Found problems: 85335
Let $ABC$ be an acute triangle with $AC>CB$ and let $M$ be the midpoint of side $AB$. Denote by $Q$ the midpoint of the big arc $AB$ which cointais $C$ and by $B_1$ the point inside $AC$ such that $BC=CB_1$. $B_1Q$ touches $BC$ in $E$ and $K$ is the intersection of $(BB_1M)$ and $(ABC)$. Prove that $KC$ bissects $B_1E$.
Prove that for every positive integer $n$ there exists a polynomial $p(x)$ of degree $n$ with real coefficients, having $n$ distinct real roots and satisfying
$$p(x)p(4-x)=p(x(4-x))$$
Real numbers $x_1,x_2,\ldots ,x_{1996}$ have the following property: For any polynomial $W$ of degree $2$ at least three of the numbers $W(x_1),W(x_2),\ldots ,W(x_{1996})$ are equal. Prove that at least three of the numbers $x_1,x_2,\ldots ,x_{1996}$ are equal.
For a non-empty finite set $A$ of positive integers, let $\text{lcm}(A)$ denote the least common multiple of elements in $A$, and let $d(A)$ denote the number of prime factors of $\text{lcm}(A)$ (counting multiplicity). Given a finite set $S$ of positive integers, and $$f_S(x)=\sum_{\emptyset \neq A \subset S} \frac{(-1)^{|A|} x^{d(A)}}{\text{lcm}(A)}.$$
Prove that, if $0 \le x \le 2$, then $-1 \le f_S(x) \le 0$.
A sequence $a_n$ is defined by
$$a_0=0,\qquad a_1=3;$$$$a_n=8a_{n-1}+9a_{n-2}+16\text{ for }n\ge2.$$Find the least positive integer $h$ such that $a_{n+h}-a_n$ is divisible by $1999$ for all $n\ge0$.
Let $N$ be a positive integer, and consider an $N \times N$ grid. A [i]right-down path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A [i]right-up path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell above the previous cell in the sequence.
Prove that the cells of the $N \times N$ grid cannot be partitioned into less than $N$ right-down or right-up paths. For example, the following partition of the $5 \times 5$ grid uses $5$ paths.
[asy]
size(4cm);
draw((5,-1)--(0,-1)--(0,-2)--(5,-2)--(5,-3)--(0,-3)--(0,-4)--(5,-4),gray+linewidth(0.5)+miterjoin);
draw((1,-5)--(1,0)--(2,0)--(2,-5)--(3,-5)--(3,0)--(4,0)--(4,-5),gray+linewidth(0.5)+miterjoin);
draw((0,0)--(5,0)--(5,-5)--(0,-5)--cycle,black+linewidth(2.5)+miterjoin);
draw((0,-1)--(3,-1)--(3,-2)--(1,-2)--(1,-4)--(4,-4)--(4,-3)--(2,-3)--(2,-2),black+linewidth(2.5)+miterjoin);
draw((3,0)--(3,-1),black+linewidth(2.5)+miterjoin);
draw((1,-4)--(1,-5),black+linewidth(2.5)+miterjoin);
draw((4,-3)--(4,-1)--(5,-1),black+linewidth(2.5)+miterjoin);
[/asy]
[i]Proposed by Zixiang Zhou, Canada[/i]
Let $ABCD$ be an isosceles trapezoid such that $AB \parallel CD$. Suppose that there exists a point $P$ in $ABCD$ such that $\angle APB > \angle ADC$ and $\angle DPC > \angle ABC$. Prove that $$AB+CD>DA+BC.$$
Consider the $5$ by $5$ equilateral triangular grid as shown: [img]https://cdn.artofproblemsolving.com/attachments/1/2/cac43ae24fd4464682a7992e62c99af4acaf8f.png[/img]
How many equilateral triangles are there with sides along the gridlines?
Starting from a positive integer $n$, we can replace the current number with a multiple of the current number or by deleting one or more zeroes from the decimal representation of the current number. Prove that for all values of $n$, it is possible to obtain a single-digit number by applying the above algorithm a finite number of times.
There is a nice solution to this...
If $x,y$ are positive real numbers satisfying $x^3-xy+1=y^3$, find the minimum possible value of $y$.
