Found problems: 85335
Given a non-negative integer \( n \), determine the values of \( c \) for which the sequence of numbers
\[
a_n = 4^n c + \frac{4^n - (-1)^n}{5}
\]
contains at least one perfect square.
Find all the pairs of integers $(x,y)$ for which $(x^2+y)(y^2+x)=(x+1)(y+1).$
Does there exist a sequence $\displaystyle{\left( {{a_n}} \right)}$ of complex numbers such that for every positive integer $\displaystyle{p}$ we have that $\displaystyle{\sum\limits_{n = 1}^{ + \infty } {a_n^p} }$ converges if and only if $\displaystyle{p}$ is not a prime?
[i]Proposed by Tomáš Bárta, Charles University, Prague.[/i]
Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$.
Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$
The real numbers $a,b,c,d$ satisfy the equations:
$$\begin{cases} a =\sqrt{45-\sqrt{21-a}} \\ b =\sqrt{45+\sqrt{21-b}}\\ c =\sqrt{45-\sqrt{21+c}}\ \\ d=\sqrt{45+\sqrt{21+d}} \end {cases}$$
Prove that $abcd = 2004$.
Let $ H$ be the orthocenter, $ O$ the circumcenter, and $ R$ the circumradius of an acute-angled triangle $ ABC$. Consider the circles $ k_a,k_b,k_c,k_h,k$, all with radius $ R$, centered at $ A,B,C,H,M,$ respectively. Circles $ k_a$ and $ k_b$ meet at $ M$ and $ F$; $ k_a$ and $ k_c$ meet at $ M$ and $ E$; and $ k_b$ and $ k_c$ meet at $ M$ and $ D$.
$ (a)$ Prove that the points $ D,E,F$ lie on the circle $ k_h$.
$ (b)$ Prove that the set of the points inside $ k_h$ that are inside exactly one of the circles $ k_a,k_b,k_c$ has the area twice the area of $ \triangle ABC$.
[b]Form 1[/b]
Consider the statements:
$\textbf{(1)}\ p\text{ } \wedge\sim q\wedge r \qquad
\textbf{(2)}\ \sim p\text{ } \wedge\sim q\wedge r\qquad
\textbf{(3)}\ p\text{ } \wedge\sim q\text{ }\wedge \sim r \qquad
\textbf{(4)}\ \sim p\text{ } \wedge q\text{ }\wedge r $,
where $p,q,$ and $r$ are propositions. How many of these imply the truth of $(p\rightarrow q)\rightarrow r$?
$\textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1\qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 4$
[b]Form 2[/b]
Consider the statements $(1)$ $p$ and $r$ are true and $q$ is false $(2)$ $r$ is true and $p$ and $q$ are false $(3)$ $p$ is true and $q$ and $r$ are false $(4)$ $q$ and $r$ are true and $p$ is false. How many of these imply the truth of the statement
"$r$ is implied by the statement that $p$ implies $q$"?
$\textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1\qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 4$
Pete has some trouble slicing a 20-inch (diameter) pizza. His first two cuts (from center to circumference of the pizza) make a 30º slice. He continues making cuts until he has gone around the whole pizza, each time trying to copy the angle of the previous slice but in fact adding 2º each time. That is, he makes adjacent slices of 30º, 32º, 34º, and so on. What is the area of the smallest slice?
The function $f$ is increasing and convex (i.e. every straight line between two points on the graph of $f$ lies above the graph) and satisfies $f(f(x))=3^x$ for all $x\in\mathbb{R}$. If $f(0)=0.5$ determine $f(0.75)$ with an error of at most $0.025$. The following are corrent to the number of digits given:
\[3^{0.25}=1.31607,\quad 3^{0.50}=1.73205,\quad 3^{0.75}=2.27951.\]
Let $ABCD$ be a cyclic quadrilateral and let the lines $CD$ and $BA$ meet at $E$. The line through $D$ which is tangent to the circle $ADE$ meets the line $CB$ at $F$. Prove that triangle $CDF$ is isosceles.
