Point $L$ inside triangle $ABC$ is such that $CL = AB$ and $ \angle BAC + \angle BLC = 180^{\circ}$. Point $K$ on the side $AC$ is such that $KL \parallel BC$. Prove that $AB = BK$

let P be a finite set with at least 2 elements. P is a partially ordered and connected set. $p:P^3 \to P$ is a 3-variable, monotone function which satisfies p(x,x,y)=y. Prove that there exists a non-empty subset $I \subset P$ such that $\forall x \in P$ $\forall y \in I$, we have $p(x, y, y) \in I$. [P is connected means that if each element is replaced by vertices and there is an edge between 2 vertices iff the 2 elements can be compared, then the graph is connected. p is monotone means that if $x_1\leq y_1 , x_2\leq y_2 , x_3\leq y_3$ , then $p(x_1,x_2,x_3)\leq p(y_1,y_2,y_3)$.]

Let $p$ be an odd prime and $a_1, a_2,...,a_p$ be integers. Prove that the following two conditions are equivalent: 1) There exists a polynomial $P(x)$ with degree $\leq \frac{p-1}{2}$ such that $P(i) \equiv a_i \pmod p$ for all $1 \leq i \leq p$ 2) For any natural $d \leq \frac{p-1}{2}$, $$ \sum_{i=1}^p (a_{i+d} - a_i )^2 \equiv 0 \pmod p$$ where indices are taken $\pmod p$

Given $\triangle ABC$ with circumcircle $\ell$. Point $M$ in $\triangle ABC$ such that $AM$ is the angle bisector of $\angle BAC$. Circle with center $M$ and radius $MB$ intersects $\ell$ and $BC$ at $D$ and $E$ respectively, $(B \not= D, B \not= E)$. Let $P$ be the midpoint of arc $BC$ in $\ell$ that didn't have $A$. Prove that $AP$ angle bisector of $\angle DPE$ if and only if $\angle B = 90^{\circ}$.

Let $VABC$ be a regular triangular pyramid with base $ABC$, of center $O$. Points $I$ and $H$ are the center of the inscribed circle, respectively the orthocenter $\vartriangle VBC$. Knowing that $AH = 3 OI$, determine the measure of the angle between the lateral edge of the pyramid and the plane of the base.

Pentagon $ABCDE$ is such that all five diagonals $AC, BD, CE, DA,$ and $EB$ lie entirely within the pentagon. If the area of each of the triangles $ABC, BCD, CDE,$ and $DEA$ is equal to $1$ and the area of triangle $EAB$ is equal to $2$, the area of the pentagon $ABCDE$ is closest to $ \mathrm{(A) \ } 4.42 \qquad \mathrm{(B) \ } 4.44 \qquad \mathrm {(C) \ } 4.46 \qquad \mathrm{(D) \ } 4.48 \qquad \mathrm{(E) \ } 4.5$

Find the smallest constant $ C$ such that for all real $ x,y$ \[ 1\plus{}(x\plus{}y)^2 \leq C \cdot (1\plus{}x^2) \cdot (1\plus{}y^2)\] holds.

Find all real parameters $a$ for which the equation $x^8 +ax^4 +1 = 0$ has four real roots forming an arithmetic progression.

Let $ m_1$, $ m_2$, $ \cdots$, $ m_r$ (may not distinct) and $ n_1$, $ n_2$ $ \cdots$, $ n_s$ (may not distinct) be two groups of positive integers such that for any positive integer $ d$ larger than $ 1$, the numbers of which can be divided by $ d$ in group $ m_1$, $ m_2$, $ \cdots$, $ m_r$ (including repeated numbers) are no less than that in group $ n_1$, $ n_2$ $ \cdots$, $ n_s$ (including repeated numbers). Prove that $ \displaystyle \frac{m_1 \cdot m_2 \cdots m_r}{n_1 \cdot n_2 \cdots n_s}$ is integer.

A [i]weird checkerboard[/i] is a coloring of an $8\times8$ grid constructed by making some (possibly none or all) of the following $14$ cuts: [list] [*] the $7$ vertical cuts along a gridline through the entire height of the board, [*] and the $7$ horizontal cuts along a gridline through the entire width of the board. [/list] The divided rectangles are then colored black and white such that the bottom left corner of the grid is black, and no two rectangles adjacent by an edge share a color. Compute the number of weird checkerboards that have an equal amount of area colored black and white. [center] [img]https://cdn.artofproblemsolving.com/attachments/9/b/f768a7a51c9c9bc56a1d55427c33e15e4bcd74.png[/img] [/center]

Given triangle $ABC$ with $AB<AC$. The line passing through $B$ and parallel to $AC$ meets the external angle bisector of $\angle BAC$ at $D$. The line passing through $C$ and parallel to $AB$ meets this bisector at $E$. Point $F$ lies on the side $AC$ and satisfies the equality $FC=AB$. Prove that $DF=FE$.

For some $ p$, the equation $ x^3 \plus{} px^2 \plus{} 3x \minus{} 10 \equal{} 0$ has three real solutions $ a,b,c$ such that $ c \minus{} b \equal{} b \minus{} a > 0$. Determine $ a,b,c,$ and $ p$.

Quadrilateral $ABCD$ is inscribed in circle $\Gamma$. Segments $AC$ and $BD$ intersect at $E$. Circle $\gamma$ passes through $E$ and is tangent to $\Gamma$ at $A$. Suppose the circumcircle of triangle $BCE$ is tangent to $\gamma$ at $E$ and is tangent to line $CD$ at $C$. Suppose that $\Gamma$ has radius $3$ and $\gamma$ has radius $2$. Compute $BD$.