Find the number of squares such that the sides of the square are segments in the following diagram and where the dot is inside the square.
[center][img]https://snag.gy/qXBIY4.jpg[/img][/center]
Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that
\[
(2 + i)^n = a_n + b_ni
\]
for all integers $n\geq 0$, where $i = \sqrt{-1}$. What is \[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?\]
$\textbf{(A) }\frac 38\qquad\textbf{(B) }\frac7{16}\qquad\textbf{(C) }\frac12\qquad\textbf{(D) }\frac9{16}\qquad\textbf{(E) }\frac47$
Let $b \geqslant 2$ be a positive integer. Anu has an infinite collection of notes with exactly $b-1$ copies of a note worth $b^k-1$ rupees, for every integer $k\geqslant 1$. A positive integer $n$ is called payable if Anu can pay exactly $n^2+1$ rupees by using some collection of her notes. Prove that if there is a payable number, there are infinitely many payable numbers.
[i]Proposed by Shantanu Nene[/i]
Let $f:[a,b]\to[a,b]$ be a continuous function. It is known that there exist $\alpha,\beta\in (a,b)$ such that $f(\alpha)=a$ and $f(\beta)=b$. Prove that the function $f\circ f$ has at least three fixed points.
A right triangle has side lengths $a$, $b$, and $\sqrt{2016}$ in some order, where $a$ and $b$ are positive integers. Determine the smallest possible perimeter of the triangle.
What is the least possible value of $x^2 + y^2 - x - y - xy$ where $x, y$ are real numbers ?
Let \( \triangle ABC \) be an acute-angled triangle. Let \( D \) and \( E \) be the feet of the altitudes from \( B \) and \( C \), respectively. Let \( E' \) be the reflection of point \( E \) with respect to line \( BD \), which is assumed to lie on the circumcircle of triangle \( \triangle ABC \). Let \( C' \) be the reflection of point \( C \) with respect to line \( BD \). Prove that triangle \( C'AE \) is isosceles and determine the ratio \( AD : DC \).
A sequence is called an [i]arithmetic progression of the first order[/i] if the differences of the successive terms are constant. It is called an [i]arithmetic progression of the second order[/i] if the differences of the successive terms form an arithmetic progression of the first order. In general, for $k\geq 2$, a sequence is called an [i]arithmetic progression of the $k$-th order[/i] if the differences of the successive terms form an arithmetic progression of the $(k-1)$-th order.
The numbers
\[4,6,13,27,50,84\]
are the first six terms of an arithmetic progression of some order. What is its least possible order? Find a formula for the $n$-th term of this progression.
Let $A^\prime$ be the point of tangency of the excircle of a triangle $ABC$ (corrsponding to $A$) with the side $BC$. The line $a$ through $A^\prime$ is parallel to the bisector of $\angle BAC$. Lines $b$ and $c$ are analogously defined. Prove that $a, b, c$ have a common point.
Let $P_0 ,P_1 ,P_2 , ... ,P_n$ be $n+1$ points in the plane. Let $d$($d>0$) denote the minimal value of all the distances between any two points. Prove that
\[|P_0P_1|\cdot |P_0P_2|\cdot ... \cdot |P_0P_n|>(\frac{d}{3})^n\sqrt{(n+1)!}.\]
[b]p1.[/b] Evaluate $1+3+5+··· +2019$.
[b]p2.[/b] Evaluate $1^2 -2^2 +3^2 -4^2 +...· +99^2 -100^2$.
[b]p3. [/b]Find the sum of all solutions to $|2018+|x -2018|| = 2018$.
[b]p4.[/b] The angles in a triangle form a geometric series with common ratio $\frac12$ . Find the smallest angle in the triangle.
[b]p5.[/b] Compute the number of ordered pairs $(a,b,c,d)$ of positive integers $1 \le a,b,c,d \le 6$ such that $ab +cd$ is a multiple of seven.
[b]p6.[/b] How many ways are there to arrange three birch trees, four maple, and five oak trees in a row if trees of the same species are considered indistinguishable.
[b]p7.[/b] How many ways are there for Mr. Paul to climb a flight of 9 stairs, taking steps of either two or three at a time?
