Given a $3\times 3$ grid, we call the remainder of the grid an “[i]angle[/i]” when a $2\times 2$ grid is cut out from the grid. Now we place some [i]angles[/i] on a $10\times 10$ grid such that the borders of those [i]angles[/i] must lie on the grid lines or its borders, moreover there is no overlap among the [i]angles[/i]. Determine the maximal value of $k$, such that no matter how we place $k$ [i]angles[/i] on the grid, we can always place another [i]angle[/i] on the grid.

Consider the graphs $ y \equal{} Ax^2$ and and $ y^2 \plus{} 3 \equal{} x^2 \plus{} 4y$, where $ A$ is a positive constant and $ x$ and $ y$ are real variables. In how many points do the two graphs intersect? $ \textbf{(A)}\ \text{exactly } 4 \qquad \textbf{(B)}\ \text{exactly } 2$ $ \textbf{(C)}\ \text{at least } 1, \text{ but the number varies for different positive values of } A$ $ \textbf{(D)}\ 0 \text{ for at least one positive value of } A \qquad \textbf{(E)}\ \text{none of these}$

Point $ D$ is chosen on the side $ AB$ of triangle $ ABC$. Point $ L$ inside the triangle $ ABC$ is such that $ BD=LD$ and $ \angle LAB=\angle LCA=\angle DCB$. It is known that $ \angle ALD+\angle ABC=180^\circ$. Prove that $ \angle BLC=90^\circ$.

Find the maximal cardinality $|S|$ of the subset $S \subset A=\{1, 2, 3, \dots, 9\}$ given that no two sums $a+b | a, b \in S, a \neq b$ are equal.

$f$ and $g$ are real-valued functions defined on the real line. For all $x$ and $y, f(x+y)+f(x-y)=2f(x)g(y)$. $f$ is not identically zero and $|f(x)|\le1$ for all $x$. Prove that $|g(x)|\le1$ for all $x$.

On the table there are $50$ stacks of coins that have $1,2,3,…,50$ coins respectively. Ana and Beto play the following game in turns: First, Ana chooses one of the $50$ piles on the table, and Beto decides if that pile is for Ana or for him. Then, Beto chooses one of the $49$ remaining piles on the table, and Ana decides if that pile is for her or for Beto. They continue playing alternately in this way until one of the players has $25$ batteries. When that happens, the other player takes all the remaining stacks on the table and whoever has the most coins wins. Determine which of the two players has a winning strategy.

Let $ m,n\ge 2 $ and consider a rectangle formed by $ m\times n $ unit squares that are colored, either white, or either black. A [i]step[/i] is the action of selecting from it a rectangle of dimensions $ 1\times k, $ where $ k $ is an odd number smaller or equal to $ n, $ or a rectangle of dimensions $ l\times 1, $ where $ l $ is and odd number smaller than $ m, $ and coloring all the unit squares of this chosen rectangle with the color that appears the least in it. [b]a)[/b] Show that, for any $ m,n\ge 5, $ there exists a succession of [i]steps[/i] that make the rectagle to be single-colored. [b]b)[/b] What about $ m=n+1=5? $

Mia has $2$ green sticks of $\textbf{3cm}$ each one, $2$ blue sticks of $\textbf{4cm}$ each one and $2$ red sticks of $\textbf{5cm}$ each one. She wants to make a triangle using the $6$ sticks as it´s perimeter, all at once and without overlapping them. How many non-congruent triangles can make?

