Found problems: 85335
Let $a,b,c$ and $d$ be prime numbers such that $a>3b>6c>12d$ and $a^2-b^2+c^2-d^2=1749$. Determine all possible values of $a^2+b^2+c^2+d^2$ .
Enzymes convert glucose $(M=180.2)$ to ethanol $(M=46.1)$ according to the equation
\[ \text{C}_6\text{H}_{12}\text{O}_6 \rightarrow 2\text{C}_2\text{H}_2\text{OH} + 2\text{CO}_2 \]
What is the maximum mass of ethanol that can be made from $15.5$ kg of glucose?
$ \textbf{(A)}\hspace{.05in}0.256 \text{kg}\qquad\textbf{(B)}\hspace{.05in}0.512 \text{kg} \qquad\textbf{(C)}\hspace{.05in}3.96 \text{kg}\qquad\textbf{(D)}\hspace{.05in}7.93 \text{kg}\qquad$
For a given natural number $ n\ge 2, $ find all $ \text{n-tuples} $ of nonnegative real numbers which have the property that each one of the numbers forming the $ \text{n-tuple} $ is the square of the sum of the other $ n-1 $ ones.
[i]Mugur Acu[/i]
Let $P(x,y)=x^2y+xy^2$ and $Q(x,y)=x^2+xy+y^2.$ For $n=1,2,3,\dots,$ let \begin{align*}F_n(x,y)=(x+y)^n-x^n-y^n \text{ and,}\\ G_n(x,y)=(x+y)^n+x^n+y^n. \end{align*} One observes that $$G_2=2Q, F_3=3P, G_4=2Q^2, F_5=5PQ, G_6=2Q^3+3P^2.$$ Prove that, in fact, for each $n$ either $F_n$ or $G_n$ is expressible as a polynomial in $P$ and $Q$ with integer coefficients.
Two squares are said to be [i]juxtaposed [/i] if their intersection is a point or a segment. Prove that it is impossible to [i]juxtapose [/i] to a square more than eight non-overlapping squares of the same size.
Given a positive integer $n \ge 2$. Find all $n$-tuples of positive integers $(a_1,a_2,\ldots,a_n)$, such that $1<a_1 \le a_2 \le a_3 \le \cdots \le a_n$, $a_1$ is odd, and
(1) $M=\frac{1}{2^n}(a_1-1)a_2 a_3 \cdots a_n$ is a positive integer;
(2) One can pick $n$-tuples of integers $(k_{i,1},k_{i,2},\ldots,k_{i,n})$ for $i=1,2,\ldots,M$ such that for any $1 \le i_1 <i_2 \le M$, there exists $j \in \{1,2,\ldots,n\}$ such that $k_{i_1,j}-k_{i_2,j} \not\equiv 0, \pm 1 \pmod{a_j}$.
Suppose that fixed circle $C_1$ with radius $a > 0$ is tangent to the fixed line $\ell$ at $A$. Variable circle $C_2$, with center $X$, is externally tangent to $C_1$ at $B \ne A$ and $\ell$ at $C$. Prove that the set of all $X$ is a parabola minus a point
Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers such that $a_0=0$ and
$$a_{n+1}^3=a_n^2-8\quad \text{for} \quad n=0,1,2,…$$
Prove that the following series is convergent:
$$\sum_{n=0}^{\infty}{|a_{n+1}-a_n|}.$$
[i]Proposed by Orif Ibrogimov, National University of Uzbekistan[/i]
In a complete graph with $2025$ vertices, each edge has one of the colors $r_1$, $r_2$, or $r_3$. For each $i = 1,2,3$, if the $2025$ vertices can be divided into $a_i$ groups such that any two vertices connected by an edge of color $r_i$ are in different groups, find the minimum possible value of $a_1 + a_2 + a_3$.
