A parallelogram is inscribed in a quadrilateral, two opposite vertices of which are the midpoints of the opposite sides of the quadrilateral. Determine the area of ​​such a parallelogram if the area of ​​the quadrilateral is equal to $S_o$.

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

Let $ABC$ be a triangle with orthocenter $H$, $\Gamma$ its circumcircle, and $A' \neq A$, $B' \neq B$, $C' \neq C$ points on $\Gamma$. Define $l_a$ as the line that passes through the projections of $A'$ over $AB$ and $AC$. Define $l_b$ and $l_c$ similarly. Let $O$ be the circumcenter of the triangle determined by $l_a$, $l_b$ and $l_c$ and $H'$ the orthocenter of $A'B'C'$. Show that $O$ is midpoint of $HH'$.

[list=1] [*] Let $G$ be a $(4, 4)$ unoriented graph, 2-regulate, containing a cycle with the length 3. Find the characteristic polynomial $P_G (\lambda)$ , its spectrum $Spec (G)$ and draw the graph $G$. [*] Let $G'$ be another 2-regulate graph, having its characteristic polynomial $P_{G'} (\lambda) = \lambda^4 - 4\lambda^2 + \alpha, \alpha \in \mathbb{R}$. Find the spectrum $Spec(G')$ and draw the graph $G'$. [*] Are the graphs $G$ and $G'$ cospectral or isomorphic? [/list]

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 \[\frac{a-d}{b-c}.\]

In isosceles triangle $ABC$, $AB=AC$. Extend segment $BC$ through $C$ to $P$. Points $X$ and $Y$ lie on lines $AB$ and $AC$, respectively, such that $PX \parallel AC$ and $PY \parallel AB$. Point $T$ lies on the circumcircle of triangle $ABC$ such that $PT \perp XY$. Prove that $\angle BAT = \angle CAT$.

There is a hole in the roof with dimensions $23 \times 19$ cm. Can August fill the the roof with tiles of dimensions $5 \times 24 \times 30$ cm?

The irrational number $\alpha>1$ satisfies $\alpha^2-3\alpha-1=0$. Given that there is a fraction $\frac{m}{n}$ such that $n<500$ and $\left|\alpha-\frac{m}{n}\right|<3\cdot10^{-6}$, find $m$. [i]2018 CCA Math Bonanza Team Round #10[/i]

In triangle $ABC$, $\angle ABC = 120^{\circ}$, $AB = 3$ and $BC = 4$. If perpendiculars constructed to $\overline{AB}$ at $A$ and to $\overline{BC}$ at $C$ meet at $D$, then $CD = $ $ \textbf{(A)}\ 3\qquad\textbf{(B)}\ \frac{8}{\sqrt{3}}\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ \frac{11}{2}\qquad\textbf{(E)}\ \frac{10}{\sqrt{3}} $

Let $A = \{1,2,...,m + n\}$, where $m,n$ are positive integers, and let the function f : $A \to A$ be defined by: $f(m) = 1$, $f(m+n) = m+1$ and $f(i) = i+1$ for all the other $i$. (a) Prove that if $m$ and $n$ are odd, then there exists a function $g : A \to A$ such that $g(g(a)) = f(a)$ for all $a \in A$. (b) Prove that if $m$ is even, then there is a function $g : A\to A$ such that $g(g(a))=f(a)$ for all $a \in A$ is and only if $n = m$.

In $\vartriangle ABC, PQ // BC$ where $P$ and $Q$ are points on $AB$ and $AC$ respectively. The lines $PC$ and $QB$ intersect at $G$. It is also given $EF//BC$, where $G \in EF, E \in AB$ and $F\in AC$ with $PQ = a$ and $EF = b$. Find value of $BC$.

For every natural number $n$, let $Q(n)$ denote the sum of the decimal digits of $n$. Prove that there are infinitely many positive integers $k$ with $Q(3^k) \ge Q(3^{k+1})$.

There is a list of seven numbers. The average of the first four numbers is $5$, and the average of the last four numbers is $8$. If the average of all seven numbers is $6\frac{4}{7}$, then the number common to both sets of four numbers is $\text{(A)}\ 5\frac{3}{7} \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 6\frac{4}{7} \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 7\frac{3}{7}$

The three-digit prime number $p$ is written in base $2$ as $p_2$ and in base $5$ as $p_5$, and the two representations share the same last $2$ digits. If the ratio of the number of digits in $p_2$ to the number of digits in $p_5$ is $5$ to $2$, find all possible values of $p$.

Find all pairs \((x, y)\) of real numbers that satisfy the system \[ (x + 1)(x^2 + 1) = y^3 + 1 \] \[ (y + 1)(y^2 + 1) = x^3 + 1 \]

Let $a, b$ and $c$ be real numbers, and let $f (x) = ax^2 + bx + c$ and $g (x) = cx^2 + bx + a$ functions such that $| f (-1) | \le 1$, $| f (0) | \le 1$ and $| f (1) | \le 1$. Show that if $-1 \le x \le 1$, then $| f (x) | \le \frac54$ and $| g (x) | \le 2$.

Consider a unit circle with center $O$. Let $P$ be a point outside the circle such that the two line segments passing through $P$ and tangent to the circle form an angle of $60^\circ$. Compute the length of $OP$.

