Found problems: 85335
Solve the system of equations:
$$\begin{cases}\sqrt{\frac{y(t-y)}{t-x}-\frac4x}+\sqrt{\frac{z(t-z)}{t-x}-\frac4x}=\sqrt x\\\sqrt{\frac{z(t-z)}{t-y}-\frac4y}+\sqrt{\frac{x(t-x)}{t-y}-\frac4y}=\sqrt y\\\sqrt{\frac{x(t-x)}{t-z}-\frac4z}+\sqrt{\frac{y(t-y)}{t-z}-\frac4z}=\sqrt z\\x+y+z=2t\end{cases}$$
if the following conditions are satisfied: $0<x<t$, $0<y<t$, $0<z<t$.
[i]H. Lesov[/i]
Prove that all the heights of a tetrahedron intersect at one point if and only if the sums of the squares of the opposite edges are equal.
In isosceles triangle $ABC$ with $AB=AC$, consider point $D$ on the base $BC$ and point $E$ on side $AC$ such that $ \angle BAD = 2 \angle CDE$. Prove that $AD=AE$.
Let $n=3k$ be a positive integer (with $k\geq 2$). An equilateral triangle is divided in $n^2$ unit equilateral triangles with sides parallel to the initial, forming a grid. We will call "trapezoid" the trapezoid which is formed by three equilateral triangles (one base is equal to one and the other is equal to two). We colour the points of the grid with three colours (red, blue and green) such that each two neighboring points have different colour.
Finally, the colour of a "trapezoid" will be the colour of the midpoint of its big base.
Find the number of all "trapezoids" in the grid (not necessarily disjoint) and determine the number of red, blue and green "trapezoids".
How many ten-digit positive integers consist of ten different digits and are divisible by $99$?
Find all the continuous functions $f : \mathbb{R} \mapsto\mathbb{R}$ such that $\forall x,y \in \mathbb{R}$,
$(1+f(x)f(y))f(x+y)=f(x)+f(y)$.
$AD$ is the altitude on side $BC$ of triangle $ABC$. If $BC+AD-AB-AC = 0$, find the range of $\angle BAC$.
[i]Alternative formulation.[/i] Let $AD$ be the altitude of triangle $ABC$ to the side $BC$. If $BC+AD=AB+AC$, then find the range of $\angle{A}$.
For a given natural number, $ n\ge 2, $ consider two matrices $ A,B\in\mathcal{M}_n(\mathbb{C}) $ that commute and such that $ A $ is invertible and that the function $ M:\mathbb{C}\longrightarrow\mathbb{C} ,M(x)=\det (A+xB) $ is bounded above or below.
Prove that $ B^n=0. $
[i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]
[u]Round 5[/u]
[b]p13.[/b] Two closed disks of radius $\sqrt2$ are drawn centered at the points $(1,0)$ and $(-1, 0)$. Let P be the
region belonging to both disks. Two congruent non-intersecting open disks of radius $r$ have all of
their points in $P$ . Find the maximum possible value of $r$ .
[b]p14.[/b] A rectangle has positive integer side lengths. The sum of the numerical values of its perimeter and area is $2017$. Find the perimeter of the rectangle.
[b]p15.[/b] Find all ordered triples of real numbers $(a,b,c)$ which satisfy $$a +b +c = 6$$
$$a \cdot (b +c) = 6$$
$$(a +b) \cdot c = 6$$
[u]Round 6[/u]
[b]p16.[/b] A four digit positive integer is called confused if it is written using the digits $2$, $0$, $1$, and $7$ in some order, each exactly one. For example, the numbers $7210$ and $2017$ are confused. Find the sum of all confused numbers.
[b]p17.[/b] Suppose $\vartriangle ABC$ is a right triangle with a right angle at $A$. Let $D$ be a point on segment $BC$ such that $\angle BAD = \angle CAD$. Suppose that $AB = 20$ and $AC = 17$. Compute $AD$.
[b]p18.[/b] Let $x$ be a real number. Find the minimum possible positive value of $\frac{|x -20|+|x -17|}{x}$.
