Found problems: 85335
Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$.
Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.
Let $a, b, c$ be non-negative reals with $a + b + c \le 2$. prove that $$\sqrt{b^2+ac} + \sqrt{a^2+bc} + \sqrt{c^2+ab} \le 3$$
Given a sequence $\{a_n\}$: $a_1=1, a_{n+1}=\left\{
\begin{array}{lcr}
a_n+n,\quad a_n\le n, \\
a_n-n,\quad a_n>n,
\end{array}
\right.
\quad n=1,2,\cdots.$
Find the number of positive integers $r$ satisfying $a_r<r\le 3^{2017}$.
Let $f$ be a real-valued function with $n+1$ derivatives at each point of $\mathbb R$. Show that for each pair of real numbers $a$, $b$, $a<b$, such that
$$\ln\left( \frac{f(b)+f'(b)+\cdots + f^{(n)} (b)}{f(a)+f'(a)+\cdots + f^{(n)}(a)}\right)=b-a$$
there is a number $c$ in the open interval $(a,b)$ for which
$$f^{(n+1)}(c)=f(c)$$
[u]Fermi Questions[/u]
[b]p1.[/b] What is $\sin (1000)$? (note: that's $1000$ radians, not degrees)
[b]p2.[/b] In liters, what is the volume of $10$ million US dollars' worth of gold?
[b]p3.[/b] How many trees are there on Earth?
[b]p4.[/b] How many prime numbers are there between $10^8$ and $10^9$?
[b]p5.[/b] What is the total amount of time spent by humans in spaceflight?
[b]p6.[/b] What is the global domestic product (total monetary value of all goods and services produced in a country's borders in a year) of Bangladesh in US dollars?
[b]p7.[/b] How much time does the average American spend eating during their lifetime, in hours?
[b]p8.[/b] How many CHMMC-related emails did the directors receive or send in the last month?
[u]Suspiciously Familiar. . .[/u]
[b]p9.[/b] Suppose a farmer learns that he will die at the end of the year (day $365$, where today is day $0$) and that he has $100$ sheep. He decides to sell all his sheep on one day, and that his utility is given by $ab$ where $a$ is the money he makes by selling the sheep (which always have a fixed price) and $b$ is the number of days he has left to enjoy the profit; i.e., $365 - k$ where $k$ is the day number. If every day his sheep breed and multiply their numbers by $(421 + b)/421$ (yes, there are small, fractional sheep), on which day should he sell out?
[b]p10.[/b] Suppose in your sock drawer of $14$ socks there are $5$ different colors and $3$ different lengths present. One day, you decide you want to wear two socks that have either different colors or different lengths but not both. Given only this information, what is the maximum number of choices you might have?
[u]I'm So Meta Even This Acronym[/u]
[b]p11.[/b] Let $\frac{s}{t}$ be the answer of problem $13$, written in lowest terms. Let $\frac{p}{q}$ be the answer of problem $12$, written in lowest terms.
If player $1$ wins in problem $11$, let $n = q$. Otherwise, let $n = p$.
Two players play a game on a connected graph with $n$ vertices and $t$ edges. On each player's turn, they remove one edge of the graph, and lose if this causes the graph to become disconnected. Which player (first or second) wins?
[b]p12.[/b] Let $\frac{s}{t}$ be the answer of problem $13$, written in lowest terms.
If player $1$ wins in problem $11$, let $n = t$. Otherwise, let $n = s$.
Find the maximum value of
$$\frac{x^n}{1 + \frac12 x + \frac14 x^2 + ...+ \frac{1}{2^{2n}} x^{2n}}$$ for $x > 0$.
[b]p13.[/b] Let $\frac{p}{q}$ be the answer of problem $12$, written in lowest terms.
Let $y$ be the largest integer such that $2^y$ divides $p$.
If player $1$ wins in problem $11$, let $z = q$. Otherwise, let $z = p$.
Suppose that $a_1 = 1$ and $$a_{n+1} = a_n -\frac{z}{n + 2}+\frac{2z}{n + 1}-\frac{z}{n}$$
What is $a_y$?
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
If $f(x)=x^n-7x^{n-1}+17x^{n-2}+a_{n-3}x^{n-3}+\ldots+a_0$ is a real-valued function of degree $n>2$ with all real roots, prove that no root has value greater than $4$ and at least one root has value less than $0$ or greater than $2$.
Prove that for every positive integer $n$ there exists a unique ordered pair $(a,b)$ of positive integers such that
\[ n = \frac{1}{2}(a + b - 1)(a + b - 2) + a . \]
For any natural numbers $n$, ( $n \geq 3$), let $f(n)$ denote the number of congruent integer-sided triangles with perimeter $n$. Show that
(i) $f(1999) > f (1996)$;
(ii) $f(2000) = f(1997)$.
Let $XY Z$ be a triangle with $\angle XY Z = 40^o$ and $\angle Y ZX = 60^o$. A circle $\Gamma$, centered at the point $I$, lies inside triangle $XY Z$ and is tangent to all three sides of the triangle. Let $A$ be the point of tangency of $\Gamma$ with $Y Z$, and let ray $\overrightarrow{XI}$ intersect side $Y Z$ at $B$. Determine the measure of $\angle AIB$.
Do there exist distinct positive integers $a$, $b$, $c$ such that $a$, $b$, $c$, $-a+b+c$, $a-b+c$, $a+b-c$, $a+b+c$ form an arithmetic progression (in some order).
Is there an infinite periodic sequence of digits for which the following condition condition is fulfilled: for any natural number $n{}$ a natural number divisible by $2^n{}$ can be cut from this sequence of digits (as a word)?
