle of the three consecutive cells with the largest sum in that column. If there is a tie, multiple lines are drawn in the row or column. A cell can have both lines drawn, with the appearance of a plus sign. [asy] // Change this to see the solution bool DRAW_SOLUTION = true; int n = 6; real LINE_WIDTH = 0.3; void drawHLine(int x, int y) { fill((x,y+0.5-LINE_WIDTH/2)--(x,y+0.5+LINE_WIDTH/2)--(x+1,y+0.5+LINE_WIDTH/2)--(x+1,y+0.5-LINE_WIDTH/2)--cycle, gray(0.8)); } void drawVLine(int x, int y) { fill((x+0.5-LINE_WIDTH/2,y)--(x+0.5+LINE_WIDTH/2,y)--(x+0.5+LINE_WIDTH/2,y+1)--(x+0.5-LINE_WIDTH/2,y+1)--cycle, gray(0.8)); } void drawNum(int x, int y, int num) { label(scale(1.5)*string(num), (x+0.5,y+0.5)); } void drawSolNum(int x, int y, int num) { if (DRAW_SOLUTION) { drawNum(x, y, num); } } drawHLine(2,0); drawHLine(4,1); drawHLine(1,2); drawHLine(3,2); drawHLine(4,3); drawHLine(2,4); drawHLine(3,5); drawVLine(0,4); drawVLine(1,3); drawVLine(2,1); drawVLine(2,3); drawVLine(3,4); drawVLine(4,1); drawVLine(5,2); drawNum(0, 0, 5); drawNum(4, 0, 3); drawNum(1, 2, 2); drawNum(3, 3, 4); for(int i = 0; i <= 6; i += 1) { draw((i,0)--(i,6)); draw((0,i)--(6,i)); } [/asy]

Let the set $A=${$ 1,2,3, \dots ,48n+24$ } , where $ n \in \mathbb {N^*}$ . Prove that there exist a subset $B $ of $A $ with $24n+12$ elements with the property : the sum of the squares of the elements of the set $B $ is equal to the sum of the squares of the elements of the set $A$ \ $B $ .

To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x,y,z$ respectively, and $y<0$, then the following operation is allowed: $x,y,z$ are replaced by $x+y,-y,z+y$ respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.

The number $2019$ has the following nice properties: (a) It is the sum of the fourth powers of fuve distinct positive integers. (b) It is the sum of six consecutive positive integers. In fact, $2019 = 1^4 + 2^4 + 3^4 + 5^4 + 6^4$ (1) $2019 = 334 + 335 + 336 + 337 + 338 + 339$ (2) Prove that $2019$ is the smallest number that satises [b]both [/b] (a) and (b). (You may assume that (1) and (2) are correct!)

Let $ABCD$ be a quadrilateral and $M$ the midpoint of the segment $AB$. Outside of the quadrilateral are constructed the equilateral triangles $BCE$, $CDF$ and $DAG$. Let $P$ and $N$ be the midpoints of the segments $GF$ and $EF$. Prove that the triangle $MNP$ is equilateral.

Prove that if $x,y,z \in \mathbb{R}^+$ such that $xy,yz,zx$ are sidelengths of a triangle and $k$ $\in$ $[-1,1]$, then \begin{align*} \sum \frac{\sqrt{xy}}{\sqrt{xz+yz+kxy}} \geq 2 \sqrt{1-k} \end{align*} Determine the equality condition too.

The plane is divided in $1\times1$ squares. In each square there is a real number such that it is the arithmetic mean of the four adjacent squares (with a common side). In a square there is $2024.$ Is it possible for $2024^{2024}$ to be written in another square if all the numbers are: a) nonnegative integers; b) integers?

3. 3 red birds for 4 days eat 36 grams of seed, 5 blue birds for 3 days eat 60 gram of seed. For how many days could be feed 2 red birds and 4 blue birds with 88 gr seed?