[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide] [b]D1.[/b] A Giant Hopper is $200$ meters away from you. It can hop $50$ meters. How many hops would it take for it to reach you? [b]D2.[/b] A rope of length $6$ is used to form the edges of an equilateral triangle (a triangle with equal side lengths). What is the length of one of these edges? [b]D3 / Z1.[/b] Point $E$ is on side $AB$ of rectangle $ABCD$. Find the area of triangle $ECD$ divided by the area of rectangle $ABCD$. [b]D4 / Z2.[/b] Garb and Grunt have two rectangular pastures of area $30$. Garb notices that his has a side length of $3$, while Grunt’s has a side length of $5$. What’s the positive difference between the perimeters of their pastures? [b]D5.[/b] Let points $A$ and $B$ be on a circle with radius $6$ and center $O$. If $\angle AOB = 90^o$, find the area of triangle $AOB$. [b]D6 / Z3.[/b] A scalene triangle (the $3$ side lengths are all different) has integer angle measures (in degrees). What is the largest possible difference between two angles in the triangle? [b]D7.[/b] Square $ABCD$ has side length $6$. If triangle $ABE$ has area $9$, find the sum of all possible values of the distance from $E$ to line $CD$. [b]D8 / Z4.[/b] Let point $E$ be on side $\overline{AB}$ of square $ABCD$ with side length $2$. Given $DE = BC+BE$, find $BE$. [b]Z5.[/b] The two diagonals of rectangle $ABCD$ meet at point $E$. If $\angle AEB = 2\angle BEC$, and $BC = 1$, find the area of rectangle $ABCD$. [b]Z6.[/b] In $\vartriangle ABC$, let $D$ be the foot of the altitude from $A$ to $BC$. Additionally, let $X$ be the intersection of the angle bisector of $\angle ACB$ and $AD$. If $BD = AC = 2AX = 6$, find the area of $ABC$. [b]Z7.[/b] Let $\vartriangle ABC$ have $\angle ABC = 40^o$. Let $D$ and $E$ be on $\overline{AB}$ and $\overline{AC}$ respectively such that DE is parallel to $\overline{BC}$, and the circle passing through points $D$, $E$, and $C$ is tangent to $\overline{AB}$. If the center of the circle is $O$, find $\angle DOE$. [b]Z8.[/b] Consider $\vartriangle ABC$ with $AB = 3$, $BC = 4$, and $AC = 5$. Let $D$ be a point of $AC$ other than $A$ for which $BD = 3$, and $E$ be a point on $BC$ such that $\angle BDE = 90^o$. Find $EC$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the cube $ABCDA'B'C'D'$ ($ABCD$ and $A'B'C'D'$ are the upper and lower bases, repsectively, and edges $AA', BB', CC', DD'$ are parallel). The point $X$ moves at a constant speed along the perimeter of the square $ABCD$ in the direction $ABCDA$, and the point $Y$ moves at the same rate along the perimiter of the square $B'C'CB$ in the direction $B'C'CBB'$. Points $X$ and $Y$ begin their motion at the same instant from the starting positions $A$ and $B'$, respectively. Determine and draw the locus of the midpionts of the segments $XY$.

It is known about two triangles that for each of them the sum of the lengths of any two of its sides is equal to the sum of the lengths of any two sides of the other triangle. Are triangles necessarily congruent?

An interstellar hotel has $100$ rooms with capacities $101,102,\ldots, 200$ people. These rooms are occupied by $n$ people in total. Now a VIP guest is about to arrive and the owner wants to provide him with a personal room. On that purpose, the owner wants to choose two rooms $A$ and $B$ and move all guests from $A$ to $B$ without exceeding its capacity. Determine the largest $n$ for which the owner can be sure that he can achieve his goal no matter what the initial distribution of the guests is.

Given a sequence of numbers $ 13, 25, 43, \ldots $ whose $ n $-th term is defined by the formula $$a_n =3(n^2 + n) + 7$$ Prove that this sequence has the following properties: 1) Of every five consecutive terms of the sequence, exactly one is divisible by $ 5 $, 2( No term of the sequence is the cube of an integer.

Consider a triangle $\Delta ABC$ and three points $D,E,F$ such that: $B$ and $E$ are on different side of the line $AC$, $C$ and $D$ are on different sides of $AB$, $A$ and $F$ are on the same side of the line $BC$. Also $\Delta ADB \sim \Delta CEA \sim \Delta CFB$. Let $M$ be the middle point of $AF$. Prove that: a)$\Delta BDF \sim \Delta FEC$. b) $M$ is the middle point of $DE$. [i]Dan Branzei[/i]

The numbers $1, 2,. . . , 33$ are written on the board . A student performs the following procedure: choose two numbers from those written on the board so that one of them is a multiple of the other number; after the election he deletes the two numbers and writes on the board their number. The student repeats the procedure so many times until only numbers without multiples remain on the board. Determine how many numbers they remain on the board in the situation where the student can no longer repeat the procedure.

Two unequal circles intersect at $X$ and $Y$. Their common tangents intersect at $Z$. One of the tangents touches the circles at $P$ and $Q$. Show that $ZX$ is tangent to the circumcircle of $PXQ$.