Found problems: 85335
Determine all natural numbers $x, y$ and $z$ such that the number $2^x +4^y +8^z +16^2$ is a power of $2$.
A billiards table is in the figure of regular hexagon $ABCDEF$. $P$ is the midpoint of $AB$. We shut the ball at $P$, then it touches $Q$ on side $BC$, then it touches side $CD,DE,EF,FA$. Finally, the ball touches side $AB$ again. Let $\theta=\angle BPQ$, find the value range of $\theta$.
If $ f(a)\equal{}a\minus{}2$ and $ F(a,b)\equal{}b^2\plus{}a$, then $ F(3,f(4))$ is:
$ \textbf{(A)}\ a^2\minus{}4a\plus{}7 \qquad\textbf{(B)}\ 28 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 11$
A student of the National Technical University was reading advanced mathematics last summer for 37 days according to the following rules :
(a) He was reading at least one hour every day.
(b) He was reading an integer number of hours, but not more than 12, each day.
(c) He had to read at most 60 hours in total.
Prove that there were some successive days during which the student was reading exactly 13 hours in total.
Let $S = \{1,2,3,4,5,6,7\}$. Compute the number of sets of subsets $T = \{A, B, C\}$ with $A, B, C \in S$ such that $A \cup B \cup C = S$, $(A \cap C) \cup (B \cap C) = \emptyset$, and no subset contains two consecutive integers.
Points $A, M_1, M_2$ and $C$ are on a line in this order. Let $k_1$ the circle with center $M_1$ passing through $A$ and $k_2$ the circle with center $M_2$ passing through $C$. The two circles intersect at points $E$ and $F$. A common tangent of $k_1$ and $k_2$, touches $k_1$ at $B$ and $k_2$ at $D$. Show that the lines $AB, CD$ and $EF$ intersect at one point.
Let $ABC$ be a triangle, $H$ its orthocenter, $O$ its circumcenter, and $R$ its circumradius. Let $D$ be the reflection of the point $A$ across the line $BC$, let $E$ be the reflection of the point $B$ across the line $CA$, and let $F$ be the reflection of the point $C$ across the line $AB$. Prove that the points $D$, $E$ and $F$ are collinear if and only if $OH=2R$.
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$$
f\left(x^2\right) - yf(y) = f(x + y) (f(x) - y)
$$
for all real numbers $x$ and $y$.
Find the number of points $P \in Z^2$ that satisfy the following two conditions:
1) If $Q$ is a point on the circle of radius $\sqrt{2020}$ centered at the origin such that the line $PQ$ is tangent to the circle at $Q$, then $PQ$ has integral length.
2) The x-coordinate of $P$ is $38$.
The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at $D, E$, and $F$ respectively. Let $\omega_a, \omega_b$ and $\omega_c$ be the circumcircles of triangles $EAF, DBF$, and $DCE$, respectively. The lines $DE$ and $DF$ cut $\omega_a$ at $E_a\neq{E}$ and $F_a\neq{F}$, respectively. Let $r_A$ be the line $E_{a}F_a$. Let $r_B$ and $r_C$ be defined analogously. Show that the lines $r_A$, $r_B$, and $r_C$ determine a triangle with its vertices on the sides of triangle $ABC$.
Two players, Aurelio and Bernardo, play the following game. Aurelio begins by writing the number $1$. Next it is Bernardo's turn, who writes number $2$. From then on, each player chooses whether to add $1$ to the number just written by the previous player, or whether multiply that number by $2$. Then write the result and it's the other player's turn. The first player to write a number greater than $ 2007$ loses the game. Determine if one of the players can ensure victory no matter what the other does.
[b]9.[/b] Let $w=f(x)$ be regular in $ \left | z \right |\leq 1$. For $0\leq r \leq 1$, denote by c, the image by $f(z)$ of the circle $\left | z \right | = r$. Show that if the maximal length of the chords of $c_{1}$ is $1$, then for every $r$ such that $0\leq r \leq 1$, the maximal length of the chords of c, is not greater than $r$. [b](F. 1)[/b]
A positive integer $n$ is said to be a [i]perfect power[/i] if $n=a^b$ for some integers $a,b$ with $b>1$.
$(\text{a})$ Find $2004$ perfect powers in arithmetic progression.
