Found problems: 85335
Find the ratio between the sum of the areas of the circles and the area of the fourth circle that are shown in the figure
Each circle passes through the center of the previous one and they are internally tangent.
[img]https://cdn.artofproblemsolving.com/attachments/d/2/29d2be270f7bcf9aee793b0b01c2ef10131e06.jpg[/img]
$(HUN 5)$ Let $r$ and $m (r \le m)$ be natural numbers and $Ak =\frac{2k-1}{2m}\pi$. Evaluate $\frac{1}{m^2}\displaystyle\sum_{k=1}^{m}\displaystyle\sum_{l=1}^{m}\sin(rA_k)\sin(rA_l)\cos(rA_k-rA_l)$
An isosceles trapezoid with bases $a$ and $c$ and altitude $h$ is given.
a) On the axis of symmetry of this trapezoid, find all points $P$ such that both legs of the trapezoid subtend right angles at $P$;
b) Calculate the distance of $p$ from either base;
c) Determine under what conditions such points $P$ actually exist. Discuss various cases that might arise.
In a triangle $ABC$ assume $AB = AC$, and let $D$ and $E$ be points on the extension of segment $BA$ beyond $A$ and on the segment $BC$, respectively, such that the lines $CD$ and $AE$ are parallel. Prove $CD \ge \frac{4h}{BC}CE$, where $h$ is the height from $A$ in triangle $ABC$. When does equality hold?
A natural number is written on the blackboard. In each step, we erase the units digit and add four times the erased digit to the remaining number, and write the result on the blackboard instead of the initial number. Starting with the number $13^{2006}$, is it possible to obtain the number $2006^{13}$ by repeating this step finitely many times?
$A, B$ and $C$ are points on the circumference of a circle with centre $O$. Tangents are drawn to the circumcircles of triangles $OAB$ and $OAC$ at $P$ and $Q$ respectively, where $P$ and $Q$ are diametrically opposite $O$. The two tangents intersect at $K$. The line $CA$ meets the circumcircle of $\triangle OAB$ at $A$ and $X$. Prove that $X$ lies on the line $KO$.
Set $S=\{1,2,...,2004\}$. We denote by $d_1$ the number of subset of $S$ such that the sum of elements of the subset has remainder $7$ when divided by $32$. We denote by $d_2$ the number of subset of $S$ such that the sum of elements of the subset has remainder $14$ when divided by $16$. Compute $\frac{d_1}{d_2}$.
All $10$-digit numbers consisting of digits $1, 2$ and $3$ are written one under the other. Each number has one more digit added to the right. $1$, $2$ or $3$, and it turned out that to the number $111. . . 11$ added $1$ to the number $ 222. . . 22$ was assigned $2$, and the number $333. . . 33$ was assigned $3$. It is known that any two numbers that differ in all ten digits have different digits assigned to them. Prove that the assigned column of numbers matches with one of the ten columns written earlier.
\[\int_0^{\pi/2} \frac{\sin(x)}{2-\sin(x)\cos(x)}\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
Find all real numbers $x$ for which the number $$a =\frac{2x + 1}{x^2 + 2x + 3}$$ is an integer.
We would like to give a present to one of $100$ children. We do this by throwing a biased coin $k$ times, after predetermining who wins in each possible outcome of this lottery.
Prove that we can choose the probability $p$ of throwing heads, and the value of $k$ such that, by distributing the $2^k$ different outcomes between the children in the right way, we can guarantee that each child has the same probability of winning.
Tom has a collection of $13$ snakes, $4$ of which are purple and $5$ of which are happy. He observes that
$\bullet$ all of his happy snakes can add
$\bullet$ none of his purple snakes can subtract, and
$\bullet$ all of his snakes that can’t subtract also can’t add
Which of these conclusions can be drawn about Tom’s snakes?
$\textbf{(A)}$ Purple snakes can add.
$\textbf{(B)}$ Purple snakes are happy.
$\textbf{(C)}$ Snakes that can add are purple.
$\textbf{(D)}$ Happy snakes are not purple.
$\textbf{(E)}$ Happy snakes can't subtract.
