What is the largest possible area of a quadrilateral with sides $1,4,7,8$ ? $ \textbf{(A)}\ 7\sqrt 2 \qquad\textbf{(B)}\ 10\sqrt 3 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 12\sqrt 3 \qquad\textbf{(E)}\ 9\sqrt 5 $

In a trapezoid $ABCD$ with $AD \parallel BC , O_{1}, O_{2}, O_{3}, O_{4}$ denote the circles with diameters $AB, BC, CD, DA$, respectively. Show that there exists a circle with center inside the trapezoid which is tangent to all the four circles $O_{1},..., O_{4}$ if and only if $ABCD$ is a parallelogram.

Let $x, y$ be numbers in the interval (0,1) such that for some $a>0, a \neq 1$ $$\log _{x} a+\log _{y} a=4 \log _{x y} a$$Prove that $x=y$

An integer which is the area of a right-angled triangle with integer sides is called [i]Pythagorean[/i]. Prove that for every positive integer $n > 12$ there exists a Pythagorean number between $n$ and $2n.$

In the triangle $ABC$, for which $AC <AB <BC$, on the sides $AB$ and $BC$ the points $K$ and $N$ were chosen, respectively, that $KA = AC = CN$. The lines $AN$ and $CK$ intersect at the point $O$. From the point $O$ held the segment $OM \perp AC $ ($M \in AC$) . Prove that the circles inscribed in triangles $ABM$ and $CBM$ are tangent. (Igor Nagel)

Fawzi cuts a spherical cheese completely into (at least three) slices of equal thickness. He starts at one end, making successive parallel cuts, working through the cheese until the slicing is complete. The discs exposed by the first two cuts have integral areas. [list](i) Prove that all the discs that he cuts have integral areas. (ii) Prove that the original sphere had integral surface area if, and only if, the area of the second disc that he exposes is even.[/list]

