a) Given an acute triangle $ABC(AB>AC).$ The circle $(O)$ with diameter $BC$ intersects $AB,AC$ at $F,E$, respectively. Let $H$ be the intersection point of $BE,CF,$ the line $AH$ intersects the line $BC$ at $D,$ the line $EF$ intersects the line $BC$ at $K.$ The line passing through $D$ and parallel to $EF$ intersects $AB,AC$ at $M,N,$ respectively. Prove that: $M,O,N,K$ are on the same circle. b) Given $\triangle ABC, \angle BAC=\angle BCA=30^{\circ}.$ $D,E,F$ are moving points on the side $AB,BC,CA$ such that: $\angle BFD=\angle BFE=60^{\circ}.$ Let $p,p_1$ be the perimeter of $\triangle ABC,\triangle DEF,$ respectively. Prove that: $p\le 2p_1.$

For any real value of $ x$ the maximum value of $ 8x \minus{} 3x^2$ is: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ \frac83 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ \frac{16}{3}$

We call a number $n$ [i]interesting [/i]if for each permutation $\sigma$ of $1,2,\ldots,n$ there exist polynomials $P_1,P_2,\ldots ,P_n$ and $\epsilon > 0$ such that: $i)$ $P_1(0)=P_2(0)=\ldots =P_n(0)$ $ii)$ $P_1(x)>P_2(x)>\ldots >P_n(x)$ for $-\epsilon<x<0$ $iii)$ $P_{\sigma (1)} (x)>P_{\sigma (2)}(x)> \ldots >P_{\sigma (n)} (x) $ for $0<x<\epsilon$ Find all [i]interesting [/i]$n$. [i]Proposed by Mojtaba Zare Bidaki[/i]

For each positive integer $n$, let $c(n)$ be the number of digits of $n$. Let $A$ be a set of positive integers with the following property: If $a$ and $b$ are two distinct elements in $A$, then $c(a +b)+2 > c(a)+c(b)$. Find the largest number of elements that $A$ can have. PS. In the original wording: c(n) = ''cantidad de dıgitos''

Given triangle $ ABC$ with sidelengths $ a,b,c$. Tangents to incircle of $ ABC$ that parallel with triangle's sides form three small triangle (each small triangle has 1 vertex of $ ABC$). Prove that the sum of area of incircles of these three small triangles and the area of incircle of triangle $ ABC$ is equal to $ \frac{\pi (a^{2}\plus{}b^{2}\plus{}c^{2})(b\plus{}c\minus{}a)(c\plus{}a\minus{}b)(a\plus{}b\minus{}c)}{(a\plus{}b\plus{}c)^{3}}$ (hmm,, looks familiar, isn't it? :wink: )

Let $P(x)$ be a polynomial with positive integer coefficients such that $deg(P)=699$. Prove that if $P(1) \le 2022$, then there exist some consecutive coefficients such that their sum is $22$, $55$, or $77$.

Suppose $x_1, x_2, x_3$ are the roots of polynomial $P(x) = x^3 - 6x^2 + 5x + 12$ The sum $|x_1| + |x_2| + |x_3|$ is (A): $4$ (B): $6$ (C): $8$ (D): $14$ (E): None of the above.

Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, and $AC = 15$. Let $D$ and $E$ be the midpoints of $BC$ and $AB$, respectively. If $AD$ and $CE$ intersect at $G$, compute the area of quadrilateral $BEGD$.

It takes $A$ algebra books (all the same thickness) and $H$ geometry books (all the same thickness, which is greater than that of an algebra book) to completely fill a certain shelf. Also, $S$ of the algebra books and $M$ of the geometry books would fill the same shelf. Finally, $E$ of the algebra books alone would fill this shelf. Given that $A, H, S, M, E$ are distinct positive integers, it follows that $E$ is $ \textbf{(A)}\ \frac{AM+SH}{M+H} \qquad\textbf{(B)}\ \frac{AM^2+SH^2}{M^2+H^2} \qquad\textbf{(C)}\ \frac{AH-SM}{M-H} \qquad\textbf{(D)}\ \frac{AM-SH}{M-H} \qquad\textbf{(E)}\ \frac{AM^2-SH^2}{M^2-H^2} $

Given a triangle $ ABC$. A circle passes through vertices $ B$ and $ C$ and intersects sides $ AB$ and $ AC$ at points $ D$ and $ E$, respectively. Segments $ CD$ and $ BE$ intersect at point $ O$. Denote the incenters of triangles $ ADE$ and $ ODE$ by $ M$ and $ N$, respectiely. Prove that the midpoint of the smaller arc $ DE$ lies on line $ MN$.