For any finite set $X$, let $|X|$ denote the number of elements in $X.$ Define $$S_n = \sum |A \cap B|,$$ where the sum is taken over all ordered pairs $(A, B)$ such that $A$ and $B$ are subsets of $\{1, 2, 3, …, n\}$ with $|A| = |B|.$ For example, $S_2 = 4$ because the sum is taken over the pairs of subsets $$(A, B) \in \{ (\emptyset, \emptyset), (\{1\}, \{1\}), (\{1\}, \{2\}), (\{2\}, \{1\}), (\{2\}, \{2\}), (\{1, 2\}, \{1, 2\})\},$$ giving $S_2 = 0 + 1 + 0 + 0 + 1 + 2 = 4.$ Let $\frac{S_{2022}}{S_{2021}} = \frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find the remainder when $p + q$ is divided by $1000.$
For a positive integer $n$ and any real number $c$, define $x_k$ recursively by :
\[ x_0=0,x_1=1 \text{ and for }k\ge 0, \;x_{k+2}=\frac{cx_{k+1}-(n-k)x_k}{k+1} \]
Fix $n$ and then take $c$ to be the largest value for which $x_{n+1}=0$. Find $x_k$ in terms of $n$ and $k,\; 1\le k\le n$.
Let $p(x)$ be a polynomial with real coefficients such that $p(0)=p(n)$. Prove that there are at least $n$ pairs of real numbers $(x,y)$ where $p(x)=p(y)$ and $y-x$ is a positive integer
$ABCD$ is a convex quadrilateral; half-lines $DA$ and $CB$ meet at point $Q$; half-lines $BA$ and $CD$ meet at point $P$. It is known that $\angle AQB=\angle APD$. The bisector of angle $\angle AQB$ meets the sides $AB$ and $CD$ of the quadrilateral at points $X$ and $Y$, respectively; the bisector of angle $\angle APD$ meets the sides $AD$ and $BC$ at points $Z$ and $T$, respectively.
The circumcircles of triangles $ZQT$ and $XPY$ meet at point $K$ inside the quadrilateral.
Prove that $K$ lies on the diagonal $AC$.
[i]Proposed by S. Berlov[/i]
Given a regular tetrahedron $OABC$. Take points $P,\ Q,\ R$ on the sides $OA,\ OB,\ OC$ respectively. Note that $P,\ Q,\ R$ are different from the vertices of the tetrahedron $OABC$. If $\triangle{PQR}$ is an equilateral triangle, then prove that three sides $PQ,\ QR,\ RP$ are pararell to three sides $AB,\ BC,\ CA$ respectively.
30 points
Compute the sum $$\sum_{k=0}^{1273}\frac{1}{1 + tan^{2548}\left(\frac{k\pi}{2548}\right)}$$
In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.
Integers $a,b$ and $c$ satisfy the equality $a+b+c=0$. Denote $S=ab+bc+ac$, $A=a^2+a+1$, $B=b^2+b+1$ and $C=c^2+c+1$.
Prove that the number $(S+A)(S+B)(S+C)$ is a perfect square.
Let $n\geqslant 2$ be a positive integer and $a_1,a_2, \ldots ,a_n$ be real numbers such that \[a_1+a_2+\dots+a_n=0.\]
Define the set $A$ by
\[A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}\]
Prove that, if $A$ is not empty, then
\[\sum_{(i, j) \in A} a_{i} a_{j}<0.\]
Let $P=[a_{ij}]_{3\times 9}$ be a $3\times 9$ matrix where $a_{ij}\ge 0$ for all $i,j$. The following conditions are given:
[list][*]Every row consists of distinct numbers;
[*]$\sum_{i=1}^{3}x_{ij}=1$ for $1\le j\le 6$;
[*]$x_{17}=x_{28}=x_{39}=0$;
[*]$x_{ij}>1$ for all $1\le i\le 3$ and $7\le j\le 9$ such that $j-i\not= 6$.
[*]The first three columns of $P$ satisfy the following property $(R)$: for an arbitrary column $[x_{1k},x_{2k},x_{3k}]^T$, $1\le k\le 9$, there exists an $i\in\{1,2,3\}$ such that $x_{ik}\le u_i=\min (x_{i1},x_{i2},x_{i3})$.[/list]
Prove that:
a) the elements $u_1,u_2,u_3$ come from three different columns;
b) if a column $[x_{1l},x_{2l},x_{3l}]^T$ of $P$, where $l\ge 4$, satisfies the condition that after replacing the third column of $P$ by it, the first three columns of the newly obtained matrix $P'$ still have property $(R)$, then this column uniquely exists.