Before the math competition, Dmytro overheard Olena and Mykola talking about their birthdays. [b]О[/b]: "The day and month of my birthday are half as large as the day and month of Mykola's birthday." [b]М[/b]: "Also, the day of Olena's birth and the month of my birth are consecutive positive integers." [b]О[/b]: "And the sum of all these four numbers is a multiple of $17$." Can Dmitro determine the day and month of Olena's birth? [i]Proposed by Olena Artemchuk and Mykola Moroz[/i]

Two circles $\omega_1$ and $\omega_2$ with radii $r_1$ and $r_2$, $r_2>r_1$, are externally tangent. The line $t_1$ is tangent to the circles $\omega_1$ and $\omega_2$ at points $A$ and $D$ respectively. The parallel line $t_2$ to the line $t_1$ is tangent to the circle $\omega_1$ and intersects the circle $\omega_2$ at points $E$ and $F$. The line $t_3$ passing through $D$ intersects the line $t_2$ and the circle $\omega_2$ in $B$ and $C$ respectively, both different of $E$ and $F$ respectively. Prove that the circumcircle of the triangle $ABC$ is tangent to the line $t_1$. [i]Dinu Serbanescu[/i]

The graph shows the price of five gallons of gasoline during the first ten months of the year. By what percent is the highest price more than the lowest price? [asy] import graph; size(12.5cm); real lsf=2; pathpen=linewidth(0.5); pointpen=black; pen fp = fontsize(10); pointfontpen=fp; real xmin=-1.33,xmax=11.05,ymin=-9.01,ymax=-0.44; pen ycycyc=rgb(0.55,0.55,0.55); pair A=(1,-6), B=(1,-2), D=(1,-5.8), E=(1,-5.6), F=(1,-5.4), G=(1,-5.2), H=(1,-5), J=(1,-4.8), K=(1,-4.6), L=(1,-4.4), M=(1,-4.2), N=(1,-4), P=(1,-3.8), Q=(1,-3.6), R=(1,-3.4), S=(1,-3.2), T=(1,-3), U=(1,-2.8), V=(1,-2.6), W=(1,-2.4), Z=(1,-2.2), E_1=(1.4,-2.6), F_1=(1.8,-2.6), O_1=(14,-6), P_1=(14,-5), Q_1=(14,-4), R_1=(14,-3), S_1=(14,-2), C_1=(1.4,-6), D_1=(1.8,-6), G_1=(2.4,-6), H_1=(2.8,-6), I_1=(3.4,-6), J_1=(3.8,-6), K_1=(4.4,-6), L_1=(4.8,-6), M_1=(5.4,-6), N_1=(5.8,-6), T_1=(6.4,-6), U_1=(6.8,-6), V_1=(7.4,-6), W_1=(7.8,-6), Z_1=(8.4,-6), A_2=(8.8,-6), B_2=(9.4,-6), C_2=(9.8,-6), D_2=(10.4,-6), E_2=(10.8,-6), L_2=(2.4,-3.2), M_2=(2.8,-3.2), N_2=(3.4,-4), O_2=(3.8,-4), P_2=(4.4,-3.6), Q_2=(4.8,-3.6), R_2=(5.4,-3.6), S_2=(5.8,-3.6), T_2=(6.4,-3.4), U_2=(6.8,-3.4), V_2=(7.4,-3.8), W_2=(7.8,-3.8), Z_2=(8.4,-2.8), A_3=(8.8,-2.8), B_3=(9.4,-3.2), C_3=(9.8,-3.2), D_3=(10.4,-3.8), E_3=(10.8,-3.8); filldraw(C_1--E_1--F_1--D_1--cycle,ycycyc); filldraw(G_1--L_2--M_2--H_1--cycle,ycycyc); filldraw(I_1--N_2--O_2--J_1--cycle,ycycyc); filldraw(K_1--P_2--Q_2--L_1--cycle,ycycyc); filldraw(M_1--R_2--S_2--N_1--cycle,ycycyc); filldraw(T_1--T_2--U_2--U_1--cycle,ycycyc); filldraw(V_1--V_2--W_2--W_1--cycle,ycycyc); filldraw(Z_1--Z_2--A_3--A_2--cycle,ycycyc); filldraw(B_2--B_3--C_3--C_2--cycle,ycycyc); filldraw(D_2--D_3--E_3--E_2--cycle,ycycyc); D(B--A,linewidth(0.4)); D(H--(8,-5),linewidth(0.4)); D(N--(8,-4),linewidth(0.4)); D(T--(8,-3),linewidth(0.4)); D(B--(8,-2),linewidth(0.4)); D(B--S_1); D(T--R_1); D(N--Q_1); D(H--P_1); D(A--O_1); D(C_1--E_1); D(E_1--F_1); D(F_1--D_1); D(D_1--C_1); D(G_1--L_2); D(L_2--M_2); D(M_2--H_1); D(H_1--G_1); D(I_1--N_2); D(N_2--O_2); D(O_2--J_1); D(J_1--I_1); D(K_1--P_2); D(P_2--Q_2); D(Q_2--L_1); D(L_1--K_1); D(M_1--R_2); D(R_2--S_2); D(S_2--N_1); D(N_1--M_1); D(T_1--T_2); D(T_2--U_2); D(U_2--U_1); D(U_1--T_1); D(V_1--V_2); D(V_2--W_2); D(W_2--W_1); D(W_1--V_1); D(Z_1--Z_2); D(Z_2--A_3); D(A_3--A_2); D(A_2--Z_1); D(B_2--B_3); D(B_3--C_3); D(C_3--C_2); D(C_2--B_2); D(D_2--D_3); D(D_3--E_3); D(E_3--E_2); D(E_2--D_2); label("0",(0.52,-5.77),SE*lsf,fp); label("\$ 5",(0.3,-4.84),SE*lsf,fp); label("\$ 10",(0.2,-3.84),SE*lsf,fp); label("\$ 15",(0.2,-2.85),SE*lsf,fp); label("\$ 20",(0.2,-1.85),SE*lsf,fp); label("$\mathrm{Price}$",(-.65,-3.84),SE*lsf,fp); label("$1$",(1.45,-5.95),SE*lsf,fp); label("$2$",(2.44,-5.95),SE*lsf,fp); label("$3$",(3.44,-5.95),SE*lsf,fp); label("$4$",(4.46,-5.95),SE*lsf,fp); label("$5$",(5.43,-5.95),SE*lsf,fp); label("$6$",(6.42,-5.95),SE*lsf,fp); label("$7$",(7.44,-5.95),SE*lsf,fp); label("$8$",(8.43,-5.95),SE*lsf,fp); label("$9$",(9.44,-5.95),SE*lsf,fp); label("$10$",(10.37,-5.95),SE*lsf,fp); label("Month",(5.67,-6.43),SE*lsf,fp); D(A,linewidth(1pt)); D(B,linewidth(1pt)); D(D,linewidth(1pt)); D(E,linewidth(1pt)); D(F,linewidth(1pt)); D(G,linewidth(1pt)); D(H,linewidth(1pt)); D(J,linewidth(1pt)); D(K,linewidth(1pt)); D(L,linewidth(1pt)); D(M,linewidth(1pt)); D(N,linewidth(1pt)); D(P,linewidth(1pt)); D(Q,linewidth(1pt)); D(R,linewidth(1pt)); D(S,linewidth(1pt)); D(T,linewidth(1pt)); D(U,linewidth(1pt)); D(V,linewidth(1pt)); D(W,linewidth(1pt)); D(Z,linewidth(1pt)); D(E_1,linewidth(1pt)); D(F_1,linewidth(1pt)); D(O_1,linewidth(1pt)); D(P_1,linewidth(1pt)); D(Q_1,linewidth(1pt)); D(R_1,linewidth(1pt)); D(S_1,linewidth(1pt)); D(C_1,linewidth(1pt)); D(D_1,linewidth(1pt)); D(G_1,linewidth(1pt)); D(H_1,linewidth(1pt)); D(I_1,linewidth(1pt)); D(J_1,linewidth(1pt)); D(K_1,linewidth(1pt)); D(L_1,linewidth(1pt)); D(M_1,linewidth(1pt)); D(N_1,linewidth(1pt)); D(T_1,linewidth(1pt)); D(U_1,linewidth(1pt)); D(V_1,linewidth(1pt)); D(W_1,linewidth(1pt)); D(Z_1,linewidth(1pt)); D(A_2,linewidth(1pt)); D(B_2,linewidth(1pt)); D(C_2,linewidth(1pt)); D(D_2,linewidth(1pt)); D(E_2,linewidth(1pt)); D(L_2,linewidth(1pt)); D(M_2,linewidth(1pt)); D(N_2,linewidth(1pt)); D(O_2,linewidth(1pt)); D(P_2,linewidth(1pt)); D(Q_2,linewidth(1pt)); D(R_2,linewidth(1pt)); D(S_2,linewidth(1pt)); D(T_2,linewidth(1pt)); D(U_2,linewidth(1pt)); D(V_2,linewidth(1pt)); D(W_2,linewidth(1pt)); D(Z_2,linewidth(1pt)); D(A_3,linewidth(1pt)); D(B_3,linewidth(1pt)); D(C_3,linewidth(1pt)); D(D_3,linewidth(1pt)); D(E_3,linewidth(1pt)); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy] $\textbf{(A)}\ 50 \qquad \textbf{(B)}\ 62 \qquad \textbf{(C)}\ 70 \qquad \textbf{(D)}\ 89 \qquad \textbf{(E)}\ 100$