[i]Proposed by P. Kozhevnikov[/i]
Prove that the numbers $1,2,...,1998$ cannot be separated into three classes whose sums of elements are divisible by $2000,3999$, and $5998$, respectively.
Let $ \Delta ABC$ be a triangle, with $ m(\angle A)=90^{\circ}$ and $ m(\angle B)=30^{\circ}.$
If $M$ is the middle of $[AB],$ $N$ is the middle of $[BC],$ and $P\in[BC],\ Q\in[MN],$ such that
\[\frac{PB}{PC}=4\cdot\frac{QM}{QN}+3,\]
prove that $ \Delta APQ$ is an equilateral triangle.
[b]Author: MARIN BANCOȘ[/b]
[b]Regional Mathematical Contest GRIGORE MOISIL, Romania, Baia Mare, 24.03.2012, 7th grade[/b]
A point $P$ is chosen inside a convex quadrilateral $ABCD$. Could it happen that$$PA = AB, \quad PB = BC, \quad PC = CD \quad \text{and} \quad PD = DA?$$
The digits of a calculator (with the exception of 0) are shown in the form indicated by the figure below, where there is also a button ``+":
[img]6965[/img]
Two players $A$ and $B$ play in the following manner: $A$ turns on the calculator and presses a digit, and then presses the button ``+". $A$ passes the calculator to $B$, which presses a digit in the same row or column with the one pressed by $A$ that is not the same as the last one pressed by $A$; and then presses + and returns the calculator to $A$, repeating the operation in this manner successively. The first player that reaches or exceeds the sum of 31 loses the game. Which of the two players have a winning strategy and what is it?
Define the sequence $\{a_i\}$ by $a_0=1$, $a_1=4$, and $a_{n+1}=5a_n-a_{n-1}$ for all $n\geq 1$. Show that all terms of the sequence are of the form $c^2+3d^2$ for some integers $c$ and $d$.
In a triangle $\vartriangle ABC$, $\angle B = 2\angle C$. Let $P$ and $Q$ be points on the perpendicular bisector of segment $BC$ such that rays $AP$ and $AQ$ trisect $\angle A$. Prove that $PQ$ is smaller than $AB$ if and only if $\angle B$ is obtuse.
Let $C$ be the class of all real valued continuously differentiable functions $f$ on the interval $[0,1]$ with $f(0)=0$
and $f(1)=1 .$ Determine the largest real number $u$ such that
$$u \leq \int_{0}^{1} |f'(x) -f(x) | \, dx $$
for all $f$ in $C.$
In a company of seven people, any six can sit at a round table so that every two neighbors turn out to be acquaintances. Prove that the whole company can be seated at a round table so that every two neighbors turn out to be acquaintances.
[b]p1.[/b] Prove that for every point inside a regular polygon, the average of the distances to the sides equals the radius of the inscribed circle. The distance to a side means the shortest distance from the point to the line obtained by extending the side.
[b]p2.[/b] Suppose we are given positive (not necessarily distinct) integers $a_1, a_2,..., a_{1997}$ . Show that it is possible to choose some numbers from this list such that their sum is a multiple of $1997$.
[b]p3.[/b] You have Blue blocks, Green blocks and Red blocks. Blue blocks and green blocks are $2$ inches thick. Red blocks are $1$ inch thick. In how many ways can you stack the blocks into a vertical column that is exactly $12$ inches high? (For example, for height $3$ there are $5$ ways: RRR, RG, GR, RB, BR.)
[b]p4.[/b] There are $1997$ nonzero real numbers written on the blackboard. An operation consists of choosing any two of these numbers, $a$ and $b$, erasing them, and writing $a+b/2$ and $b-a/2$ instead of them. Prove that if a sequence of such operations is performed, one can never end up with the initial collection of numbers.
[b]p5.[/b] An $m\times n$ checkerboard (m and n are positive integers) is covered by nonoverlapping tiles of sizes $2\times 2$ and $1\times 4$. One $2\times 2$ tile is removed and replaced by a $1\times 4$ tile. Is it possible to rearrange the tiles so that they cover the checkerboard?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $x,y,z\geq0$ be real numbers such that $x+y+z=1$ Define $f(x,y,z)$ in this way :
\[f(x,y,z)=\frac{x(2y-z)}{1+x+3y}+\frac{y(2z-x)}{1+y+3z}+\frac{z(2x-y)}{1+z+3x}\]
Find the minimum value and maximum value of $f(x,y,z)$ .
Let $\triangle ABC$ be a triangle with $AB<AC$. Let the angle bisector of $\angle BAC$ meet $BC$ at $D$, and let $M$ be the midpoint of $\overline{BC}$. Let $P$ be the foot of the perpendicular from $B$ to $\overline{AD}$. Extend $\overline{BP}$ to meet $\overline{AM}$ at $Q$. Show that $\overline{DQ}$ is parallel to $\overline{AB}$.
We are given a triangle $ABC$. Points $D$ and $E$ on the line $AB$ are such that $AD=AC$ and $BE=BC$, with the arrangment of points $D - A - B - E$. The circumscribed circles of the triangles $DBC$ and $EAC$ meet again at the point $X\neq C$, and the circumscribed circles of the triangles $DEC$ and $ABC$ meet again at the point $Y\neq C$. Find the measure of $\angle ACB$ given the condition $DY+EY=2XY$.
Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that
$$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$
for all positive integers $n$. Show that $a_{2022}\leq 1$.
Let $a$ and $b$ be positive real numbers with the sum equal to $1$. Prove that if $a^3$ and $b^3$ are rational, so are $a$ and $b$.