Circles $\omega_1$ and $\omega_2$ are externally tangent to each other at $T$. Let $X$ be a point on circle $\omega_1$. Line $l_1$ is tangent to circle $\omega_1$ and $X$, and line $l$ intersects circle $\omega_2$ at $A$ and $B$. Line $XT$ meets circle $\omega$ at $S$. Point $C$ lies on arc $TS$ (of circle $\omega_2$, not containing points $A$ and $B$). Point $Y$ lies on circle $\omega_1$ and line $YC$ is tangent to circle $\omega_1$. Let $I$ be the intersection of lines $XY$ ad $SC$. Prove that... a) points $C$, $T$, $Y$, $I$ lie on a circle (B) $I$ is an excenter of triangle $ABC$.

Evaluate the following definite integrals. (a) $\int_1^2 \frac{x-1}{x^2-2x+2}\ dx$ (b) $\int_0^1 \frac{e^{4x}}{e^{2x}+2}\ dx$ (c) $\int_1^e x\ln \sqrt{x}\ dx$ (d) $\int_0^{\frac{\pi}{3}} \left(\cos ^ 2 x\sin 3x-\frac 14\sin 5x\right)\ dx$

Consider parallelogram $ABCD$, with acute angle at $A$, $AC$ and $BD$ intersect at $E$. Circumscribed circle of triangle $ACD$ intersects $AB$, $BC$ and $BD$ at $K$, $L$ and $P$ (in that order). Then, circumscribed circle of triangle $CEL$ intersects $BD$ at $M$. Prove that: $$KD*KM=KL*PC$$

How many positive two-digit integers are factors of $2^{24} -1$? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14 $

Let $ ABC$ be an acute triangle, $ CC_1$ its bisector, $ O$ its circumcenter. The perpendicular from $ C$ to $ AB$ meets line $ OC_1$ in a point lying on the circumcircle of $ AOB$. Determine angle $ C$.

Let $ n $ be an integer greater than or equal to $ 2 $. Prove that if the real numbers $ a_1 , a_2 , \cdots , a_n $ satisfy $ a_1 ^2 + a_2 ^2 + \cdots + a_n ^ 2 = n $, then \[\sum_{1 \le i < j \le n} \frac{1}{n- a_i a_j} \le \frac{n}{2} \] must hold.

Call a $9$-digit number a [i]cassowary[/i] if it uses each of the digits $1$ through $9$ exactly once. Compute the number of cassowaries that are prime.

Let $b\geq 3$ be a positive integer and let $\sigma$ be a nonidentity permutation of the set $\{0,1,\ldots,b-1\}$ such that $\sigma(0)=0.$ The [i]substitution cipher[/i] $C_\sigma$ encrypts every positive integer $n$ by replacing each digit $a$ in the representation of $n$ in base $b$ with $\sigma(a).$ Let $d$ be any positive integer such that $b$ does not divide $d.$ We say that $C_\sigma$ [i]complies[/i] with $d$ if $C_\sigma$ maps every multiple of $d$ onto a multiple of $d,$ and we say that $d$ is [i]cryptic[/i] if there exists some $C_\sigma$ such that $C_\sigma$ complies with $d.$ Let $k$ be any positive integer, and let $p=2^k+1.$ a) Find the greatest power of $2$ that is cryptic in base $2p,$ and prove that there is only one substitution cipher complying with it. b) Find the greatest power of $p$ that is cryptic in base $2p,$ and prove that there is only one substitution cipher complying with it. c) Suppose, furthermore, that $p$ is a prime number. Find the greatest cryptic positive integer in base $2p$ and prove that there is only one substitution cipher that complies with it. [i]Proposed by Nikolai Beluhov, Bulgaria[/i]

Show that the radian measure of an acute angle is less than the arithmetic mean of its sine and its tangent.

Let $A$ and $B$ be two triangles on the plane such that the interior of both contains the origin and for each circle $C_r$ centered at the origin $|C_r \cap A|=|C_r\cap B|$ (where $|\cdot |$ is the arc-length measure). Prove that $A$ and $B$ are congruent. Does this statement remain true if the origin is on the border of $A$ or $B$? (translated by Miklós Maróti)

In a certain group there are $n \ge 5$ people, with every two people who do not know each other exactly having one mutual friend and no one knows everyone else. Prove $5$ of $n$ people, may sit at a circle around the table so that each of them sits between a) friends, b) strangers.

Let $c$ be the probability that the cards are neither from the same suit or the same rank. Compute $\lfloor 1000c\rfloor$.

If $a,b,c$ are real numbers and $(a+b-5)^2+(b+2c+3)^2+(c+3a-10)^2=0$ find the integer nearest to $a^3+b^3+c^3$.

Determine the number of positive integral values of $p$ for which there exists a triangle with sides $a,b,$ and $c$ which satisfy $$a^2 + (p^2 + 9)b^2 + 9c^2 - 6ab - 6pbc = 0.$$

In a plane with a square grid, where the side length of the base square is $1$, lies a right triangle. All its vertices are lattice points and all side lengths are integer. Prove that the center of the incircle is also a lattice point.