Prove that there exist infinitely many positive integers $n$ such that $\sqrt{n}$ is not an integer and $n$ is divisible by $[\sqrt{n}] $.
Prove the equality
$$\frac{a_1}{a_2(a_1+a_2)}+\frac{a_2}{a_3(a_2+a_3)}+...+\frac{a_n}{a_1(a_n+a_1)}=\frac{a_2}{a_1(a_1+a_2)}+\frac{a_3}{a_2(a_2+a_3)}+...+\frac{a_1}{a_n(a_n+a_1)} $$
(R. Zhenodarov)
[b]p1.[/b] Chris goes to Matt's Hamburger Shop to buy a hamburger. Each hamburger must contain exactly one bread, one lettuce, one cheese, one protein, and at least one condiment. There are two kinds of bread, two kinds of lettuce, three kinds of cheese, three kinds of protein, and six different condiments: ketchup, mayo, mustard, dill pickles, jalape~nos, and Matt's Magical Sunshine Sauce. How many different hamburgers can Chris make?
[b]p2.[/b] The degree measures of the interior angles in convex pentagon $NICKY$ are all integers and form an increasing arithmetic sequence in some order. What is the smallest possible degree measure of the pentagon's smallest angle?
[b]p3.[/b] Daniel thinks of a two-digit positive integer $x$. He swaps its two digits and gets a number $y$ that is less than $x$. If $5$ divides $x-y$ and $7$ divides $x+y$, find all possible two-digit numbers Daniel could have in mind.
[b]p4.[/b] At the Lio Orympics, a target in archery consists of ten concentric circles. The radii of the circles are $1$, $2$, $3$, $...$, $9$, and $10$ respectively. Hitting the innermost circle scores the archer $10$ points, the next ring is worth $9$ points, the next ring is worth 8 points, all the way to the outermost ring, which is worth $1$ point. If a beginner archer has an equal probability of hitting any point on the target and never misses the target, what is the probability that his total score after making two shots is even?
[b]p5.[/b] Let $F(x) = x^2 + 2x - 35$ and $G(x) = x^2 + 10x + 14$. Find all distinct real roots of $F(G(x)) = 0$.
[b]p6.[/b] One day while driving, Ivan noticed a curious property on his car's digital clock. The sum of the digits of the current hour equaled the sum of the digits of the current minute. (Ivan's car clock shows $24$-hour time; that is, the hour ranges from $0$ to $23$, and the minute ranges from $0$ to $59$.) For how many possible times of the day could Ivan have observed this property?
[b]p7.[/b] Qi Qi has a set $Q$ of all lattice points in the coordinate plane whose $x$- and $y$-coordinates are between $1$ and $7$ inclusive. She wishes to color $7$ points of the set blue and the rest white so that each row or column contains exactly $1$ blue point and no blue point lies on or below the line $x + y = 5$. In how many ways can she color the points?
[b]p8.[/b] A piece of paper is in the shape of an equilateral triangle $ABC$ with side length $12$. Points $A_B$ and $B_A$ lie on segment $AB$, such that $AA_B = 3$, and $BB_A = 3$. Define points $B_C$ and $C_B$ on segment $BC$ and points $C_A$ and $A_C$ on segment $CA$ similarly. Point $A_1$ is the intersection of $A_CB_C$ and $A_BC_B$. Define $B_1$ and $C_1$ similarly. The three rhombi - $AA_BA_1A_C$,$BB_CB_1B_A$, $CC_AC_1C_B$ - are cut from triangle $ABC$, and the paper is folded along segments $A_1B_1$, $B_1C_1$, $C_1A_1$, to form a tray without a top. What is the volume of this tray?
[b]p9.[/b] Define $\{x\}$ as the fractional part of $x$. Let $S$ be the set of points $(x, y)$ in the Cartesian coordinate plane such that $x + \{x\} \le y$, $x \ge 0$, and $y \le 100$. Find the area of $S$.
[b]p10.[/b] Nicky likes dolls. He has $10$ toy chairs in a row, and he wants to put some indistinguishable dolls on some of these chairs. (A chair can hold only one doll.) He doesn't want his dolls to get lonely, so he wants each doll sitting on a chair to be adjacent to at least one other doll. How many ways are there for him to put any number (possibly none) of dolls on the chairs? Two ways are considered distinct if and only if there is a chair that has a doll in one way but does not have one in the other.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].