For positive integers $n,d$, define $n \% d$ to be the unique value of the positive integer $r < d$ such that $n = qd + r$, for some positive integer $q$. What is the smallest value of $n$ not divisible by $5,7,11,13$ for which $n^2 \% 5 < n^2 \% 7 < n^2 \% 11 < n^2 \% 13$?
Let $ m > 1$ be an integer, $ n$ is an odd number satisfying $ 3\le n < 2m,$ number $ a_{i,j} (i,j\in N, 1\le i\le m, 1\le j\le n)$ satisfies $ (1)$ for any $ 1\le j\le n, a_{1,j},a_{2,j},\cdots,a_{m,j}$ is a permutation of $ 1,2,3,\cdots,m; (2)$ for any $ 1 < i\le m, 1\le j\le n \minus{} 1, |a_{i,j} \minus{} a_{i,{j \plus{} 1}}|\le 1$ holds. Find the minimal value of $ M$, where $ M \equal{} max_{1 < i < m}\sum_{j \equal{} 1}^n{a_{i,j}}.$
A black unit equilateral triangle is drawn on the plane. How can we place nine tiles, each a unit equilateral triangle, on the plane so that they do not overlap, and each tile covers at least one interior point of the black triangle?
(Folklore)
A square with sidelength $1$ is covered by $3$ congruent disks. Find the smallest possible value of the radius of the disks.
The system of equations
\begin{eqnarray*}\log_{10}(2000xy) - (\log_{10}x)(\log_{10}y) & = & 4 \\
\log_{10}(2yz) - (\log_{10}y)(\log_{10}z) & = & 1 \\
\log_{10}(zx) - (\log_{10}z)(\log_{10}x) & = & 0 \\
\end{eqnarray*}
has two solutions $ (x_{1},y_{1},z_{1})$ and $ (x_{2},y_{2},z_{2}).$ Find $ y_{1} + y_{2}.$
$a_1, a_2, a_3, ...$ are positive reals such that $a_n^2 \ge a_1 + a_2 +... + a_{n-1}$.
Show that for some $C > 0$ we have $a_n \ge C n$ for all $n$.
A circle of radius $5$ is inscribed in an isosceles right triangle, $ABC$. The length of the hypotenuse of $ABC$ can be expressed as $a+a\sqrt{2}$ for some $a$. What is $a$?
Let $ABC$ be a scalene triangle and $\omega$ is your incircle. The sides $BC,CA$ and $AB$ are tangents to $\omega$ in $X,Y,Z$ respectively. Let $M$ be the midpoint of $BC$ and $D$ is the intersection point of $BC$ with the angle bisector of $\angle BAC$. Prove that $\angle BAX=\angle MAC$ if and only if $YZ$ passes by the midpoint of $AD$.
Consider a pyramid whose base is an equilateral triangle $BCD$ and whose other faces are triangles isosceles, right at the common vertex $A$. An ant leaves the vertex $B$ arrives at a point $P$ of the $CD$ edge, from there goes to a point $Q$ of the edge $AC$ and returns to point $B$. If the path you made is minimal, how much is the angle $PQA$ ?
Find the value of $\left\lfloor \frac{1}{6}\right\rfloor+\left\lfloor\frac{4}{6}\right\rfloor+\left\lfloor\frac{9}{6}\right\rfloor+\dots+\left\lfloor\frac{1296}{6}\right\rfloor$.
[i]Proposed by Zachary Perry[/i]
[b]p1.[/b] There exist real numbers $B$, $M$, and $T$ such that $B + M + T = 23$ and $B - M - T = 20$. Compute $M + T$.
[b]p2.[/b] Kaity has a rectangular garden that measures $10$ yards by $12$ yards. Austin’s triangular garden has side lengths $6$ yards, $8$ yards, and $10$ yards. Compute the ratio of the area of Kaity’s garden to the area of Austin’s garden.
