Found problems: 85335
Define $f(x)=|| x|-\tfrac{1}{2}|$ and $g(x)=|| x|-\tfrac{1}{4}|$. Find the number of intersections of the graphs of $$y=4 g(f(\sin (2 \pi x))) \quad\text{ and }\quad x=4 g(f(\cos (3 \pi y))).$$
Let $p$ be a prime number. Find all polynomials $P$ with integer coefficients with the following properties:
$(a)$ $P(x)>x$ for all positive integers $x$.
$(b)$ The sequence defined by $p_0:=p$, $p_{n+1}:=P(p_n)$ for all positive integers $n$, satisfies the property that for all positive integers $m$ there exists some $l\geq 0$ such that $m\mid p_l$.
In a circle, two non-intersecting chords $AB,CD$ are drawn.On the chord $AB$,a point $E$ (different from $A$,$B$) is taken Consider the arc $AB$ that does not contain the points $C,D$. With a compass and a straighthedge, find all possible point $F$ on that arc such that $\dfrac{PE}{EQ}=\dfrac{1}{2}$, where $P$ and $Q$ are the points in which the chord $AB$ meets the segment $FC$ and $FD$.
[i](tatari/nightmare)[/i]
Let $ABC$ be a triangle and let $D, E$ and $F$ be the midpoints of the sides $AB, BC$ and $CA$ respectively. Suppose that the angle bisector of $\angle BDC$ meets $BC$ at the point $M$ and the angle bisector of $\angle ADC$ meets $AC$ at the point $N$. Let $MN$ and $CD$ intersect at $O$ and let the line $EO$ meet $AC$ at $P$ and the line $FO$ meet $BC$ at $Q$. Prove that $CD = PQ$.
Map the interior of the unit disc conformally onto the first quadrant.
The sequence $\{x_n\}$ is defined by $x_1=5$ and $x_{k+1}=x_k^2-3x_k+3$ for $k=1,2,3\cdots$. Prove that $x_k>3^{2^{k-1}}$ for any positive integer $k$.
An ordered pair of integers $(m,n)$ with $1<m<n$ is said to be a [i]Benelux couple[/i] if the following two conditions hold: $m$ has the same prime divisors as $n$, and $m+1$ has the same prime divisors as $n+1$.
(a) Find three Benelux couples $(m,n)$ with $m\leqslant 14$.
(b) Prove that there are infinitely many Benelux couples
Given are on a line three points $A, B, C$ such that $AB = 1$ and $BC = x$. Consider the circles $\Omega_a, \Omega_b$ and $\Omega_c$ which are tangent to the given line at the points $A, B, C$ respectively, and such that $\Omega_b$ is tangent externally with both $\Omega_a$ and $\Omega_c$ in points $M, N$ respectively. Find all values of the radius of the circle $\Omega_b$ for which the triangle $BMN$ is isosceles.
Given the triangle $ABC$, let $L$, $M$ and $N $be the midpoints of $BC$, $CA$ and $AB$ respectively. The lines $LM$ and $LN$ cut the tangent to the circumcircle at $A$ at $P$ and $Q$ respectively . Prove that $CP \parallel BQ$.
In quadrilateral $ABCD$, $AB=7, BC=24, CD=15, DA=20,$ and $AC=25$. Let segments $AC$ and $BD$ intersect at $E$. What is the length of $EC$?
[i]Proposed by James Lin[/i]
Define a $good~word$ as a sequence of letters that consists only of the letters $A,$ $B,$ and $C$ $-$ some of these letters may not appear in the sequence $-$ and in which $A$ is never immediately followed by $B,$ $B$ is never immediately followed by $C,$ and $C$ is never immediately followed by $A.$ How many seven-letter good words are there?
Let Pascal’s triangle be constructed where each $\tbinom{n}{i}$ is written inside its own cell in row $n.$ Colby colors the cells red for $1 \le n \le 63$ when $\tbinom{n}{i}$ is divisible by $4.$ How many cells does he color red?
