How many $f:\mathbb{R} \rightarrow \mathbb{R}$ are there satisfying $f(x)f(y)f(z)=12f(xyz)-16xyz$ for every real $x,y,z$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 0 \qquad \textbf{(E)}\ \text{None}$

The number obtained from the last two nonzero digits of $ 90!$ is equal to $ n$. What is $ n$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 68$

For positive integers $ n$ let $ D(n)$ denote the set of positive integers that divide $ n$ and let $ S(n)\equal{}\Sigma_{k \in D(n)} \frac{1}{k}$. What is $ S(2006)$? Answer with a fraction reduced to lowest terms.

Upper content of a subset $E$ of the plane $\mathbb{R}^{2}$ is defined as $$\mathcal{C}(E)=\inf\{\sum_{i=1}^{n} \text{diam}(E_{i})\}$$ where $\inf$ is taken over all finite families of sets $E_{1},\dots,E_{n}$ $n\in \mathbb{N}$, in $\mathbb{R}^{2}$ such that $E\subset \bigcup_{i=1}^{n}E_{i}$. Lower content of $E$ is defined as $$\mathcal{K}(E)=\sup\{\text{length}(L) |\, L \text{ is a closed line segment onto which $E$ can be contracted}\}$$. Prove that i) $\mathcal{C}(L)=\text{length}(L)$ if $L$ is a closed line segment; ii) $\mathcal{C}(E) \geq \mathcal{K}(E)$; iii) the equality in ii) is not always true even if $E$ is compact.

A finite set of line segments, of total length $18$, belongs to a square of unit side length (we assume that the square includes its boundary and that a line segment includes its end points). The line segments are parallel to the sides of the square and may intersect one another. Prove that among the regions into which the square is divided by the line segments, at least one of these must have area not less than $0.01$. (A Berzinsh, Riga)

On the sides $AB, BC$ and $CA$ of triangle $ABC$, points $L, M$ and $N$ are chosen, respectively, such that the lines $CL, AM$ and $BN$ intersect at a common point O inside the triangle and the quadrilaterals $ALON, BMOL$ and $CNOM$ have incircles. Prove that $$\frac{1}{AL\cdot BM} +\frac{1}{BM\cdot CN} +\frac{1}{CN \cdot AL} =\frac{1}{AN\cdot BL} +\frac{1}{BL\cdot CM} +\frac{1}{CM\cdot AN} $$

Points $E,F$ are chosen on the sides $CA,AB$ of triangle $ABC$. Let $(K)$ be the circumcircle of triangle $AEF$. The tangents at $E, F$ of $(K)$ intersect at $T$ . Prove that (a) $T$ is on $BC$ if and only if $BE$ meets $CF$ at a point on the circle $(K)$, (b) $EF, PQ,BC$ are concurrent given that $BE$ meets $FT$ at $M, CF$ meets $ET$ at $N, AM$ and $AN$ intersects $(K)$ at $P,Q$ distinct from $A$. Trần Quang Hùng

[b]p7.[/b] An ant starts at the point $(0, 0)$. It travels along the integer lattice, at each lattice point choosing the positive $x$ or $y$ direction with equal probability. If the ant reaches $(20, 23)$, what is the probability it did not pass through $(20, 20)$? [b]p8.[/b] Let $a_0 = 2023$ and $a_n$ be the sum of all divisors of $a_{n-1}$ for all $n \ge 1$. Compute the sum of the prime numbers that divide $a_3$. [b]p9.[/b] Five circles of radius one are stored in a box of base length five as in the following diagram. How far above the base of the box are the upper circles touching the sides of the box? [img]https://cdn.artofproblemsolving.com/attachments/7/c/c20b5fa21fbd8ce791358fd888ed78fcdb7646.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Positive integer numbers a and b satisfy $(a^2- 9b^2)^2 - 33b = 1$. a) Prove $|a -3b|\ge 1$. b) Find all pairs of positive integers $(a, b)$ satisfying the equality.

Let $ ABC$ be a triangle, and $ D$ the midpoint of the side $ BC$. On the sides $ AB$ and $ AC$ we consider the points $ M$ and $ N$, respectively, both different from the midpoints of the sides, such that \[ AM^2\plus{}AN^2 \equal{}BM^2 \plus{} CN^2 \textrm{ and } \angle MDN \equal{} \angle BAC.\] Prove that $ \angle BAC \equal{} 90^\circ$.

For every positive integer $n$, let $\sigma(n)$ denote the sum of all positive divisors of $n$ ($1$ and $n$, inclusive). Show that a positive integer $n$, which has at most two distinct prime factors, satisfies the condition $\sigma(n)=2n-2$ if and only if $n=2^k(2^{k+1}+1)$, where $k$ is a non-negative integer and $2^{k+1}+1$ is prime.