[b]p3.[/b] Nikhil’s mom and brother both have ages under $100$ years that are perfect squares. His mom is $33$ years older than his brother. Compute the sum of their ages.
[b]p4.[/b] Madison wants to arrange $3$ identical blue books and $2$ identical pink books on a shelf so that each book is next to at least one book of the other color. In how many ways can Madison arrange the books?
[b]p5.[/b] Two friends, Anna and Bruno, are biking together at the same initial speed from school to the mall, which is $6$ miles away. Suddenly, $1$ mile in, Anna realizes that she forgot her calculator at school. If she bikes $4$ miles per hour faster than her initial speed, she could head back to school and still reach the mall at the same time as Bruno, assuming Bruno continues biking towards the mall at their initial speed. In miles per hour, what is Anna and Bruno’s initial speed, before Anna has changed her speed? (Assume that the rate at which Anna and Bruno bike is constant.)
[b]p6.[/b] Let a number be “almost-perfect” if the sum of its digits is $28$. Compute the sum of the third smallest and third largest almost-perfect $4$-digit positive integers.
[b]p7.[/b] Regular hexagon $ABCDEF$ is contained in rectangle $PQRS$ such that line $\overline{AB}$ lies on line $\overline{PQ}$, point $C$ lies on line $\overline{QR}$, line $\overline{DE}$ lies on line $\overline{RS}$, and point $F$ lies on line $\overline{SP}$. Given that $PQ = 4$, compute the perimeter of $AQCDSF$.
[img]https://cdn.artofproblemsolving.com/attachments/6/7/5db3d5806eaefa00d7fc90fb786a41c0466a90.png[/img]
[b]p8.[/b] Compute the number of ordered pairs $(m, n)$, where $m$ and $n$ are relatively prime positive integers and $mn = 2520$. (Note that positive integers $x$ and $y$ are relatively prime if they share no common divisors other than $1$. For example, this means that $1$ is relatively prime to every positive integer.)
[b]p9.[/b] A geometric sequence with more than two terms has first term $x$, last term $2023$, and common ratio $y$, where $x$ and $y$ are both positive integers greater than $1$. An arithmetic sequence with a finite number of terms has first term $x$ and common difference $y$. Also, of all arithmetic sequences with first term $x$, common difference $y$, and no terms exceeding $2023$, this sequence is the longest. What is the last term of the arithmetic sequence?
[b]p10.[/b] Andrew is playing a game where he must choose three slips, uniformly at random and without replacement, from a jar that has nine slips labeled $1$ through $9$. He wins if the sum of the three chosen numbers is divisible by $3$ and one of the numbers is $1$. What is the probability Andrew wins?
[b]p11.[/b] Circle $O$ is inscribed in square $ABCD$. Let $E$ be the point where $O$ meets line segment $\overline{AB}$. Line segments $\overline{EC}$ and $\overline{ED}$ intersect $O$ at points $P$ and $Q$, respectively. Compute the ratio of the area of triangle $\vartriangle EPQ$ to the area of triangle $\vartriangle ECD$.
[b]p12.[/b] Define a recursive sequence by $a_1 = \frac12$ and $a_2 = 1$, and $$a_n =\frac{1 + a_{n-1}}{a_{n-2}}$$ for n ≥ 3. The product $a_1a_2a_3 ... a_{2023}$ can be expressed in the form $a^b \cdot c^d \cdot e^f$ , where $a$, $b$, $c$, $d$, $e$, and $f$ are positive (not necessarily distinct) integers, and a, c, and e are prime. Compute $a + b + c + d + e + f$.
[b]p13.[/b] An increasing sequence of $3$-digit positive integers satisfies the following properties:
$\bullet$ Each number is a multiple of $2$, $3$, or $5$.
$\bullet$ Adjacent numbers differ by only one digit and are relatively prime. (Note that positive integers x and y are relatively prime if they share no common divisors other than $1$.)