Find the difference between the greatest and least values of $lcm (a,b,c)$, where $a$, $b$, and $c$ are distinct positive integers between $1$ and $10$, inclusive.

For a given integer $n\ge 2$, let $\{a_1,a_2,…,a_m\}$ be the set of positive integers less than $n$ that are relatively prime to $n$. Prove that if every prime that divides $m$ also divides $n$, then $a_1^k+a_2^k + \dots + a_m^k$ is divisible by $m$ for every positive integer $k$. [i]Proposed by Ivan Borsenco[/i]

Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ for which the equality \[f(f(x)\minus{}y)\equal{}f(x)\plus{}f(f(y)\minus{}f(\minus{}x))\plus{}x\] holds for all real $x,y$.

Altitudes $AA_1$ and $CC_1$ of an acute-angled triangle $ABC$ intersect at point $H$. A straight line passing through point $H$ parallel to line $A_1C_1$ intersects the circumscribed circles of triangles $AHC_1$ and $CHA_1$ at points $X$ and $Y$, respectively. Prove that points $X$ and $Y$ are equidistant from the midpoint of segment $BH$.

Let $ABCD$ be a trapezium, $AD\parallel BC$, and let $E,F$ be points on the sides$AB$ and $CD$, respectively. The circumcircle of $AEF$ meets $AD$ again at $A_1$, and the circumcircle of $CEF$ meets $BC$ again at $C_1$. Prove that $A_1C_1,BD,EF$ are concurrent.

Let $k$ be any fixed positive integer. Let's look at integer pairs $(a, b)$, for which the quadratic equations $x^2 - 2ax + b = 0$ and $y^2 + 2ay + b = 0$ are real solutions (not necessarily different), which can be denoted by $x_1, x_2$ and $y_1, y_2$, respectively, in such an order that the equation $x_1 y_1 - x_2 y_2 = 4k$. a) Find the largest possible value of the second component $b$ of such a pair of numbers ($a, b)$. b) Find the sum of the other components of all such pairs of numbers.

Let $a, b, c, d$ be the roots of the quartic polynomial $f(x) = x^4 + 2x + 4$. Find the value of $$\frac{a^2}{a^3 + 2} + \frac{b^2}{b^3 + 2} + \frac{c^2}{c^3 + 2} + \frac{d^2}{d^3 + 2}.$$

On the "battleship" field (a square of $10\times 10$ cells), $10$ "ships" are placed in the following sequence: first one "ship" of size $1\times 4$, then two - of size $1\times 3$, three - of size $1\times 2$, and, finally, four - $1\times 1$. The rules do not allow "ships" to touch each other even with their tops. Can it happen that when part of the "ships" have already been displayed, there is nowhere to place the next one?