[u]Round 5[/u] [b]p13.[/b] Mandy is baking cookies. Her recipe calls for $N$ grams of flour, where $N$ is the number of perfect square divisors of $20! + 24!$. Find $N$. [b]p14.[/b] Consider a circular table with center $R$. Beef-loving Bryan places a steak at point $I$ on the circumference of the table. Then he places a bowl of rice at points $C$ and $E$ on the circumference of the table such that $CE \parallel IR$ and $\angle ICE = 25^o$. Find $\angle CIE$. [b]p15.[/b] Enya writes the $4$-letter words $LEEK$, $BEAN$, $SOUP$, $PEAS$, $HAMS$, and $TACO$ on the board. She then thinks of one of these words and gives Daria, Ava, Harini, and Tiffany a slip of paper containing exactly one letter from that word such that if they ordered the letters on their slips correctly, they would form the word. Each person announces at the same time whether they know the word or not. Ava, Harini, and Tiffany all say they do not know the word, while Daria says she knows the word. After hearing this, Ava, Harini, and Tiffany all know the word. Assuming all four girls are perfect logicians and they all thought of the same correct word, determine Daria’s letter. [u]Round 6[/u] [b]p16.[/b] Michael receives a cheese cube and a chocolate octahedron for his 5th birthday. On every day after, he slices off each corner of his cheese and chocolate with a knife. Each slice cuts off exactly one corner. He then eats each corner sliced off. Find the difference between the total number of cheese and chocolate pieces he has eaten by the end of his $6$th birthday. (Michael’s $5$th and $6$th birthdays do not occur on leap years.) [b]p17.[/b] Let $D$ be the average of all positive integers n satisfying $$lcm (gcd (n, 2000), gcd (n, 24)) = gcd (lcm (n, 2000), lcm (n, 24)).$$ Find $3D$. [b]p18.[/b] The base $\vartriangle ABC$ of the triangular pyramid $PABC$ is an equilateral triangle with a side length of $3$. Given that $PA = 3$, $PB = 4$, and $PC = 5$, find the circumradius of $PABC$. [u]Round 7[/u] [b]p19.[/b] $2049300$ points are arranged in an equilateral triangle point grid, a smaller version of which is shown below, such that the sides contain $2024$ points each. Peter starts at the topmost point of the grid. At $9:00$ am each day, he moves to an adjacent point in the row below him. Derrick wants to prevent Peter from reaching the bottom row, so at $12:00$ pm each day, he selects a point on the bottom row and places a rock at that point. Peter stops moving as soon as he is guaranteed to end up at a point with a rock on it. At least how many moves will Peter complete, no matter how Derrick places the rocks? [img]https://cdn.artofproblemsolving.com/attachments/f/a/346d25a5d7bb7a5fbefae7edad727965312b25.png[/img] [b]p20.[/b] There are $N$ stones in a pile, where $N$ is a positive integer. Ava and Anika take turns playing a game, with Ava moving first. If there are n stones in the pile, a move consists of removing $x$ stones, where $1 < gcd(x, n) \le x < n$. Whoever first has no possible moves on their turn wins. Both Ava and Anika play optimally. Find the $2024$th smallest value of $N$ for which Ava wins. [b]p21.[/b] Alan is bored and alone, so he plays a fun game with himself. He writes down all quadratic polynomials with leading coefficient $1$ whose coefficients are integers between $-10$ and $10$, inclusive, on a blackboard. He then erases all polynomials which have a non-integer root. Alan defines the size of a polynomial $P(x)$ to be $P(1)$ and spends an hour adding up the sizes of all the polynomials remaining on the blackboard. Assuming Alan does computation perfectly, find the sum Alan obtains. [u]Round 8[/u] [b]p22.[/b] A prime number is a positive integer with exactly two distinct divisors. You must submit a prime number for this problem. If you do not submit a prime number, you gain $0$ points, and your submission will not be considered valid. The median of all valid submitted numbers is $M$ (duplicates are counted). Estimate $2M$. If your team’s absolute difference between $2M$ and your submission is the $i$th smallest absolute difference among all teams, you gain max$(23 - 2i, 0)$ points. All teams who did not submit any number gain $0$ points. (In the case of a tie, all teams that tied gain the same amount of points.) [b]p23.[/b] Ribbotson the Frog is at the point $(0, 0)$ and wants to reach the point $(18, 18)$ in $36$ steps. Each step, he either moves one unit in the $+x$ direction or one unit in the $+y$ direction. However, Ribbotson hates turning, so he must make at least two steps in any direction before switching directions. If $m$ is the number of different paths Ribbotson the Frog can make, estimate $m$. If $N$ is your team’s submitted number, your team earns points equal to the closest integer to $21\left(1 -\left|\log_{10}\frac{N}{m} \right|^2\right)$. [b]p24.[/b] Let $M = \pi^{\pi^{\pi^{\pi}}}$. Estimate $k$, where $M = 10^{10^{k}}$. If $N$ is your team’s submitted number, your team earns points equal to the closest integer to $21 \cdot 1.01^{(-|N-k|^3)}$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3248729p29808138]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $k$ be a circle with centre $M$ and let $AB$ be a diameter of $k$. Furthermore, let $C$ be a point on $k$ such that $AC = AM$. Let $D$ be the point on the line $AC$ such that $CD = AB$ and $C$ lies between $A$ and $D$. Let $E$ be the second intersection of the circumcircle of $BCD$ with line $AB$ and $F$ be the intersection of the lines $ED$ and $BC$. The line $AF$ cuts the segment $BD$ in $X$. Determine the ratio $BX/XD$.

Consider five points in the plane, with no three of them collinear. Every pair of points among them is joined by a line. In how many ways can we color these lines by red or blue, so taht no three of the points form a triangle with lines of the same colour.

Determine all functions $f : \mathbb {N} \rightarrow \mathbb {N}$ such that $f$ is increasing (not necessarily strictly) and the numbers $f(n)+n+1$ and $f(f(n))-f(n)$ are both perfect squares for every positive integer $n$.

Find all polynomials $P(x)$ with real coefficients such that \[(x-2010)P(x+67)=xP(x) \] for every integer $x$.