[u]Round 7[/u]
[b]p19.[/b] Find the sum of all real numbers $0 < x < 1$ that satisfy $\{2017x\} = \{x\}$.
[b]p20.[/b] Let $a_1,a_2, ,,, ,a_{10}$ be real numbers which sum to $20$ and satisfy $\{a_i\} <0.5$ for $1 \le i\le 10$. Find the sum of all possible values of $\sum_{ 1 \le i <j\le 10} \lfloor a_i +a_j \rfloor .$
Here, $\lfloor x \rfloor$ denotes the greatest integer $x_0$ such that $x_0 \le x$ and $\{x\} =x -\lfloor x \rfloor$.
[b]p21.[/b] Compute the remainder when $20^{2017}$ is divided by $17$.
[u]Round 8[/u]
[b]p22.[/b] Let $\vartriangle ABC$ be a triangle with a right angle at $B$. Additionally, letM be the midpoint of $AC$. Suppose the circumcircle of $\vartriangle BCM$ intersects segment $AB$ at a point $P \ne B$. If $CP = 20$ and $BP = 17$, compute $AC$.
[b]p23.[/b] Two vertices on a cube are called neighbors if they are distinct endpoints of the same edge. On a cube, how many ways can a nonempty subset $S$ of the vertices be chosen such that for any vertex $v \in S$, at least two of the three neighbors of $v$ are also in $S$? Reflections and rotations are considered distinct.
[b]p24.[/b] Let $x$ be a real number such that $x +\sqrt[4]{5-x^4}=2$. Find all possible values of $x\sqrt[4]{5-x^4}$.
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3158491p28715220]here[/url].and 9-12 [url=https://artofproblemsolving.com/community/c3h3162362p28764144]here[/url] Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Prove or disprove that there exist two tetrahedra $T_1$ and $T_2$ such that:
(i) the volume of $T_1$ is greater than that of $T_2$;
(ii) the area of any face of $T_1$ does not exceed the area of any face of $T_2$.
Problem-2:
Given a circle with center $O$, the two tangent lines from a point $S$ outside the circle touch the circle at points $P$ and $Q$. Line $SO$ intersects the circle at $A$ and $B$, with $B$ closer to $S$. Let $X$ be an interior point of minor arc $PB$, and let line $OS$ intersect lines $QX$ and $PX$ at $C$ and $D$, respectively. Prove that
$\frac{1}{\left| AC \right|}+\frac{1}{\left| AD \right|}=\frac{2}{\left| AB \right|}$.
Alice and Bob are playing a year number game.
There will be two game numbers $19$ and $20$ and one starting number from the set $\{9, 10\}$ used. Alice chooses independently her game number and Bob chooses the starting number. The other number is given to Bob. Then Alice adds her game number to the starting number, Bob adds his game number to the result, Alice adds her number of games to the result, etc. The game continues until the number $2019$ is reached or exceeded.
Whoever reaches the number $2019$ wins. If $2019$ is exceeded, the game ends in a draw.
$\bullet$ Show that Bob cannot win.
$\bullet$ What starting number does Bob have to choose to prevent Alice from winning?
(Richard Henner)
Determine the largest value the expression
$$ \sum_{1\le i<j\le 4} \left( x_i+x_j \right)\sqrt{x_ix_j} $$
may achieve, as $ x_1,x_2,x_3,x_4 $ run through the non-negative real numbers, and add up to $ 1. $ Find also the specific values of this numbers that make the above sum achieve the asked maximum.
Let $\,S\,$ be a finite set of points in three-dimensional space. Let $\,S_{x},\,S_{y},\,S_{z}\,$ be the sets consisting of the orthogonal projections of the points of $\,S\,$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that \[ \vert S\vert^{2}\leq \vert S_{x} \vert \cdot \vert S_{y} \vert \cdot \vert S_{z} \vert, \] where $\vert A \vert$ denotes the number of elements in the finite set $A$.
