In the triangle $A,B,C$, let $D$ be the middle of $BC$ and $E$ the projection of $C$ on $AD$. Suppose $\angle ACE = \angle ABC$. Show that the triangle $ABC$ is isosceles or rectangle.

Let a function $f : \Bbb{R}^+ \to \Bbb{R}$ satisfy: (i) $f$ is strictly increasing, (ii) $f(x) > -1/x$ for all $x > 0$, (iii)$ f(x)f (f(x) + 1/x) = 1$ for all $x > 0$. Determine $f(1)$.

Tom Sawyer must whitewash a circular fence consisting of $N$ planks. He whitewashes the fence going clockwise and following the rule: He whitewashes the first plank, skips two planks, whitewashes one, skips three, and so on. Some planks may be whitewashed several times. Tom believes that all planks will be whitewashed sooner or later, but aunt Polly is sure that some planks will remain unwhitewashed forever. Prove that Tom is right if $N$ is a power of two, otherwise aunt Polly is right.

Two persons are dividing a piece of cheese. The first person cuts it into two pieces, then the second person cuts one of these pieces into two, then again the first person cuts one of the pieces into two, and so until they have 5 pieces. After that the first person chooses one of the pieces, then the second person chooses one of remaining pieces and so on until all pieces are taken. For each of the players, what is the maximal amount of cheese he can get for certain, regardless of the other's actions?

Let $(a_i,b_i)$ be pairwise distinct pairs of positive integers for $1\le i\le n$. Prove that \[(a_1+a_2+\ldots+a_n)(b_1+b_2+\ldots+b_n)>\frac29 n^3,\] and show that the statement is sharp, i.e. for an arbitrary $c>\frac29$ it is possible that \[(a_1+a_2+\ldots+a_n)(b_1+b_2+\ldots+b_n)<cn^3.\] [i]Submitted by Péter Pál Pach, Budapest, based on an OKTV problem[/i]

[b]p1.[/b] How many ordered triples of integers $(a, b, c)$ where $1 \le a, b, c \le 10$ are such that for every natural number, the equation $(a + n)x^2 + (b + 2n)x + c + n = 0$ has at least one real root? [b]p2.[/b] Find the smallest integer $n$ such that we can cut a $n \times n$ grid into $5$ rectangles with distinct side lengths in $\{1, 2, 3..., 10\}$. Every value is used exactly once. [b]p3.[/b] A plane is flying at constant altitude along a circle of radius $12$ miles with center at a point $A$.The speed of the aircraft is v. At some moment in time, a missile is fired at the aircraft from the point $A$, which has speed v and is guided so that its velocity vector always points towards the aircraft. How far does the missile travel before colliding with the aircraft? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Compute the smallest real $t$ such that there exist constants $a$, $b$ for which the roots of $x^3-ax^2+bx - \frac{ab}{t}$ are the side lengths of a right triangle

Find all triplets of positive integers $(x, y, z)$ such that $x^2 + y^2 + x + y + z = xyz + 1$. [i]Proposed by Viktor Simjanoski[/i]

Let $D$, $E$, $F$ be the midpoints of the arcs $BC$, $CA$, $AB$ on the circumcircle of a triangle $ABC$ not containing the points $A$, $B$, $C$, respectively. Let the line $DE$ meets $BC$ and $CA$ at $G$ and $H$, and let $M$ be the midpoint of the segment $GH$. Let the line $FD$ meet $BC$ and $AB$ at $K$ and $J$, and let $N$ be the midpoint of the segment $KJ$. a) Find the angles of triangle $DMN$; b) Prove that if $P$ is the point of intersection of the lines $AD$ and $EF$, then the circumcenter of triangle $DMN$ lies on the circumcircle of triangle $PMN$.

Non-zero polynomials $P(x)$, $Q(x)$, and $R(x)$ with real coefficients satisfy the identities $$ P(x) + Q(x) + R(x) = P(Q(x)) + Q(R(x)) + R(P(x)) = 0. $$ Prove that the degrees of the three polynomials are all even.