An $ 8$-foot by $ 10$-foot floor is tiled with square tiles of size $ 1$ foot by $ 1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $ 1/2$ foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded? [asy]unitsize(2cm); defaultpen(linewidth(.8pt)); fill(unitsquare,gray); filldraw(Arc((0,0),.5,0,90)--(0,0)--cycle,white,black); filldraw(Arc((1,0),.5,90,180)--(1,0)--cycle,white,black); filldraw(Arc((1,1),.5,180,270)--(1,1)--cycle,white,black); filldraw(Arc((0,1),.5,270,360)--(0,1)--cycle,white,black);[/asy]$ \textbf{(A)}\ 80\minus{}20\pi \qquad \textbf{(B)}\ 60\minus{}10\pi \qquad \textbf{(C)}\ 80\minus{}10\pi \qquad \textbf{(D)}\ 60\plus{}10\pi \qquad \textbf{(E)}\ 80\plus{}10\pi$

A particle moves in a straight line with monotonically decreasing acceleration. It starts from rest and has velocity $v$ a distance $d$ from the start. What is the maximum time it could have taken to travel the distance $d$?

Let $\Phi$ and $\Psi$ denote the Euler totient and Dedekind‘s totient respectively. Determine all $n$ such that $\Phi(n)$ divides $n +\Psi (n)$.

A semicircle with diameter $d$ is contained in a square whose sides have length $8$. Given the maximum value of $d$ is $m- \sqrt{n}$, find $m+n$.

Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Let $k$ be the circle inscribed in convex quadrilateral $ABCD$. Lines $AD$ and $BC$ meet at $P$ ,and circumcircles of $\triangle PAB$ and $\triangle PCD$ meet in $X$ . Prove that tangents from $X$ to $k$ form equal angles with lines $AX$ and $CX$ .

A grandpa has a finite number of boxes in his attic. Each box is a straight rectangular prism with integer edge lengths. For every box its width is greater or equal to its height and its length is greater or equal to its width. A box can be put inside another box if and only if all of its dimensions are respectively smaller than the other one's. You can put two or more boxes in a bigger box only if the smaller boxes are all already inside one of the boxes. The grandpa decided to put the boxes in each other so that there would be a minimal number of visible boxes in the attic (boxes that have not been put inside another). He decided to use the following algorithm: at each step he finds the longest sequence of boxes so that the first can be put in the second, the second can be put in the third, etc., and then he puts them inside each other in the aforementioned order. The grandpa used the algorithm until no box could be put inside another. It is known that at each step the longest sequence of boxes was unique, e.g., at no moment were there two different sequences with the same length. The grandpa now claims that he has the minimal possible number of visible boxes in his attic. Is the claim necessarily true?

Consider two functions $f , \,g \,:\mathbb{R} \to \mathbb{R}$ such that from some $a>0$ holds $g(x)=f(x+a)$ for any $x \in \mathbb{R}$. If $f$ is even and $g$ is odd, prove that both functions are periodic.

Let there be $320$ points arranged on a circle, labeled $1$, $2$, $3$, $\dots$, $8$, $1$, $2$, $3$, $\dots$, $8$, $\dots$ in order. Line segments may only be drawn to connect points labelled with the same number. What the largest number of non-intersecting line segments one can draw? (Two segments sharing the same endpoint are considered to be intersecting).

The hexagon $ABLCDK$ is inscribed and the line $LK$ intersects the segments $AD, BC, AC$ and $BD$ in points $M, N, P$ and $Q$, respectively. Prove that $NL \cdot KP \cdot MQ = KM \cdot PN \cdot LQ$.

The sequence $(a_n)$ of complex numbers is considered in the complex plane, in which is: $$a_0 = 1, \,\,\, a_n = a_{n-1} +\frac{1}{n}(\cos 45^o + i \sin 45^o )^n.$$ Prove that the sequence of the real parts of the terms of $(a_n)$ is convergent and its limit is a number between $0.85$ and $1.15$.

A triangle has altitudes of length 5 and 7. What is the maximum length of the third altitude?

Given a triangle $ABC$, let $D$ be the point where the incircle of the triangle $ABC$ touches the side $BC$. A circle through the vertices $B$ and $C$ is tangent to the incircle of triangle $ABC$ at the point $E$. Show that the line $DE$ passes through the excentre of triangle $ABC$ corresponding to vertex $A$.

Compute the sum \[S=1+2+3-4-5+6+7+8-9-10+\dots-2010\] where every three consecutive $+$ are followed by two $-$.