Found problems: 85335
Prove that there is no positive integers $x,y,z$ satisfying
\[ x^2 y^4 - x^4 y^2 + 4x^2 y^2 z^2 +x^2 z^4 -y^2 z^4 =0 \]
[b]p16.[/b] When not writing power rounds, Eric likes to climb trees. The strength in his arms as a function of time is $s(t) = t^3 - 3t^2$. His climbing velocity as a function of the strength in his arms is $v(s) = s^5 + 9s^4 + 19s^3 - 9s^2 - 20s$. At how many (possibly negative) points in time is Eric stationary?
[b]p17[/b]. Consider a triangle $\vartriangle ABC$ with angles $\angle ACB = 60^o$, $\angle ABC = 45^o$. The circumcircle around $\vartriangle ABH$, where $H$ is the orthocenter of $\vartriangle ABC$, intersects $BC$ for a second time in point $P$, and the center of that circumcircle is $O_c$. The line $PH$ intersects $AC$ in point $Q$, and $N$ is center of the circumcircle around $\vartriangle AQP$. Find $\angle NO_cP$.
[b]p18.[/b] If $x, y$ are positive real numbers and $xy^3 = \frac{16}{9}$ , what is the minimum possible value of $3x + y$?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
A square with a side 1 is colored in 3 colors. What’s the greatest real number $a$ such that there can always be found 2 points of the same color at a distance $a$?
Let $\mathbb N = {1, 2, 3, . . .}$. For real $x, y$, set $S(x, y) = \{s | s = [nx+y], n \in \mathbb N\}$. Prove that if $r > 1$ is a rational number, there exist real numbers $u$ and $v$ such that
\[S(r, 0) \cap S(u, v) = \emptyset, S(r, 0) \cup S(u, v) = \mathbb N.\]
A $3\times 3\times 3$ cube is made of 27 unit cube pieces. Each piece contains a lamp, which can be on or off. Every time a piece is pressed (the center piece cannot be pressed), the state of that piece and the pieces that share a face with it changes. Initially all lamps are off. Determine which of the following states are achievable:
(1) All lamps are on.
(2) All lamps are on except the central one.
(3) Only the central lamp is on.
Given two sides of the triangle. Construct that triangle, if medians to those sides are orthogonal.
A quadrilateral is rotated clockwise, and the sides are extended its length in the direction of the movement. It turns out the endpoints of the segments constructed form a square. Prove the initial quadrilateral must also be a square.
[b]Generalization:[/b]
Prove that if the same process is applied to any $n$-gon and the result is a regular $n$-gon, then the intial $n$-gon must also be regular.
Let $a_0=2$ and for $n\geq 1$, $a_n=\frac{\sqrt3 a_{n-1}+1}{\sqrt3-a_{n-1}}$. Find the value of $a_{2002}$ in the form $p+q\sqrt3$ where $p$ and $q$ are rational numbers
Let $ABCD$ be a convex plane quadrilateral and let $A_1$ denote the circumcenter of $\triangle BCD$. Define $B_1, C_1,D_1$ in a corresponding way.
(a) Prove that either all of $A_1,B_1, C_1,D_1$ coincide in one point, or they are all distinct. Assuming the latter case, show that $A_1$, C1 are on opposite sides of the line $B_1D_1$, and similarly,$ B_1,D_1$ are on opposite sides of the line $A_1C_1$. (This establishes the convexity of the quadrilateral $A_1B_1C_1D_1$.)
(b) Denote by $A_2$ the circumcenter of $B_1C_1D_1$, and define $B_2, C_2,D_2$ in an analogous way. Show that the quadrilateral $A_2B_2C_2D_2$ is similar to the quadrilateral $ABCD.$
Let $ABC$ be an acute triangle, $H$ its orthocentre, $D$ a point on the side $[BC]$, and $P$ a point such that $ADPH$ is a parallelogram.
Show that $\angle BPC > \angle BAC$.
Prove that \[\displaystyle \sum_{\{i,j,k\}=\{1,2,3\}} \csc ^{13} \frac{2^i \pi}{7}\csc ^{14} \frac{2^j \pi}{7}\csc ^{2005} \frac{2^k\pi}{7}\] is rational. Here, $(i,j,k)$ is summed over all possible permutations of $(1,2,3)$.
A positive integer $n$ is said to be base-able if there exists positive integers $a$ and $b,$ with $b>1,$ such that $n=a^b.$ How many positive integer divisors of $729000000$ are base-able?