$(\text{b})$ Prove that perfect powers cannot form an infinite arithmetic progression.
[b]p1[/b]. Your Halloween took a bad turn, and you are trapped on a small rock above a sea of lava. You are on rock $1$, and rocks $2$ through $12$ are arranged in a straight line in front of you. You want to get to rock $12$. You must jump from rock to rock, and you can either (1) jump from rock $n$ to $n + 1$ or (2) jump from rock $n$ to $n + 2$. Unfortunately, you are weak from eating too much candy, and you cannot do (2) twice in a row. How many different sequences of jumps will take you to your destination?
[b]p2.[/b] Find the number of ordered triples $(p; q; r)$ such that $p, q, r$ are prime, $pq + pr$ is a perfect square and $p + q + r \le 100$.
[b]p3.[/b] Let $x, y, z$ be nonzero complex numbers such that $\frac{1}{x}+\frac{1}{y} + \frac{1}{z} \ne 0$ and
$$x^2(y + z) + y^2(z + x) + z^2(x + y) = 4(xy + yz + zx) = -3xyz.$$ Find $\frac{x^3 + y^3 + z^3}{x^2 + y^2 + z^2}$ .
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $ABC$ be an acute triangle with $AB\neq AC$. The angle bisector of $\angle BAC$ intersects with $BC$ at a point $D$. $BE,CF$ are the altitudes of the triangle and $Ap_1,Ap_2$ are the isodynamic points of triangle $ABC$.Let the $A$-median of $ABC$ intersect $EF$ at $T$. Show that the line connecting $T$ with the nine-point center of $ABC$ is perpendicular to $BC$ if and only if $\angle Ap_1DAp_2=90^\circ$.
Let $a$ be the largest root of the equation $x^3 - 3x^2 + 1 = 0$.
Find the first $200$ decimal digits for the number $a^{2000}$.
Let $ABCD$ be a convex quadrilateral. The perpendicular bisectors of the sides $[AD]$ and $[BC]$ intersect at a point $P$ inside the quadrilateral and the perpendicular bisectors of the sides $[AB]$ and $[CD]$ also intersect at a point $Q$ inside the quadrilateral. Show that, if $\angle APD = \angle BPC$ then $\angle AQB = \angle CQD$
Given a positive integer $n$, we define $\lambda (n)$ as the number of positive integer solutions of $x^2-y^2=n$. We say that $n$ is [i]olympic[/i] if $\lambda (n) = 2021$. Which is the smallest olympic positive integer? Which is the smallest olympic positive odd integer?
In the garden of Wonderland, there are $2016$ apples, $2017$ bananas and $2018$ oranges.Two monkeys Adu and Bakar play the following game: alternatively each of them takes and eats one fruit of any kind except for the one that he took in previous turn (in the first turn, each of them can take a fruit of any kind). Who can not take a fruit is the loser. Which monkey has the winning strategy if Adu plays first?
Does there exist a sequence of natural numbers in which every natural number occurs exactly once, such that for each $k = 1, 2, 3, \dots$ the sum of the first $k$ terms of the sequence is divisible by $k$?
[i]A. Shapovalov[/i]
Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$, and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$, evaluate the angle $NMB$.
Let $ABC$ be a triangle with $AB = 26$, $BC = 51$, and $CA = 73$, and let $O$ be an arbitrary point in the interior of $\vartriangle ABC$. Lines $\ell_1$, $\ell_2$, and $\ell_3$ pass through $O$ and are parallel to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. The intersections of $\ell_1$, $\ell_2$, and $\ell_3$ and the sides of $\vartriangle ABC$ form a hexagon whose area is $A$. Compute the minimum value of $A$.
Find all pairs of positive integers (m, n) for which it is possible to
paint each unit square of an m*n chessboard either black or white
in such way that, for any unit square of the board, the number
of unit squares which are painted the same color as that square
and which have at least one common vertex with it (including the
square itself) is even.
Suzie flips a fair coin 6 times. The probability that Suzie flips 3 heads in a row but not 4 heads in a row is given by $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Determine all functions $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{}$ which satisfy \[ f \left(\frac {f(x)}{yf(x) \plus{} 1}\right) \equal{} \frac {x}{xf(y)\plus{}1} \quad \forall x,y > 0\]