Let $ABCD$ be a cyclic convex quadrilateral and $\Gamma$ be its circumcircle. Let $E$ be the intersection of the diagonals of $AC$ and $BD$. Let $L$ be the center of the circle tangent to sides $AB$, $BC$, and $CD$, and let $M$ be the midpoint of the arc $BC$ of $\Gamma$ not containing $A$ and $D$. Prove that the excenter of triangle $BCE$ opposite $E$ lies on the line $LM$.

Let $p$ be a prime, and let $a_1, \dots, a_p$ be integers. Show that there exists an integer $k$ such that the numbers \[a_1 + k, a_2 + 2k, \dots, a_p + pk\] produce at least $\tfrac{1}{2} p$ distinct remainders upon division by $p$. [i]Proposed by Ankan Bhattacharya[/i]

Let $\Gamma$ be a circle and $AB$ be a diameter. Let $l$ be a line outside the circle, and is perpendicular to $AB$. Let $X$, $Y$ be two points on $l$. If $X'$, $Y'$ are two points on $l$ such that $AX$, $BX'$ intersect on $\Gamma$ and such that $AY$, $BY'$ intersect on $\Gamma$. Prove that the circumcircles of triangles $AXY$ and $AX'Y'$ intersect at a point on $\Gamma$ other than $A$, or the three circles are tangent at $A$.

When simplified, $\log{8} \div \log{\frac{1}{8}}$ becomes: ${{{ \textbf{(A)}\ 6\log{2} \qquad\textbf{(B)}\ \log{2} \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0}\qquad\textbf{(E)}\ -1}} $

Ana and Bob are playing the following game. [list] [*] First, Bob draws triangle $ABC$ and a point $P$ inside it. [*] Then Ana and Bob alternate, starting with Ana, choosing three different permutations $\sigma_1$, $\sigma_2$ and $\sigma_3$ of $\{A, B, C\}$. [*] Finally, Ana draw a triangle $V_1V_2V_3$. [/list] For $i=1,2,3$, let $\psi_i$ be the similarity transformation which takes $\sigma_i(A), \sigma_i(B)$ and $\sigma_i(C)$ to $V_i, V_{i+1}$ and $ X_i$ respectively (here $V_4=V_1$) where triangle $\Delta V_iV_{i+1}X_i$ lies on the outside of triangle $V_1V_2V_3$. Finally, let $Q_i=\psi_i(P)$. Ana wins if triangles $Q_1Q_2Q_3$ and $ABC$ are similar (in some order of vertices) and Bob wins otherwise. Determine who has the winning strategy.

Given that \[\frac{a-b}{c-d}=2\quad\text{and}\quad\frac{a-c}{b-d}=3\] for certain real numbers $a,b,c,d$, determine the value of \[\frac{a-d}{b-c}.\]

On a plane are given finitely many red and blue lines, no two parallel, such that any intersection point of two lines of the same color also lies on another line of the other color. Prove that all the lines pass through a single point.

Having three sets $ A,B\subset C, $ solve the set equation $ (X\cup (C\setminus A))\cap ((C\setminus X)\cup A)=B. $

Let \(ABC\) be an acute scalene triangle with orthocenter \(H\). Line \(BH\) intersects \(\overline{AC}\) at \(E\) and line \(CH\) intersects \(\overline{AB}\) at \(F\). Let \(X\) be the foot of the perpendicular from \(H\) to the line through \(A\) parallel to \(\overline{EF}\). Point \(B_1\) lies on line \(XF\) such that \(\overline{BB_1}\) is parallel to \(\overline{AC}\), and point \(C_1\) lies on line \(XE\) such that \(\overline{CC_1}\) is parallel to \(\overline{AB}\). Prove that points \(B\), \(C\), \(B_1\), \(C_1\) are concyclic. [i]Proposed by Luke Robitaille[/i]

Let $\mathbb{N}$ denote the set of natural numbers, and let $\left( a_i, b_i \right)$, $1 \leq i \leq 9$, be nine distinct tuples in $\mathbb{N} \times \mathbb{N}$. Show that there are three distinct elements in the set $\{ 2^{a_i} 3^{b_i} \colon 1 \leq i \leq 9 \}$ whose product is a perfect cube.