What is the maximum possible length of the sequence?
[b]p14.[/b] Circles $O_A$ and $O_B$ with centers $A$ and $B$, respectively, have radii $3$ and $8$, respectively, and are internally tangent to each other at point $P$. Point $C$ is on circle $O_A$ such that line $\overline{BC}$ is tangent to circle $OA$. Extend line $\overline{PC}$ to intersect circle $O_B$ at point $D \ne P$. Compute $CD$.
[b]p15.[/b] Compute the product of all real solutions $x$ to the equation $x^2 + 20x - 23 = 2
\sqrt{x^2 + 20x + 1}$.
[b]p16.[/b] Compute the number of divisors of $729, 000, 000$ that are perfect powers. (A perfect power is an integer that can be written in the form $a^b$, where $a$ and $b$ are positive integers and $b > 1$.)
[b]p17.[/b] The arithmetic mean of two positive integers $x$ and $y$, each less than $100$, is $4$ more than their geometric mean. Given $x > y$, compute the sum of all possible values for $x + y$. (Note that the geometric mean of $x$ and $y$ is defined to be $\sqrt{xy}$.)
[b]p18.[/b] Ankit and Richard are playing a game. Ankit repeatedly writes the digits $2$, $0$, $2$, $3$, in that order, from left to right on a board until Richard tells him to stop. Richard wins if the resulting number, interpreted as a base-$10$ integer, is divisible by as many positive integers less than or equal to $12$ as possible. For example, if Richard stops Ankit after $7$ digits have been written, the number would be $2023202$, which is divisible by $1$ and $2$. Richard wants to win the game as early as possible. Assuming Ankit must write at least one digit, after how many digits should Richard stop Ankit?
[b]p19.[/b] Eight chairs are set around a circular table. Among these chairs, two are red, two are blue, two are green, and two are yellow. Chairs that are the same color are identical. If rotations and reflections of arrangements of chairs are considered distinct, how many arrangements of chairs satisfy the property that each pair of adjacent chairs are different colors?
[b]p20.[/b] Four congruent spheres are placed inside a right-circular cone such that they are all tangent to the base and the lateral face of the cone, and each sphere is tangent to exactly two other spheres. If the radius of the cone is $1$ and the height of the cone is $2\sqrt2$, what is the radius of one of the spheres?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $a, b, c$ be integers. Prove that there exist integers $p_1, q_1, r_1, p_2, q_2$ and $r_2$, satisfying $a = q_1r_2 - q_2r_1, b = r_1p_2 - r_2p_1$ and $c = p_1q_2 - p_2q_1.$
Triangle $ABC$ with $AB=4$, $BC=5$, $CA=6$ has circumcircle $\Omega$ and incircle $\omega$. Let $\Gamma$ be the circle tangent to $\Omega$ and the sides $AB$, $BC$, and let $X=\Gamma \cap \Omega$. Let $Y$, $Z$ be distinct points on $\Omega$ such that $XY$, $YZ$ are tangent to $\omega$. Find $YZ^2$.
[i]The following fact may be useful: if $\triangle{ABC}$ has incircle $w$ with incenter $I$ and radius $r$, and $\triangle{DEF}$ is the intouch triangle (i.e. $D$, $E$, $F$ are intersections of incircle with $BC$, $CA$, $AB$, respectively) and $H$ is the orthocenter of $\triangle{DEF}$, then the inversion of $X$ about $\omega$ (i.e. the point $X'$ on ray $IX$ such that $IX' \cdot IX=r^2$) is the midpoint of $DH$.[/i]
Let $ABCDE$ be a convex pentagon with five equal sides and right angles at $C$ and $D$. Let $P$ denote the intersection point of the diagonals $AC$ and $BD$. Prove that the segments $PA$ and $PD$ have the same length.
(Gottfried Perz)
Do there exist 2021 points with integer coordinates on the plane such that the pairwise distances between them are pairwise distinct consecutive integers?