Found problems: 85335
The circles in the figure have their centers at $C$ and $D$ and intersect at $A$ and $B$. Let $\angle ACB =60$, $\angle ADB =90^o$ and $DA = 1$ . Find the length of $CA$.
[img]https://cdn.artofproblemsolving.com/attachments/0/1/950a55984283091d15083fadcf35e8b95cb229.png[/img]
Let $n$ be a natural number. Solve the equation
$$||\cdots|||x-1|-2|-3|-\ldots-(n-1)|-n|=0.$$
On a table near the sea, there are $N$ glass boxes where $N<2021$, each containing exactly $2021$ balls. Sowdha and Rafi play a game by taking turns on the boxes where Sowdha takes the first turn. In each turn, a player selects a non-empty box and throws out some of the balls from it into the sea. If a player wants, he can throw out all of the balls in the selected box. The player who throws out the last ball wins. Let $S$ be the sum of all values of $N$ for which Sowdha has a winning strategy and let $R$ be the sum of all values of $N$ for which Rafi has a winning strategy. What is the value of $\frac{R-S}{10}$?
If point $M(x,y)$ lies on the line with equation $y=x+2$ and $1<y<3$, calculate the value of
$A=\sqrt{y^2-8x}+\sqrt{y^2+2x+5}$
[b]Q14.[/b] Let be given a trinagle $ABC$ with $\angle A=90^o$ and the bisectrices of angles $B$ and $C$ meet at $I$. Suppose that $IH$ is perpendicular to $BC$ ($H$ belongs to $BC$). If $HB=5 \text{cm}, \; HC=8 \text{cm}$, compute the area of $\triangle ABC$.
Given two non-intersecting chords $AB$ and $CD$ of a circle $k$ and a length $a <CD$. Determine a point $X$ on $k$ with the following property: If lines $XA$ and $XB$ intersect $CD$ at points $P$ and $Q$ respectively, then $PQ = a$. Show how to construct all such points $X$ and prove that the obtained points indeed have the desired property.
For any natural $n$ prove the inequality
$$\sqrt{2\sqrt{2}{\sqrt{3}\sqrt{4 ...\sqrt{n-1\sqrt{n}}}}} <3$$
There are two piles of stones: $1703$ stones in one pile and $2022$ in the other. Sasha and Olya
play the game, making moves in turn, Sasha starts. Let before the player's move the heaps contain $a$ and $b$ stones, with $a \geq b$. Then, on his own move, the player is allowed take from the pile with $a$ stones any number of stones from $1$ to $b$. A player loses if he can't make a move. Who wins?
Remark: For 10.4, the initial numbers are $(444,999)$
A triangle has no angle greater than $90^o$. Show that the sum of the medians is greater than four times the circumradius.
Let $ABC$ be a triangle, and let $BCDE$, $CAFG$, $ABHI$ be squares that do not overlap the triangle with centers $X$, $Y$, $Z$ respectively. Given that $AX=6$, $BY=7$, and $CA=8$, find the area of triangle $XYZ$.
What is the correct ordering of the three numbers $\frac5{19}$, $\frac7{21}$, and $\frac9{23}$, in increasing order?
$\textbf{(A)}\hspace{.05in}\dfrac9{23} < \dfrac7{21} < \dfrac5{19}$
$\textbf{(B)}\hspace{.05in}\dfrac5{19} < \dfrac7{21} < \dfrac9{23}$
$\textbf{(C)}\hspace{.05in}\dfrac9{23} < \dfrac5{19} < \dfrac7{21}$
$\textbf{(D)}\hspace{.05in}\dfrac5{19} < \dfrac9{23} < \dfrac7{21}$
$\textbf{(E)}\hspace{.05in}\dfrac7{21} < \dfrac5{19} < \dfrac9{23}$
Let $p>0$ be a given parameter. Determine all real $x$ such that \[\frac{1}{\,x+\sqrt{p-x^2\,}\,}+\frac{1}{\,x-\sqrt{p-x^2\,}\,}\ge\frac{1}{\,p\,}.\]
Let $n\geq3$ an integer. Mario draws $20$ lines in the plane, such that there are not two parallel lines.