[hide="Note"] Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane. [/hide]
Let $a, b, c, x$ be reals with $(a + b)(b + c)(c + a) \ne 0$ that satisfy
$$\frac{a^2}{a + b}=\frac{a^2}{a + c}+ 20, \,\,\, \frac{b^2}{b + c}=\frac{b^2}{b + a}+ 14, \text{and}\,\,\, \frac{c^2}{c + a}=\frac{c^2}{c + b}+ x.$$
Compute $x$.
Simplify the product
\[
\prod_{m=1}^{100}\prod_{n=1}^{100}\frac{x^{n+m}+x^{n+m+2}+x^{2n+1}+x^{2m+1}}{x^{2n}+2x^{n+m}+x^{2m}}
\]
Express your answer in terms of $x$.
Given two intersecting planes $\alpha$ and $\beta$ and a point $A$ on the line of their intersection. Prove that of all lines belonging to $\alpha$ and passing through $A$ the line which is perpendicular to the intersection line of $\alpha$ and $\beta$ forms the greatest angle with $\beta$.
For any function $f: \mathbb{R}^2\to \mathbb{R}$ consider the function $\Phi_f:\mathbb{R}^2\to [-\infty,\infty]$ for which $\Phi_f(x,y)=\limsup_{ z \to y} f(x,z)$ for any $(x,y) \in \mathbb{R}^2$.
[list=a]
[*]Is it true that if $f$ is Lebesgue measurable then $\Phi_f$ is also Lebesgue measurable?[/*]
[*]Is it true that if $f$ is Borel measurable then $\Phi_f$ is also Borel measurable?[/*]
[/list]
For each positive integer $k$, let $a_k$ be the greatest integer not exceeding $\sqrt{k}$ and let $b_k$ be the greatest integer not exceeding $\sqrt[3]{k}$. Calculate $$\sum_{k=1}^{2003} (a_k-b_k).$$
A sequence $ a_1, a_2, \ldots$ of non-negative integers is defined by the rule $ a_{n \plus{} 2} \equal{} |a_{n \plus{} 1} \minus{} a_n|$ for $ n\ge 1$. If $ a_1 \equal{} 999, a_2 < 999,$ and $ a_{2006} \equal{} 1$, how many different values of $ a_2$ are possible?
$ \textbf{(A) } 165 \qquad \textbf{(B) } 324 \qquad \textbf{(C) } 495 \qquad \textbf{(D) } 499 \qquad \textbf{(E) } 660$
For every subset $\mathcal{P}$ of the plane let $S(\mathcal{P})$ denote the set of circles and lines that intersect $\mathcal{P}$ in at least three points. Find all sets $\mathcal{P}$ consisting of 2024 points such that for any two distinct elements of $S(\mathcal{P}),$ their intersection points all belong to $\mathcal{P}{}.$
Quadrilateral $ABCD$ is circumscribed about a circle $\Gamma$ and $K,L,M,N$ are points of tangency of sides $AB,BC,CD,DA$ with $\Gamma$ respectively. Let $S\equiv KM\cap LN$. If quadrilateral $SKBL$ is cyclic then show that $SNDM$ is also cyclic.
Find all primes $p$ such that there exist positive integers $q$ and $r$ such that $p \nmid q$, $3 \nmid q$, $p^3 = r^3 - q^2$.
Which figure can be the images of equations $mx+ny^2=0$ and $mx^2+ny^2=1$$(m,n\neq0)$?
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZC9kLzUzZTAwZjU1YzEyN2I3ZDJjNjcwNDQ2ZmQ5MDBmYWZlODAwNGU0LnBuZw==&rn=YWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYS5wbmc=[/img]
Given an acute triangle $ABC$ let $M$ be the midpoint of $AB$. Point $K$ is given on the other side of line $AC$ from that of point $B$ such that $\angle KMC = 90 ^ \circ $ and $\angle KAC = 180^\circ - \angle ABC$. The tangent to circumcircle of triangle $ABC$ at $A$ intersects line $CK$ at $E$. Prove that the reflection of line $BC$ with respect to $CM$ passes through the midpoint of line segment $ME$.