Found problems: 1671
Let $x,y$ and $z$ be positive reals such that $x + y + z = 3$. Prove that
\[ 2 \sqrt{x + \sqrt{y}} + 2 \sqrt{y + \sqrt{z}} + 2 \sqrt{z + \sqrt{x}} \le \sqrt{8 + x - y} + \sqrt{8 + y - z} + \sqrt{8 + z - x} \]
Cauchy the magician has a new card trick. He takes a standard deck(which consists of 52 cards with 13 denominations in each 4 suits) and let Schwartz to shuffle randomly. Schwartz is told to take $ m $ cards not more than $ \frac{1}{3} $ form the top of the deck. Then, Cauchy take $ 18 $ cards one by one from the top of the remaining deck and show it to Schwartz with the second card is placed in front of the first card (from Schwartz view) and so on. He ask Schwartz to memorize the $ m-th $ card when showing the cards. Let it be $ C_1 $. After that, Cauchy places the $ 18 $ cards and the $ m $ cards on the bottom of the deck with the $ m $ cards are placed lower than the $ 18 $ cards. Now, Cauchy distributes and flip the cards on the table from the top of the deck while shouting the numbers $ 10 $ until $ 1 $ with the following operation:
a) When a card flipped has the number which is same as the number shouted by Cauchy, stop the distribution and continue with another set.\\
b) When $ 10 $ cards are flipped and none of the cards flipped has the number which is same as the number shouted by Cauchy, take a card from the top of the deck and place it on top of the set with backside(the site which has no value) facing up. Then continue with another set.\\
Cauchy stops when 3 sets of cards are placed. Then, he adds up all the numbers on top of each sets of cards( backside is consider $ 0 $ ). Let $ k $ be the sum. He placed another $ k $ cards to the table from the top of the remaining deck. Finally, he shows the first card on top of the remaining deck to Schwartz. Let it be $ C_2 $.
Show that $ C_1 = C_2 $.
Let $ n$ be an even positive integer. Let $ A_1, A_2, \ldots, A_{n \plus{} 1}$ be sets having $ n$ elements each such that any two of them have exactly one element in common while every element of their union belongs to at least two of the given sets. For which $ n$ can one assign to every element of the union one of the numbers 0 and 1 in such a manner that each of the sets has exactly $ \frac {n}{2}$ zeros?
Let $ ABC$ be a triangle. Points $ D$ and $ E$ are on sides $ AB$ and $ AC,$ respectively, and point $ F$ is on line segment $ DE.$ Let $ \frac {AD}{AB} \equal{} x,$ $ \frac {AE}{AC} \equal{} y,$ $ \frac {DF}{DE} \equal{} z.$ Prove that
(1) $ S_{\triangle BDF} \equal{} (1 \minus{} x)y S_{\triangle ABC}$ and $ S_{\triangle CEF} \equal{} x(1 \minus{} y) (1 \minus{} z)S_{\triangle ABC};$
(2) $ \sqrt [3]{S_{\triangle BDF}} \plus{} \sqrt [3]{S_{\triangle CEF}} \leq \sqrt [3]{S_{\triangle ABC}}.$
Let $a,b,c$ be positive real numbers such that $abc=1$. Determine the minimum value of
$E(a,b,c) = \sum \dfrac{a^3+5}{a^3(b+c)}$ .
The sequence $a_1, a_2, \ldots, a_n$ of positive real numbers satisfies the following conditions:
\begin{align*}
\sum_{i=1}^n \frac{1}{a_i} \le 1 \ \ \ \ \hbox{and} \ \ \ \ a_i \le a_{i-1}+1
\end{align*}
for all $i\in \lbrace 1, 2, \ldots, n \rbrace$, where $a_0$ is an integer. Prove that
\begin{align*}
n \le 4a_0 \cdot \sum_{i=1}^n \frac{1}{a_i}
\end{align*}
Let $\, a_1, a_2, a_3, \ldots \,$ be a sequence of positive real numbers satisfying $\, \sum_{j=1}^n a_j \geq \sqrt{n} \,$ for all $\, n \geq 1$. Prove that, for all $\, n \geq 1, \,$ \[ \sum_{j=1}^n a_j^2 > \frac{1}{4} \left( 1 + \frac{1}{2} + \cdots + \frac{1}{n} \right). \]
Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the equation $$f(x^{2019} + y^{2019}) = x(f(x))^{2018} + y(f(y))^{2018}$$ for all real numbers $x$ and $y$.