Let $f\left(x\right)=x^2-14x+52$ and $g\left(x\right)=ax+b$, where $a$ and $b$ are positive. Find $a$, given that $f\left(g\left(-5\right)\right)=3$ and $f\left(g\left(0\right)\right)=103$. $\text{(A) }2\qquad\text{(B) }5\qquad\text{(C) }7\qquad\text{(D) }10\qquad\text{(E) }17$

If $ 2x \plus{} 1 \equal{} 8$, then $ 4x \plus{} 1 \equal{}$ $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 17 \qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 19$

Let $a$ be a positive real number. Then prove that the polynomial \[ p(x)=a^3x^3+a^2x^2+ax+a \] has integer roots if and only if $a=1$ and determine those roots.

Let $ABCD$ be a convex quadrilateral with $AB=5$, $AD=17$, and $CD=6$. If the angle bisector of $\angle{BAD}$ and $\angle{ADC}$ intersect at the midpoint of $BC$, find the area of $ABCD$.

Cut a square into eight acute-angled triangles.

Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of its opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube? $ \textbf{(A)}\ \frac {1}{8}\qquad \textbf{(B)}\ \frac {3}{16}\qquad \textbf{(C)}\ \frac {1}{4} \qquad \textbf{(D)}\ \frac {3}{8}\qquad \textbf{(E)}\ \frac {1}{2}$

Prove that there exists a positive integer $n$ such that $\int_0^1 x\sin\ (x^2-x+1)dx\geq \frac {n}{n+1}\sin \frac{n+2}{n+3}.$

Let a triangle $ABC$ inscribed in a circle $\Gamma$ with center $O$. Let $I$ the incenter of triangle $ABC$ and $D, E, F$ the contact points of the incircle with sides $BC, AC, AB$ of triangle $ABC$ respectively . Let also $S$ the foot of the perpendicular line from $D$ to the line $EF$.Prove that line $SI$ passes from the antidiametric point $N$ of $A$ in the circle $\Gamma$.( $AN$ is a diametre of the circle $\Gamma$).

Find all integers $n \geq 3$ such that among any $n$ positive real numbers $a_1, a_2, \hdots, a_n$ with $\text{max}(a_1,a_2,\hdots,a_n) \leq n \cdot \text{min}(a_1,a_2,\hdots,a_n)$, there exist three that are the side lengths of an acute triangle.

Let $ABCD$ be a rectangle inscribed in circle $\omega$ with center $O$. Line $l$ passes trough $O$ and intersects lines $BC$ and $AD$ at points $E$ and $F$ respectively. Points $K$ and $L$ are the intersection points of $l$ and $\omega$ and points $K, E, F, L$ lie in this order on the line $l$. Lines tangent to $w$ in $K$ and $L$ intersect $CD$ at $M$ and $N$ respectively. Prove that $E, F, M, N$ lie on a common circle.

Let $ABCD$ be a convex cyclic quadilateral. Suppose $P$ is a point in the plane of the quadilateral such that the sum of its distances from the vertices of $ABCD$ is the least. If $$\{PC, PB, PC, PD\} = \{3, 4, 6, 8\}$$, what is the maxumum possible area of $ABCD$?

A rectangular box $\mathcal{P}$ has distinct edge lengths $a, b,$ and $c$. The sum of the lengths of all $12$ edges of $\mathcal{P}$ is $13$, the sum of the areas of all $6$ faces of $\mathcal{P}$ is $\frac{11}{2}$, and the volume of $\mathcal{P}$ is $\frac{1}{2}$. What is the length of the longest interior diagonal connecting two vertices of $\mathcal{P}$? $\textbf{(A)}~2\qquad\textbf{(B)}~\frac{3}{8}\qquad\textbf{(C)}~\frac{9}{8}\qquad\textbf{(D)}~\frac{9}{4}\qquad\textbf{(E)}~\frac{3}{2}$

Define the product $P_n=1! \cdot 2!\cdot 3!\cdots (n-1)!\cdot n!$ a) Find all positive integers $m$, such that $\frac {P_{2020}}{m!}$ is a perfect square. b) Prove that there are infinite many value(s) of $n$, such that $\frac {P_{n}}{m!}$ is a perfect square, for at least two positive integers $m$.

Solve in the integers the diophantine equation $$x^4-6x^2+1 = 7 \cdot 2^y.$$

In an exam with k questions, n students are taking part. A student fails the exam if he answers correctly less than half of all questions. Call a question easy if more than half of all students answer it correctly. For which pairs (k, n) of positive integers is it possible that (a) all students fail the exam although all questions are easy; (b) no student fails the exam although no question is easy?