Given a quadrilateral $ABCD$ with $AB = BD = DC$ and $AC = BC$. On $BC$ lies point $E$ such that $AE = AB$. Prove that $ED = EB$.

A spatial quadrilateral is circumscribed around a sphere. Prove that all the tangent points lie in one plane.

Let $P$ and $Q$ be points selected uniformly and independently at random inside a regular hexagon $ABCDEF.$ Compute the probability that segment $\overline{PQ}$ is entirely contained in at least one of the quadrilaterals $ABCD,$ $BCDE,$ $CDEF,$ $DEFA,$ $EFAB,$ or $FABC.$

Find the sum of all $4$-digit numbers using the digits $2,3,4,5,6$ without a repetition of any of those digits.

Alice and Bob stand atop two different towers in the Arctic. Both towers are a positive integer number of meters tall and are a positive (not necessarily integer) distance away from each other. One night, the sea between them has frozen completely into reflective ice. Alice shines her flashlight directly at the top of Bob's tower, and Bob shines his flashlight at the top of Alice's tower by first reflecting it off the ice. The light from Alice's tower travels $16$ meters to get to Bob's tower, while the light from Bob's tower travels $26$ meters to get to Alice's tower. Assuming that the lights are both shown from exactly the top of their respective towers, what are the possibilities for the height of Alice's tower?

$\triangle ABC$ is given with $|AB|=7, |BC|=12$, and $|CA|=13$. Let $D$ be a point on $[BC]$ such that $|BD|=5$. Let $r_1$ and $r_2$ be the inradii of $\triangle ABD$ and $\triangle ACD$, respectively. What is $r_1/r_2$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ \frac{13}{12} \qquad \textbf{(C)}\ \frac{7}{5} \qquad \textbf{(D)}\ \frac{3}{2} \qquad \textbf{(E)}\ \text{None}$

If the sum of the lengths of the six edges of a trirectangular tetrahedron $ PABC$ (i.e., $ \angle APB \equal{} \angle BPC \equal{} \angle CPA \equal{} 90^\circ$) is $ S$, determine its maximum volume.

The two legs of a right triangle, which are altitudes, have lengths $2\sqrt3$ and $6$. How long is the third altitude of the triangle? $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 $

Let $n$ be a natural number. A sequence is $k-$complete if it contains all residues modulo $n^k$. Let $Q(x)$ be a polynomial with integer coefficients. For $k\ge 2$, define $Q^k(x)=Q(Q^{k-1}(x))$, where $Q^1(x)=Q(x)$. Show that if $$0,Q(0),Q^2(0),Q^3(0),\ldots $$is $2018-$complete, then it is $k-$complete for all positive integers $k$. [i]Proposed by Ma Zhao Yu[/i]

Given that the expression $\frac{20^{21}}{20^{20}} +\frac{20^{20}}{20^{21}}$ can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers, find $m +n$. [i]Proposed by Ada Tsui[/i]

Is it true that for every $n\ge 2021$ there exist $n$ integer numbers such that the square of each number is equal to the sum of all other numbers, and not all the numbers are equal? (O. Rudenko)

In an acute-angled triangle $ABC$, $AM$ is a median, $AL$ is a bisector and $AH$ is an altitude ($H$ lies between $L$ and $B$). It is known that $ML=LH=HB$. Find the ratios of the sidelengths of $ABC$.

Prove that for any positive integer $k$, \[(k^2)!\cdot\displaystyle\prod_{j=0}^{k-1}\frac{j!}{(j+k)!}\]is an integer.

Find all monotonically increasing or monotonically decreasing functions $f: \mathbb{R}_+\to\mathbb{R}_+$ which satisfy the equation $f\left(xy\right)\cdot f\left(\frac{f\left(y\right)}{x}\right)=1$ for any two numbers $x$ and $y$ from $\mathbb{R}_+$. Hereby, $\mathbb{R}_+$ is the set of all positive real numbers. [i]Note.[/i] A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically increasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\geq f\left(y\right)$. A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically decreasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\leq f\left(y\right)$.