Let $x$, $y$, $z$ be arbitrary positive numbers such that $xy+yz+zx=x+y+z$.
Prove that $$\frac{1}{x^2+y+1} + \frac{1}{y^2+z+1} + \frac{1}{z^2+x+1} \leq 1$$.
When does equality occur?
[i]Proposed by Marko Radovanovic[/i]
For all pairwise distinct positive real numbers $a, b, c$ such that $abc = 1$, prove that $$\frac{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1}{(a+b+c+1)^2}+\frac{3}{8}\sqrt[3]{\frac{(a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3}{(a-b)^3+(b-c)^3+(c-a)^3}}\geq 1.$$
Let $a,b,c\geq 0$ and $a+b+c=1.$ Prove that$$\frac{a}{2a+1}+\frac{b}{3b+1}+\frac{c}{6c+1}\leq \frac{1}{2}.$$
[size=50](Marian Dinca)[/size]
Show that for all real numbers $ a,b,c,d,$ the following inequality holds:
\[ (a\plus{}b\plus{}c\plus{}d)^2 \leq 3 (a^2 \plus{} b^2 \plus{} c^2 \plus{} d^2) \plus{} 6ab\]
Find the maximal value of the following expression, if $a,b,c$ are nonnegative and $a+b+c=1$.
\[ \frac{1}{a^2 -4a+9} + \frac {1}{b^2 -4b+9} + \frac{1}{c^2 -4c+9} \]
[b]p1.[/b] Suppose that $a \oplus b = ab - a - b$. Find the value of $$((1 \oplus 2) \oplus (3 \oplus 4)) \oplus 5.$$
[b]p2.[/b] Neethin scores a $59$ on his number theory test. He proceeds to score a $17$, $23$, and $34$ on the next three tests. What score must he achieve on his next test to earn an overall average of $60$ across all five tests?
[b]p3.[/b] Consider a triangle with side lengths $28$ and $39$. Find the number of possible integer lengths of the third side.
[b]p4.[/b] Nithin is thinking of a number. He says that it is an odd two digit number where both of its digits are prime, and that the number is divisible by the sum of its digits. What is the sum of all possible numbers Nithin might be thinking of?
[b]p5.[/b] Dora sees a fire burning on the dance floor. She calls her friends to warn them to stay away. During the first pminute Dora calls Poonam and Serena. During the second minute, Poonam and Serena call two more friends each, and so does Dora. This process continues, with each person calling two new friends every minute. How many total people would know of the fire after $6$ minutes?
[b]p6.[/b] Charlotte writes all the positive integers $n$ that leave a remainder of $2$ when $2018$ is divided by $n$. What is the sum of the numbers that she writes?
[b]p7.[/b] Consider the following grid. Stefan the bug starts from the origin, and can move either to the right, diagonally in the positive direction, or upwards. In how many ways can he reach $(5, 5)$?
[img]https://cdn.artofproblemsolving.com/attachments/9/9/b9fdfdf604762ec529a1b90d663e289b36b3f2.png[/img]
[b]p8.[/b] Let $a, b, c$ be positive numbers where $a^2 + b^2 + c^2 = 63$ and $2a + 3b + 6c = 21\sqrt7$. Find
$\left( \frac{a}{c}\right)^{\frac{a}{b}} $.
[b]p9.[/b] What is the sum of the distinct prime factors of $12^5 + 12^4 + 1$?
[b]p10.[/b] Allen starts writing all permutations of the numbers $1$, $2$, $3$, $4$, $5$, $6$ $7$, $8$, $9$, $10$ on a blackboard. At one point he writes the permutation $9$, $4$, $3$, $1$, $2$, $5$, $6$, $7$, $8$, $10$. David points at the permutation and observes that for any two consecutive integers $i$ and $i+1$, all integers that appear in between these two integers in the permutation are all less than $i$. For example, $4$ and $5$ have only the numbers $3$, $1$, $2$ in between them. How many of the $10!$ permutations on the board satisfy this property that David observes?