Let $O$ be the center of a circle inscribed in a convex quadrilateral $ABCD$ and $|AB|= a$, $|CD|=$c. Prove that $$\frac{a}{c}=\frac{AO\cdot BO}{CO\cdot DO}.$$

We say that a set $\displaystyle A$ of non-zero vectors from the plane has the property $\displaystyle \left( \mathcal S \right)$ iff it has at least three elements and for all $\displaystyle \overrightarrow u \in A$ there are $\displaystyle \overrightarrow v, \overrightarrow w \in A$ such that $\displaystyle \overrightarrow v \neq \overrightarrow w$ and $\displaystyle \overrightarrow u = \overrightarrow v + \overrightarrow w$. (a) Prove that for all $\displaystyle n \geq 6$ there is a set of $\displaystyle n$ non-zero vectors, which has the property $\displaystyle \left( \mathcal S \right)$. (b) Prove that every finite set of non-zero vectors, which has the property $\displaystyle \left( \mathcal S \right)$, has at least $\displaystyle 6$ elements. [i]Mihai Baluna[/i]

We consider $5 \times 5$ tables containing a real number in each of the $25$ cells. The same number may occur in different cells, but no row or column contains five equal numbers. Such a table is [i]balanced [/i] if the number in the middle cell of every row and column is the average of the numbers in that row or column. A cell is called [i]small [/i] if the number in that cell is strictly smaller than the number in the cell in the very middle of the table. What is the least number of small cells that a balanced table can have?

Suppose $(a_1,a_2,a_3,a_4)$ is a 4-term sequence of real numbers satisfying the following two conditions: [list] [*] $a_3=a_2+a_1$ and $a_4=a_3+a_2$; [*] there exist real numbers $a,b,c$ such that \[an^2+bn+c=\cos(a_n)\] for all $n\in\{1,2,3,4\}$. [/list] Compute the maximum possible value of \[\cos(a_1)-\cos(a_4)\] over all such sequences $(a_1,a_2,a_3,a_4)$.

Is there an increasing sequence of integers $ 0 = {{a} _{0}} <{{a} _{1}} <{{a} _{2}} <\ldots $ for which the following two conditions are satisfied simultaneously: 1) any natural number can be given as $ {{a} _{i}} + {{a} _{j}} $ for some (possibly equal) $ i \ge 0 $, $ j \ge 0$ , 2) $ {{a} _ {n}}> \tfrac {{{n} ^ {2}}} {16} $ for all natural $ n $?

If $ x^4 \plus{} 4x^3 \plus{} 6px^2 \plus{} 4qx \plus{} r$ is exactly divisible by $ x^3 \plus{} 3x^2 \plus{} 9x \plus{} 3$, the value of $ (p \plus{} q)r$ is: $ \textbf{(A)}\ \minus{} 18 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 45 \qquad$

$(BEL 1)$ A parabola $P_1$ with equation $x^2 - 2py = 0$ and parabola $P_2$ with equation $x^2 + 2py = 0, p > 0$, are given. A line $t$ is tangent to $P_2.$ Find the locus of pole $M$ of the line $t$ with respect to $P_1.$

Let $ n$ be a positive integer. Consider \[ S \equal{} \left\{ (x,y,z) \mid x,y,z \in \{ 0, 1, \ldots, n\}, x \plus{} y \plus{} z > 0 \right \} \] as a set of $ (n \plus{} 1)^{3} \minus{} 1$ points in the three-dimensional space. Determine the smallest possible number of planes, the union of which contains $ S$ but does not include $ (0,0,0)$. [i]Author: Gerhard Wöginger, Netherlands [/i]

Player Zero and Player One play a game on a $n \times n$ board ($n \ge 1$). The columns of this $n \times n$ board are numbered $1,2,4,\dots,2^{n-1}$. Turn my turn, the players put their own number in one of the free cells (thus Player Zero puts a $0$ and Player One puts a $1$). Player Zero begins. When the board is filled, the game ends and each row yields a (reverse binary) number obtained by adding the values of the columns with a $1$ in that row. For instance, when $n=4$, a row with $0101$ yields the number $0 \cdot1+1 \cdot 2+0 \cdot 4+1 \cdot 8=10$. a) For which natural numbers $n$ can Player One always ensure that at least one of the row numbers is divisible by $4$? b) For which natural numbers $n$ can Player One always ensure that at least one of the row numbers is divisible by $3$?

Prove that there are no four points $A, B, C, D$ on a plane such that all triangles $\vartriangle ABC,\vartriangle BCD, \vartriangle CDA, \vartriangle DAB$ are acute ones.

Let $n \in \mathbb{N}_{\geq 2}.$ Prove that for any complex numbers $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n,$ the following statements are equivalent: a) $\sum_{k=1}^n|z-a_k|^2 \leq \sum_{k=1}^n|z-b_k|^2, \: \forall z \in \mathbb{C}.$ b) $\sum_{k=1}^na_k=\sum_{k=1}^nb_k$ and $\sum_{k=1}^n|a_k|^2 \leq \sum_{k=1}^n|b_k|^2.$

Find all values of the positive integer $k$ that has the property: There are no positive integers $a,b$ such that the expression $A(k,a,b)=\frac{a+b}{a^2+k^2b^2-k^2ab}$ is a composite positive number.