Which convex domains (figures) on a plane can contain an entire straight line?
It is assumed that the figure is flat and does not degenerate into a straight line and is closed, that is, it contains all its boundary points.
Let $ABC$ be a right triangle with right angle at $B$. Let $ACDE$ be a square drawn exterior to triangle $ABC$. If $M$ is the center of this square, find the measure of $\angle MBC$.
Show that there exists an equiangular hexagon in the plane, whose sides measure 5,8,11,14,23, and 29 units in some order.
Let $ABC$ be a triangle. $I$ is the incenter of $\Delta ABC$ and $D$ is the meet point of $AI$ and the circumcircle of $\Delta ABC$. Let $E,F$ be on $BD,CD$, respectively such that $IE,IF$ are perpendicular to $BD,CD$, respectively. If $IE+IF=\frac{AD}{2}$, find the value of $\angle BAC$.
Let $ABCDEF$ be a convex hexagon with $AB=DE$, $BC=EF$, $CD=FA$, and $\angle A-\angle D = \angle C -\angle F = \angle E -\angle B$. Prove that the diagonals $AD$, $BE$, and $CF$ are concurrent.
A non-equilateral triangle $ABC$ is inscribed in a circle $\Gamma$ with centre $O$, radius $R$ and its incircle has centre $I$ and touches $BC,CA,AB$ at $D,E, F$, respectively. A circle with centre $I$ and radius $r$ intersects the rays $[ID), [IE), [IF)$ at $A',B',C'$. Show that the orthocentre $K$ of $\vartriangle A'B'C'$ is on the line $OI$ and that $\frac{IK}{IO}=\frac{r}{R}$
Michel Bataille
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Suppose that $1985$ points are given inside a unit cube. Show that one can always choose $32$ of them in such a way that every (possibly degenerate) closed polygon with these points as vertices has a total length of less than $8 \sqrt 3.$
A cyclist travels at a constant speed of $\text{22.0 km/hr}$ except for a $20$ minute stop. The cyclist’s average speed was $\text{17.5 km/hr}$. How far did the cyclist travel?
(A) $\text{28.5 km}$
(B) $\text{30.3 km}$
(C) $\text{31.2 km}$
(D) $\text{36.5 km}$
(E) $\text{38.9 km}$
For a positive integer $n$, we define the function $f_n(x)=\sum_{k=1}^n |x-k|$ for all real numbers $x$. For any two-digit number $n$ (in decimal representation), determine the set of solutions $\mathbb{L}_n$ of the inequality $f_n(x)<41$.
[i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 2)[/i]
There are $2019$ coins on a table. Some are placed with head up and others tail up. A group of $2019$ persons perform the following operations: the first person chooses any one coin and then turns it over, the second person choses any two coins and turns them over and so on and the $2019$-th person turns over all the coins. Prove that no matter which sides the coins are up initially, the $2019$ persons can come up with a procedure for turning the coins such that all the coins have smae side up at the end of the operations.
Find all functions $f: \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that for all positive integers $m,n$ with $m\ge n$, $$f(m\varphi(n^3)) = f(m)\cdot \varphi(n^3).$$
Here $\varphi(n)$ denotes the number of positive integers coprime to $n$ and not exceeding $n$.
Which pair of numbers does NOT have a product equal to $36$?
$\text{(A)}\ \{ -4,-9\} \qquad \text{(B)}\ \{ -3,-12\} \qquad \text{(C)}\ \left\{ \dfrac{1}{2},-72\right\} \qquad \text{(D)}\ \{ 1,36\} \qquad \text{(E)}\ \left\{\dfrac{3}{2},24\right\}$
Given a regular hexagonal prism. Find a polygonal line that, starting from a vertex of the base, runs through all the lateral faces and ends at the vertex of the face top, located on the same edge as the starting vertex, and has a minimum length.