[b]p11.[/b] (Estimation) How many positive integers less than $2018$ can be expressed as the sum of $3$ square numbers?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $a,b,c>0$ such that $a+b+c=1$. Prove: \[\frac{a^{2}}b+\frac{b^{2}}c+\frac{c^{2}}a \ge 3(a^{2}+b^{2}+c^{2}) \]
Let $a,b,c$ be sides of a triangle such that $a\geq b \geq c$. prove that:
$\sqrt{a(a+b-\sqrt{ab})}+\sqrt{b(a+c-\sqrt{ac})}+\sqrt{c(b+c-\sqrt{bc})}\geq a+b+c$
Finding all quads of integers $(a, b, c, p)$ where $p \ge 5$ is prime number such that the remainders of the numbers $am^3 + bm^2 + cm$, $m = 0, 1, . . . , p - 1$, upon division of $p$ are two by two different..
Let $ a_1$, $ a_2$, $ a_3$, $ a_4$, $ a_5$ be real numbers satisfying the following equations:
$ \frac{a_1}{k^2\plus{}1}\plus{}\frac{a_2}{k^2\plus{}2}\plus{}\frac{a_3}{k^2\plus{}3}\plus{}\frac{a_4}{k^2\plus{}4}\plus{}\frac{a_5}{k^2\plus{}5} \equal{} \frac{1}{k^2}$ for $ k \equal{} 1, 2, 3, 4, 5$
Find the value of $ \frac{a_1}{37}\plus{}\frac{a_2}{38}\plus{}\frac{a_3}{39}\plus{}\frac{a_4}{40}\plus{}\frac{a_5}{41}$ (Express the value in a single fraction.)
Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=1$. Prove that\[ 6(a^3+b^3+c^3+d^3)\ge(a^2+b^2+c^2+d^2)+\frac{1}{8} \]
Given $x_1,x_2,\dots,x_n>0,n\geq 5$, show that
\[\frac{x_1x_2}{x_1^2+x_2^2+2x_3x_4}+\frac{x_2x_3}{x_2^2+x_3^2+2x_4x_5}+\cdots+\frac{x_nx_1}{x_n^2+x_1^2+2x_2x_3}\leq \frac{n-1}{2}\]
$x,y,z$ positive real numbers such that $xyz=1$
Prove that:
$\frac{1}{x+y^{20}+z^{11}}+\frac{1}{y+z^{20}+x^{11}}+\frac{1}{z+x^{20}+y^{11}}\leq1$
For $x,y,z > 0$ and $xyz=1$, prove that
\[\frac{x^{9}+y^{9}}{x^{6}+x^{3}y^{3}+y^{6}}+\frac{x^{9}+z^{9}}{x^{6}+x^{3}z^{3}+z^{6}}+\frac{y^{9}+z^{9}}{y^{6}+y^{3}z^{3}+z^{6}}\geq 2\]
Let $ n\in \mathbb{N^*}$ and $ f: [0,1]\rightarrow \mathbb{R}$ a continuos function with the prop. $ \int_{0}^{1}(1\minus{}x^n)f(x)dx\equal{}0$.
Prove that $ \int_{0}^{1}f^2(x)dx \geq 2(n\plus{}1)\left(\int_{0}^{1}f(x)dx\right)^2$
Let $a$ and $b$ be real numbers. Define $f_{a,b}\colon R^2\to R^2$ by $f_{a,b}(x;y)=(a-by-x^2;x)$. If $P=(x;y)\in R^2$, define $f^0_{a,b}(P) = P$ and $f^{k+1}_{a,b}(P)=f_{a,b}(f_{a,b}^k(P))$ for all nonnegative integers $k$.
The set $per(a;b)$ of the [i]periodic points[/i] of $f_{a,b}$ is the set of points $P\in R^2$ such that $f_{a,b}^n(P) = P$ for some positive integer $n$.
Fix $b$. Prove that the set $A_b=\{a\in R \mid per(a;b)\neq \emptyset\}$ admits a minimum. Find this minimum.
Find the minimum and maximum values of the function
$$f(x,y,z,t)=\frac{ax^2+by^2}{ax+by}+\frac{az^2+bt^2}{az+bt},~(a>0,b>0),$$given that $x+z=y+t=1$, and $x,y,z,t\ge0$.