Let $f$ be a function from nonnegative integers to nonnegative integers such that $f(0)=0$ and $$f(m)=f\left(\left\lfloor \frac{m}{2}\right\rfloor\right)+\left\lceil\frac{m}{2}\right\rceil^2$$ for all positive integers $m.$ Compute $$\frac{f(1)}{1\cdot2}+\frac{f(2)}{2\cdot3}+\frac{f(3)}{3\cdot4}+\cdots+\frac{f(31)}{31\cdot32}.$$(Here, $\lfloor z \rfloor$ is the greatest integer less than or equal to $z,$ and $\lceil z \rceil$ is the least positive integer greater than or equal to $z.$)

[i]Introductory part [/i] We call an $n$-tuple $x = (x_1, x_2, ... , x_n)$, with $x_k \in R$ (or respectively with all $x_k \in Z$) a real vector (or respectively an integer vector). The set of all real vectors (respectively all integer vectors) is usually denoted by $R^n$ (respectively $Z^n$). A vector $x$ is null if and only if $x_k = 0$, for all $k \in \{1, 2,... , n\}$. Also let $U_n$ be the set of all real vectors $x = (x_1, x_2, ... , x_n)$, such that $x^2_1 + x^2_2 + ...+ x^2_n = 1$. For two vectors $x = (x_1, ... , x_n), y = (y_1, ..., y_n)$ we define the scalar product as the real number $x\cdot y = x_1y_1 + x_2y_2 +...+ x_ny_n$. We define the norm of the vector $x$ as $||x|| =\sqrt{x^2_1 + x^2_2 + ...+ x^2_n}$ [i]The problem[/i] Let $A(k, r) = \{x \in U_n |$ for all $z \in Z^n$ we have either $|x \cdot z| \ge \frac{k}{||z||^r}$ or $z$ is null $\}$. Prove that if $r > n - 1$ the we can find a positive number $k$ such that $A(k, r)$ is not empty, and if $r < n - 1$ we cannot find such a positive number $k$.

A set $S$ of positive integers is called magic if for any two distinct members of $S, i$ and $j$, $\frac{i+ j}{GCD(i, j)}$is also a member of $S$. The $GCD$, or greatest common divisor, of two positive integers is the largest integer that divides evenly into both of them; for example, $GCD(36,80) = 4$. Find and describe all finite magic sets.

Let be given a triangle $ ABC$ with $ BC \equal{} a$, $ CA \equal{} b$, $ AB \equal{} c$. Six distinct points $ A_1$, $ A_2$, $ B_1$, $ B_2$, $ C_1$, $ C_2$ not coinciding with $ A$, $ B$, $ C$ are chosen so that $ A_1$, $ A_2$ lie on line $ BC$; $ B_1$, $ B_2$ lie on $ CA$ and $ C_1$, $ C_2$ lie on $ AB$. Let $ \alpha$, $ \beta$, $ \gamma$ three real numbers satisfy $ \overrightarrow{A_1A_2} \equal{} \frac {\alpha}{a}\overrightarrow{BC}$, $ \overrightarrow{B_1B_2} \equal{} \frac {\beta}{b}\overrightarrow{CA}$, $ \overrightarrow{C_1C_2} \equal{} \frac {\gamma}{c}\overrightarrow{AB}$. Let $ d_A$, $ d_B$, $ d_C$ be respectively the radical axes of the circumcircles of the pairs of triangles $ AB_1C_1$ and $ AB_2C_2$; $ BC_1A_1$ and $ BC_2A_2$; $ CA_1B_1$ and $ CA_2B_2$. Prove that $ d_A$, $ d_B$ and $ d_C$ are concurrent if and only if $ \alpha a \plus{} \beta b \plus{} \gamma c \neq 0$.

Let $ABCD$ be a square, and let $P$ be a point on the minor arc $CD$ of its circumcircle. The lines $PA, PB$ meet the diagonals $BD, AC$ at points $K, L$ respectively. The points $M, N$ are the projections of $K, L$ respectively to $CD$, and $Q$ is the common point of lines $KN$ and $ML$. Prove that $PQ$ bisects the segment $AB$.

If $\mathbf{A}$ and $\mathbf{B}$ are square matrices of the same size such that $\mathbf{ABAB}=\mathbf{0}$, does it follow that $\mathbf{BABA}=\mathbf{0}$.