[i]Proposed by Kyle Lee[/i]
The segment $AB$ is the diameter of the circle. The points $M$ and $C$ belong to this circle and are located in different half-planes relative to the line $AB$. From the point $M$ the perpendiculars $MN$ and $MK$ are drawn on the lines $AB$ and $AC$, respectively. Prove that the line $KN$ intersects the segment $CM$ in its midpoint.
(Igor Nagel)
Given a positive integer $ n $, let $ A, B $ be two co-prime positive integers such that $$ \frac{B}{A} = \left(\frac{n\left(n+1\right)}{2}\right)!\cdot\prod\limits_{k=1}^{n}{\frac{k!}{\left(2k\right)!}} $$ Prove that $ A $ is a power of $ 2 $.
Given is a convex quadrilateral $ ABCD$ with $ \angle ADC\equal{}\angle BCD>90^{\circ}$.
Let $ E$ be the point of intersection of the line $ AC$ with the parallel line to $ AD$ through $ B$ and $ F$ be the point of intersection of the line $ BD$ with the parallel line to $ BC$ through $ A$. Show that $ EF$ is parallel to $ CD$
Jimmy invites Kima, Lester, Marlo, Namond, and Omar to dinner. There are nine chairs at Jimmy's round dinner table. Jimmy sits in the chair nearest the kitchen. How many different ways can Jimmy's five dinner guests arrange themselves in the remaining $8$ chairs at the table if Kima and Marlo refuse to be seated in adjacent chairs?
Let $A_n$ be the area outside a regular $n$-gon of side length $1$ but inside its circumscribed circle, let $B_n$ be the area inside the $n$-gon but outside its inscribed circle. Find the limit as $n$ tends to infinity of $\dfrac{A_n}{B_n}$.
In a class with $23$ students, each pair of students have watched a movie together. Let the set of movies watched by a student be his [i]movie collection[/i]. If every student has watched every movie at most once, at least how many different movie collections can these students have?
Find the arithmetic mean of all the three-digit palindromes. (Recall that a palindrome is a number that reads the same forward and backward, such as $777$ or $383$.)
For $n$ distinct positive integers all their $n(n-1)/2$ pairwise sums are considered. For each of these sums Ivan has written on the board the number of original integers which are less than that sum and divide it. What is the maximum possible sum of the numbers written by Ivan?
Let $ABIH$,$BDEC$ and $ACFG$ be arbitrary rectangles constructed (externally) on the sides of triangle $ABC$.Choose point $S$ outside rectangle $ABIH$ (on the opposite side as triangle $ABC$) such that $\angle SHI=\angle FAC$ and $\angle HIS=\angle EBC$.Prove that the lines $FI,EH$ and $CS$ are concurrent(i.e., the three lines intersect in one point).
In the Cartesian plane, a perfectly reflective semicircular room is bounded by the upper half of the unit circle centered at $(0,0)$ and the line segment from $(-1,0)$ to $(1,0)$. David stands at the point $(-1,0)$ and shines a flashlight into the room at an angle of $46^{\circ}$ above the horizontal. How many times does the light beam reflect off the walls before coming back to David at $(-1,0)$ for the first time?
Let $P$ and $Q$ be the midpoints of non-parallel chords $k_1$ and $k_2$ of a circle $\omega$, respectively. Let the tangent lines of $\omega$ passing through the endpoints of $k_1$ intersect at $A$ and the tangent lines passing through the endpoints of $k_2$ intersect at $B$. Let the symmetric point of the orthocenter of triangle $ABP$ with respect to the line $AB$ be $R$ and let the feet of the perpendiculars from $R$ to the lines $AP, BP, AQ, BQ$ be $R_1, R_2, R_3, R_4$, respectively. Prove that
\[ \frac{AR_1}{PR_1} \cdot \frac{PR_2}{BR_2} = \frac{AR_3}{QR_3} \cdot \frac{QR_4}{BR_4} \]
Find all prime number $p$ such that there exists an integer-coefficient polynomial $f(x)=x^{p-1}+a_{p-2}x^{p-2}+…+a_1x+a_0$ that has $p-1$ consecutive positive integer roots and $p^2\mid f(i)f(-i)$, where $i$ is the imaginary unit.
Let $P(x) \in \mathbb {Z}[X]$ be a polynomial of degree $2016$ with no rational roots. Prove that there exists a polynomial $T(x) \in \mathbb {Z}[X]$ of degree $1395$ such that for all distinct (not necessarily real) roots of $P(x)$ like $(\alpha ,\beta):$
$$T(\alpha)-T(\beta) \not \in \mathbb {Q}$$
Note: $\mathbb {Q}$ is the set of rational numbers.