Found problems: 85335
Find the smallest real number $A$ such that, for every quadratic polynomial $f(x)$ satisfying $ | f(x)| \le 1$ for $0 \le x \le 1$, it holds that $f' (0) \le A$.
Let ($b_n$) be a sequence such that $b_n \geq 0 $ and $b_{n+1}^2 \geq \frac{b_1^2}{1^3}+\cdots+\frac{b_n^2}{n^3}$ for all $n \geq 1$. Prove that there exists a natural number $K$ such that
\[\sum_{n=1}^{K} \frac{b_{n+1}}{b_1+b_2+ \cdots + b_n} \geq \frac{1993}{1000}\]
Find all positive integers $n$ for which there exists a polynomial $P(x) \in \mathbb{Z}[x]$ such that for every positive integer $m\geq 1$, the numbers $P^m(1), \ldots, P^m(n)$ leave exactly $\lceil n/2^m\rceil$ distinct remainders when divided by $n$. (Here, $P^m$ means $P$ applied $m$ times.)
[i]Proposed by Carl Schildkraut, USA[/i]
Find all integers $a,b,c$ such that $a^2 = bc + 1$ and $b^2 = ac + 1$
Suppose that $a_1,a_2,\dots a_n$ are real $(n>1)$ and $$A+ \sum_{i=1}^{n} a^2_i< \frac{1}{n-1} (\sum_{i=1}^{n} a_i)^2.$$ Prove that $A<2a_ia_j$ for $1\leq i<j\leq n.$
$ABC$ is an acute-angled triangle with circumcenter $O$. The circumcircle of $ABO$ intersects$ AC$ and $BC$ at $M$ and $N$. Show that the circumradii of $ABO$ and $MNC$ are the same.
Let $\Delta ABC$ be a scalene triangle. Points $D,E$ lie on side $\overline{AC}$ in the order, $A,E,D,C$. Let the parallel through $E$ to $BC$ intersect $\odot (ABD)$ at $F$, such that, $E$ and $F$ lie on the same side of $AB$. Let the parallel through $E$ to $AB$ intersect $\odot (BDC)$ at $G$, such that, $E$ and $G$ lie on the same side of $BC$. Prove, Points $D,F,E,G$ are concyclic
Do there exist functions $f$ and $g$, $f : R \to R$, $g : R \to R$ such that $f(x + f(y)) = y^2 + g(x)$ for all real $x$ and $y$ ?
(I. Gorodnin)
Prove that there exists no in nite sequence of prime numbers $p_0, p_1, p_2,...$ such that for all positive integers $k$:
$p_k = 2p_{k-1} + 1$ or $p_k = 2p_{k-1} - 1$.
Let $\alpha$ be given positive real number, find all the functions $f: N^{+} \rightarrow R$ such that $f(k + m) = f(k) + f(m)$ holds for any positive integers $k$, $m$ satisfying $\alpha m \leq k \leq (\alpha + 1)m$.
A parallelogram $P$ can be folded over a straight line so that the resulting shape is a regular pentagon with side length $1.$ Compute the perimeter of $P.$
Find the product of all real $x$ for which \[ 2^{3x+1} - 17 \cdot 2^{2x} + 2^{x+3} = 0. \]
We consider an acute angled triangle $ABC$ (with $AB<AC$) and its circumcircle $c(O,R) $(with center $O$ and semidiametre $R$).The altitude $AD$ cuts the circumcircle at the point $E$ ,while the perpedicular bisector $(m)$ of the segment $AB$,cuts $AD$ at the point $L$.$BL$ cuts $AC$ at the point $M$ and the circumcircle $c(O,R)$ at the point $N$.Finally $EN$ cuts the perpedicular bisector $(m)$ at the point $Z$.Prove that:
\[ MZ \perp BC \iff \left(CA=CB \;\; \text{or} \;\; Z\equiv O \right) \]
Let $ P$ be an interior point of circle $ K$ other than the center of $ K$. Form all chords of $ K$ which pass through $ P$, and determine their midpoints. The locus of these midpoints is
$ \textbf{(A)}\ \text{a circle with one point deleted} \qquad$
$ \textbf{(B)}\ \text{a circle if the distance from } P \text{ to the center of } K \text{ is less than}$
$ \text{one half the radius of } K \text{; otherwise a circular arc of less than}$
$ 360^{\circ}\qquad$
$ \textbf{(C)}\ \text{a semicircle with one point deleted} \qquad$
$ \textbf{(D)}\ \text{a semicircle} \qquad$
$ \textbf{(E)}\ \text{a circle}$
[b]p1.[/b] The Evergreen School booked buses for a field trip. Altogether, $138$ people went to West Lake, while $115$ people went to East Lake. The buses all had the same number of seats, and every bus has more than one seat. All seats were occupied and everybody had a seat. How many seats were there in each bus?