When simplified, $ (x^{ \minus{} 1} \plus{} y^{ \minus{} 1})^{ \minus{} 1}$ is equal to:
$ \textbf{(A)}\ x \plus{} y \qquad\textbf{(B)}\ \frac {xy}{x \plus{} y} \qquad\textbf{(C)}\ xy \qquad\textbf{(D)}\ \frac {1}{xy} \qquad\textbf{(E)}\ \frac {x \plus{} y}{xy}$
$AB$ is a chord of $O$ and $AB$ is not a diameter of $O$. The tangent lines to $O$ at $A$ and $B$ meet at $C$. Let $M$ and $N$ be the midpoint of the segments $AC$ and $BC$, respectively. A circle passing through $C$ and tangent to $O$ meets line $MN$ at $P$ and $Q$. Prove that $\angle PCQ = \angle CAB$.
For any positive integer $ n>1$, let $ P\left(n\right)$ denote the largest prime divisor of $ n$. Prove that there exist infinitely many positive integers $ n$ for which
\[ P\left(n\right)<P\left(n\plus{}1\right)<P\left(n\plus{}2\right).\]
Let $ABCD$ be a convex quadrilateral and $P$ be a point in its interior, such that $\angle APB+\angle CPD=\angle BPC+\angle DPA$, $\angle PAD+\angle PCD=\angle PAB+\angle PCB$ and $\angle PDC+ \angle PBC= \angle PDA+\angle PBA$. Prove that the quadrilateral is circumscribed.
(a) Prove that in the set $ S=\{2008,2009,. . .,4200\}$ there are $ 5^3$ elements such that any three of them are not in arithmetic progression.
(b) Bonus: Try to find a smaller integer $ n \in (2008,4200)$ such that in the set $ S'=\{2008,2009,...,n\}$ there are $ 5^3$ elements such that any three of them are not in arithmetic progression.
If $x, 2x+2, 3x+3, \dots$ are in geometric progression, the fourth term is:
${{ \textbf{(A)}\ -27 \qquad\textbf{(B)}\ -13\frac{1}{2} \qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 13\frac{1}{2} }\qquad\textbf{(E)}\ 27 } $
[b]p1.[/b] How many primes $p < 100$ satisfy $p = a^2 + b^2$ for some positive integers $a$ and $b$?
[b]p2. [/b] For $a < b < c$, there exists exactly one Pythagorean triple such that $a + b + c = 2000$. Find $a + c - b$.
[b]p3.[/b] Five points lie on the surface of a sphere of radius $ 1$ such that the distance between any two points is at least $\sqrt2$. Find the maximum volume enclosed by these five points.
[b]p4.[/b] $ABCDEF$ is a convex hexagon with $AB = BC = CD = DE = EF = FA = 5$ and $AC = CE = EA = 6$. Find the area of $ABCDEF$.
[b]p5.[/b] Joe and Wanda are playing a game of chance. Each player rolls a fair $11$-sided die, whose sides are labeled with numbers $1, 2, ... , 11$. Let the result of the Joe’s roll be $X$, and the result of Wanda’s roll be $Y$ . Joe wins if $XY$ has remainder $ 1$ when divided by $11$, and Wanda wins otherwise. What is the probability that Joe wins?
[b]p6.[/b] Vivek picks a number and then plays a game. At each step of the game, he takes the current number and replaces it with a new number according to the following rule: if the current number $n$ is divisible by $3$, he replaces $n$ with $\frac{n}{3} + 2$, and otherwise he replaces $n$ with $\lfloor 3 \log_3 n \rfloor$. If he starts with the number $3^{2011}$, what number will he have after $2011$ steps?
Note that $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.
[b]p7.[/b] Define a sequence an of positive real numbers with a$_1 = 1$, and $$a_{n+1} =\frac{4a^2_n - 1}{-2 + \frac{4a^2_n -1}{-2+ \frac{4a^2_n -1}{-2+...}}}.$$
What is $a_{2011}$?
[b]p8.[/b] A set $S$ of positive integers is called good if for any $x, y \in S$ either $x = y$ or $|x - y| \ge 3$. How many subsets of $\{1, 2, 3, ..., 13\}$ are good? Include the empty set in your count.