For each [b]equilateral triangle[/b] formed by three of these lines, Mario receives three coins.
For each [b]isosceles[/b] and [b]non-equilateral[/b] triangle ([u]at the same time[/u]) formed by three of these lines, Mario receives a coin. How is the maximum number of coins that can Mario receive?
Prove that the number $$\bigg\lfloor\frac{2024}{1}\bigg\rfloor+\bigg\lfloor\frac{2023}{2}\bigg\rfloor+\bigg\lfloor\frac{2022}{3}\bigg\rfloor+\dots +\bigg\lfloor\frac{1013}{1012}\bigg\rfloor$$ is even.
$n$ is an odd positive integer and $x,y$ are two rational numbers satisfying $$x^n+2y=y^n+2x.$$Prove that $x=y$.
Find all triples $(l, m, n)$ of distinct positive integers satisfying \[{\gcd(l, m)}^{2}= l+m, \;{\gcd(m, n)}^{2}= m+n, \; \text{and}\;\;{\gcd(n, l)}^{2}= n+l.\]
Natural nonzero numder, which consists of $ m$ digits, is called hiperprime, if its any segment, which consists $ 1,2,...,m$ digits is prime (for example $ 53$ is hiperprime, because numbers $ 53,3,5$ are prime). Find all hiperprime numbers.
The cubic equation $x^3 + ax^2 + bx + c = 0$ has three real roots. Show that $a^2-3b\geq 0$, and that $\sqrt{a^2-3b}$ is less than or equal to the difference between the largest and smallest roots.
$5$ people stand in a line facing one direction. In every round, the person at the front moves randomly to any position in the line, including the front or the end. Suppose that $\frac{m}{n}$ is the expected number of rounds needed for the last person of the initial line to appear at the front of the line, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?
Find the last digit of $2023^{2023}$.
[i]Proposed by Yifan Kang[/i]
Joaquín and his brother Andrés go to class every day on the $62$ bus. Joaquín always pays for the tickets. Each ticket has a $5$-digit number printed on it. One day, Joaquín observes that the numbers on his tickets - his and his brother's - as well as being consecutive, are such that the sum of the ten digits is precisely $62$. Andrés asks him if the sum of the digits of any of the tickets is $35$ and, knowing the answer, he can directly say the number of each ticket. What were those numbers?
Let $S$ be a subset of $\{1,2,\dots,2017\}$ such that for any two distinct elements in $S$, both their sum and product are not divisible by seven. Compute the maximum number of elements that can be in $S$.
In a building there are $119$ inhabitants who live in $120$ apartments (several inhabitants can live in the same apartment). We call an apartment [i]overcrowded [/i] if $15$ or more people live in it. Every day in some overcrowded apartment (if there is one) its inhabitants have a fight and yes they all go to live in a different apartment (which may or may not be already inhabited). Should you always terminate this process?
Points $ A_1,A_2, ... ,A_n$ inside a circle and points $ B_1,B_2,...,B_n$ on its boundary are positioned so that the segments $ A_1B_1,A_2B_2, ... ,A_nB_n$ do not intersect. A bug can go from point $ A_i$ to $ A_j$ if the segment $ A_iA_j$ does not intersect any segment $ A_kB_k$, $ k \neq i, j$. Prove that the bug can go from any point $ A_p$ to any point $ A_q$ in a finite number of steps.
We call a function $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ [i]good[/i] if for all $x,y \in \mathbb{Q}^+$ we have: $$f(x)+f(y)\geq 4f(x+y).$$
a) Prove that for all good functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ and $x,y,z \in \mathbb{Q}^+$ $$f(x)+f(y)+f(z) \geq 8f(x+y+z)$$
b) Does there exists a good functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ and $x,y,z \in \mathbb{Q}^+$ such that $$f(x)+f(y)+f(z) < 9f(x+y+z) ?$$