[b]p2.[/b] In New Scotland there are three kinds of coins: $1$ cent, $6$ cent, and $36$ cent coins. Josh has $50$ of the $36$-cent coins (and no other coins). He is allowed to exchange a $36$ cent coin for $6$ coins of $6$ cents, and to exchange a 6 cent coin for $6$ coins of $1$ cent. Is it possible that after several exchanges Josh will have $150$ coins?
[b]p3.[/b] Pinocchio multiplied two $2$ digit numbers. But witch Masha erased some of the digits. The erased digits are the ones marked with a $*$. Could you help Pinocchio to restore all the erased digits?
$\begin{tabular}{ccccc}
& & & 9 & 5 \\
x & & & * & * \\
\hline
& & & * & * \\
+ & 1 & * & * & \\
\hline
& * & * & * & * \\
\end{tabular}$
Find all solutions.
[b]p4.[/b] There are $50$ senators and $435$ members of House of Representatives. On Friday all of them voted a very important issue. Each senator and each representative was required to vote either "yes" or "no". The announced results showed that the number of "yes" votes was greater than the number of "no" votes by $24$. Prove that there was an error in counting the votes.
[b]p5.[/b] Was there a year in the last millennium (from $1000$ to $2000$) such that the sum of the digits of that year is equal to the product of the digits?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $a, b$, and $c$ be positive integers satisfying
$a^4 + a^2b^2 + b^4 = 9633$
$2a^2 + a^2b^2 + 2b^2 + c^5 = 3605$.
What is the sum of all distinct values of $a + b + c$?
In triangle \( ABC \), \( D \), \( E \), and \( F \) are the feet of the perpendiculars from the vertices \( A \), \( B \), and \( C \), respectively. The parallel to \( EF \) through \( D \) intersects \( AB \) at \( P_B \) and \( AC \) at \( P_C \). Let \( X \) be the intersection of \( EF \) and \( BC \). Prove that the circumcircle of triangle \( P_B P_C X \) passes through the midpoint of side \( BC \).
The equation $ x^3 \minus{} 9x^2 \plus{} 8x \plus{} 2 \equal{} 0$ has three real roots $ p$, $ q$, $ r$. Find $ \frac {1}{p^2} \plus{} \frac {1}{q^2} \plus{} \frac {1}{r^2}$.
Let $A_n$ be the set of real numbers such that each element of $A_n$ can be expressed as $1+\frac{a_1}{\sqrt{2}}+\frac{a_2}{(\sqrt{2})^2}+\cdots +\frac{a_n}{(\sqrt{n})^n}$ for given $n.$ Find both $|A_n|$ and sum of the products of two distinct elements of $A_n$ where each $a_i$ is either $1$ or $-1.$
Let $f:(0,+\infty)\to(0,+\infty)$ be a decreasing function which satisfies $\int^\infty_0f(x)\text dx<+\infty$. Prove that $\lim_{x\to+\infty}xf(x)=0$.
Let $x, y$, and $z$ be positive reals that satisfy the system
$$\begin{cases} x^2 + x y + y^2 = 10 \\ x^2 + xz + z^2 = 20 \\ y^2 + yz + z^2 = 30\end{cases}$$
Find $x y + yz + xz$.
How many distinct arrangements are possible for wearing five different rings in the five fingers of the right hand? (We can wear multiple rings in one finger)
[list=1]
[*] $\frac{10!}{5!}$
[*] $5^5$
[*] $\frac{9!}{4!}$
[*] None of these
[/list]
[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide]
[b]D1.[/b] What is the solution to the equation $3 \cdot x \cdot 5 = 4 \cdot 5 \cdot 6$?
[b]D2.[/b] Mr. Rose is making Platonic solids! If there are five different types of Platonic solids, and each Platonic solid can be one of three colors, how many different colored Platonic solids can Mr. Rose make?
[b]D3.[/b] What fraction of the multiples of $5$ between $1$ and $100$ inclusive are also multiples of $20$?
[b]D4.[/b] What is the maximum number of times a circle can intersect a triangle?
[b]D5 / L1.[/b] At an interesting supermarket, the nth apple you purchase costs $n$ dollars, while pears are $3$ dollars each. Given that Layla has exactly enough money to purchase either $k$ apples or $2k$ pears for $k > 0$, how much money does Layla have?
[b]D6 / L3.[/b] For how many positive integers $1 \le n \le 10$ does there exist a prime $p$ such that the sum of the digits of $p$ is $n$?
[b]D7 / L2.[/b] Real numbers $a, b, c$ are selected uniformly and independently at random between $0$ and $1$. What is the probability that $a \ge b \le c$?