[b]p9.[/b] Find all pairs of positive integers $(a, b)$ with $a \le b$ such that $10 \cdot lcm \, (a, b) = a^2 + b^2$. Note that $lcm \,(m, n)$ denotes the least common multiple of $m$ and $n$.
[b]p10.[/b] For a natural number $n$, $g(n)$ denotes the largest odd divisor of $n$. Find $$g(1) + g(2) + g(3) + ... + g(2^{2011})$$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Suppose that for some $m,n\in\mathbb{N}$ we have $\varphi (5^m-1)=5^n-1$, where $\varphi$ denotes the Euler function. Show that $(m,n)>1$.
Let $a, b, c, d$ be an increasing arithmetic sequence of positive real numbers with common difference $\sqrt2$. Given that the product $abcd = 2021$, $d$ can be written as $\frac{m+\sqrt{n}}{\sqrt{p}}$ , where $m, n,$ and $p$ are positive integers not divisible by the square of any prime. Find $m + n + p$.
A set of positive integers is said to be [i]pilak[/i] if it can be partitioned into 2 disjoint subsets $F$ and $T$, each with at least $2$ elements, such that the elements of $F$ are consecutive Fibonacci numbers, and the elements of $T$ are consecutive triangular numbers. Find all positive integers $n$ such that the set containing all the positive divisors of $n$ except $n$ itself is pilak.
Let $ABCD$ be a convex quadrilateral. Let $E,F,G,H$ be points on segments $AB$, $BC$, $CD$, $DA$, respectively, and let $P$ be the intersection of $EG$ and $FH$. Given that quadrilaterals $HAEP$, $EBFP$, $FCGP$, $GDHP$ all have inscribed circles, prove that $ABCD$ also has an inscribed circle.
[i]Evan O'Dorney.[/i]
The bisector of angle $A$ of triangle $ABC$ ($AB > AC$) meets its circumcircle at point $P$. The perpendicular to $AC$ from $C$ meets the bisector of angle $A$ at point $K$. A cừcle with center $P$ and radius $PK$ meets the minor arc $PA$ of the circumcircle at point $D$. Prove that the quadrilateral $ABDC$ is circumscribed.
Compute the remainder when $98!$ is divided by $101$.
Let $P$ be an interior point of triangle $ABC$ such that $\angle BPC = \angle CPA =\angle APB = 120^o$. Prove that the Euler lines of triangles $APB,BPC,CPA$ are concurrent.
The ruler of Laputa will set up a train network between cities in the state, which satisfies the following conditions:
- [i]Uniformity[/i]: From one city to another, by train, possibly through exchanges.
- [i]Prohibition N[/i]: There exist no four cities $A, B, C, D$ such that there are direct routes between $A$ and $B, B$ and $C$, and $C$ and $D$, but taking a shortcut is not possible, that is, there are no direct rout between $A$ and $C, B$ and $D$, or $A$ and $D$.
In addition, a direct airliner connection will be established exactly between their city pairs, with no direct train connection.
Prove that the airline network is not connected when there is more than one city.
Along a track for cross-country skiing, $1000$ seats are placed in a row and numbered in order from $1$ to $1000$. By mistake, $n$ tickets were sold, $100 < n < 1000$, each with one of the numbers $1,2,..., 100$ printed on it. Also for each number $1,2,..., 100$ there exists at least one ticket with this number printed on it. Of course, there are tickets that have the same seat numbers. These $n$ spectators arrive one at a time.
Each goes to the seat shown on his ticket and occupies it if it is still empty. If not, he just says “Oh” and moves to the seat with the next number. This is repeated until he finds an empty seat and occupies it, saying “Oh” once for each occupied seat passed over but not at any other time. Prove that all the spectators will be seated and that the total number of the exclamations “Oh” that have been made before all the spectators are seated does not depend on the order in which the n spectators arrive, although it does depend on the distribution of numbers on the tickets.
(A Shen)