[b]D8.[/b] How many ordered pairs of positive integers $(x, y)$ satisfy $lcm(x, y) = 500$?
[b]D9 / L4.[/b] There are $50$ dogs in the local animal shelter. Each dog is enemies with at least $2$ other dogs. Steven wants to adopt as many dogs as possible, but he doesn’t want to adopt any pair of enemies, since they will cause a ruckus. Considering all possible enemy networks among the dogs, find the maximum number of dogs that Steven can possibly adopt.
[b]D10 / L7.[/b] Unit circles $a, b, c$ satisfy $d(a, b) = 1$, $d(b, c) = 2$, and $d(c, a) = 3,$ where $d(x, y)$ is defined to be the minimum distance between any two points on circles $x$ and $y$. Find the radius of the smallest circle entirely containing $a$, $b$, and $c$.
[b]D11 / L8.[/b] The numbers $1$ through $5$ are written on a chalkboard. Every second, Sara erases two numbers $a$ and $b$ such that $a \ge b$ and writes $\sqrt{a^2 - b^2}$ on the board. Let M and m be the maximum and minimum possible values on the board when there is only one number left, respectively. Find the ordered pair $(M, m)$.
[b]D12 / L9.[/b] $N$ people stand in a line. Bella says, “There exists an assignment of nonnegative numbers to the $N$ people so that the sum of all the numbers is $1$ and the sum of any three consecutive people’s numbers does not exceed $1/2019$.” If Bella is right, find the minimum value of $N$ possible.
[b]D13 / L10.[/b] In triangle $\vartriangle ABC$, $D$ is on $AC$ such that $BD$ is an altitude, and $E$ is on $AB$ such that $CE$ is an altitude. Let F be the intersection of $BD$ and $CE$. If $EF = 2FC$, $BF = 8DF$, and $DC = 3$, then find the area of $\vartriangle CDF$.
[b]D14 / L11.[/b] Consider nonnegative real numbers $a_1, ..., a_6$ such that $a_1 +... + a_6 = 20$. Find the minimum possible value of $$\sqrt{a^2_1 + 1^2} +\sqrt{a^2_2 + 2^2} +\sqrt{a^2_3 + 3^2} +\sqrt{a^2_4 + 4^2} +\sqrt{a^2_5 + 5^2} +\sqrt{a^2_6 + 6^2}.$$
[b]D15 / L13.[/b] Find an $a < 1000000$ so that both $a$ and $101a$ are triangular numbers. (A triangular number is a number that can be written as $1 + 2 +... + n$ for some $n \ge 1$.)
Note: There are multiple possible answers to this problem. You only need to find one.
[b]L6.[/b] How many ordered pairs of positive integers $(x, y)$, where $x$ is a perfect square and $y$ is a perfect cube, satisfy $lcm(x, y) = 81000000$?
[b]L12.[/b] Given two points $A$ and $B$ in the plane with $AB = 1$, define $f(C)$ to be the incenter of triangle $ABC$, if it exists. Find the area of the region of points $f(f(X))$ where $X$ is arbitrary.
[b]L14.[/b] Leptina and Zandar play a game. At the four corners of a square, the numbers $1, 2, 3$, and $4$ are written in clockwise order. On Leptina’s turn, she must swap a pair of adjacent numbers. On Zandar’s turn, he must choose two adjacent numbers $a$ and $b$ with $a \ge b$ and replace $a$ with $ a - b$. Zandar wants to reduce the sum of the numbers at the four corners of the square to $2$ in as few turns as possible, and Leptina wants to delay this as long as possible. If Leptina goes first and both players play optimally, find the minimum number of turns Zandar can take after which Zandar is guaranteed to have reduced the sum of the numbers to $2$.
[b]L15.[/b] There exist polynomials $P, Q$ and real numbers $c_0, c_1, c_2, ... , c_{10}$ so that the three polynomials $P, Q$, and $$c_0P^{10} + c_1P^9Q + c_2P^8Q^2 + ... + c_{10}Q^{10}$$ are all polynomials of degree 2019. Suppose that $c_0 = 1$, $c_1 = -7$, $c_2 = 22$. Find all possible values of $c_{10}$.
Note: The answer(s) are rational numbers. It suffices to give the prime factorization(s) of the numerator(s) and denominator(s).
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $A$ denote the set of all the positive integer divisors of $30.$ For each nonempty subset $s \subseteq A,$ define $p(s)$ to be the product of the elements in $s.$ Finally, let $B$ denote the set of all possible remainders when $p(s)$ is divided by $30.$ How many (distinct) elements are in $B?$
A circular loop with radius $r$ spins with angular frequency $\omega$ in a generated magnetic field of strength $B$. It is hooked to a resistor load $R$. How much work is done by the generator in one revolution?
[i]Problem proposed by Ahaan Rungta[/i]