[b]p8.[/b] Find the largest natural number $x$ for which $x^x$ divides $17!$
[b]p9.[/b] How many positive integers less than or equal to $2018$ have an odd number of factors?
[b]p10.[/b] Square $MAIL$ and equilateral triangle $LIT$ share side $IL$ and point $T$ is on the interior of the square. What is the measure of angle $LMT$?
[b]p11.[/b] The product of all divisors of $2018^3$ can be written in the form $2^a \cdot 2018^b$ for positive integers $a$ and $b$. Find $a +b$.
[b]p12.[/b] Find the sum all four digit palindromes. (A number is said to be palindromic if its digits read the same forwards and backwards.
[b]p13.[/b] How ways are there for an ant to travel from point $(0,0)$ to $(5,5)$ in the coordinate plane if it may only move one unit in the positive x or y directions each step, and may not pass through the point $(1, 1)$ or $(4, 4)$?
[b]p14.[/b] A certain square has area $6$. A triangle is constructed such that each vertex is a point on the perimeter of the square. What is the maximum possible area of the triangle?
[b]p15.[/b] Find the value of ab if positive integers $a,b$ satisfy $9a^2 -12ab +2b^2 +36b = 162$.
[b]p16.[/b] $\vartriangle ABC$ is an equilateral triangle with side length $3$. Point $D$ lies on the segment $BC$ such that $BD = 1$ and $E$ lies on $AC$ such that $AE = AD$. Compute the area of $\vartriangle ADE$.
[b]p17[/b]. Let $A_1, A_2,..., A_{10}$ be $10$ points evenly spaced out on a line, in that order. Points $B_1$ and $B_2$ lie on opposite sides of the perpendicular bisector of $A_1A_{10}$ and are equidistant to $l$. Lines $B_1A_1,...,B_1A_{10}$ and $B_2A_1,...· ,B_2A_{10}$ are drawn. How many triangles of any size are present?
[b]p18.[/b] Let $T_n = 1+2+3··· +n$ be the $n$th triangular number. Determine the value of the infinite sum $\sum_{k\ge 1} \frac{T_k}{2^k}$.
[b]p19.[/b] An infinitely large bag of coins is such that for every $0.5 < p \le 1$, there is exactly one coin in the bag with probability $p$ of landing on heads and probability $1- p$ of landing on tails. There are no other coins besides these in the bag. A coin is pulled out of the bag at random and when flipped lands on heads. Find the probability that the coin lands on heads when flipped again.
[b]p20.[/b] The sequence $\{x_n\}_{n\ge 1}$ satisfies $x1 = 1$ and $(4+ x_1 + x_2 +··· + x_n)(x_1 + x_2 +··· + x_{n+1}) = 1$ for all $n \ge 1$. Compute $\left \lfloor \frac{x_{2018}}{x_{2019}} \right \rfloor$.
PS. You had better use hide for answers.
Determine the type of triangle if the lengths of its sides $a, b, c$ satisfy the relation $$a^4 + b^4 + c^4 = a^2b^2 + b^2c^2 + c^2a^2$$
In a function $f (x) = (x^2 + ax + b )/ (x^2 + cx + d)$ , the quadratics $x^2 + ax + b$ and $x^2 + cx + d$ have no common roots. Prove that the next two statements are equivalent:
(i) there is a numerical interval without any values of $f(x)$ ,
(ii) $f(x)$ can be represented in the form $f (x) = f_1 (f_2( ... f_{n-1} (f_n (x))... ))$ where each of the functions $f_j$ is o f one of the three forms $k_j x + b_j, 1/x, x^2$ .
(A Kanel)
We are given two two-digit numbers , $x$ and $y$. It is known that $x$ is twice as big as $y$. One of the digits of $y$ is the sum, while the other digit of $y$ is the difference, of the digits of $x$ . Find the values of $x$ and $y$, proving that there are no others.
The integers $1, 2, 3,. . . , 2016$ are written in a board. You can choose any pair of numbers in the board and replace them with their average. For example, you can replace $1$ and $2$ with $1.5$, or you can replace $1$ and $3$ with a second copy of $2$. After such replacements, the board will have only one number.
(a) Prove that there is a sequence of substitutions that will make let the final number be $2$.
(b) Prove that there is a sequence of substitutions that will make let the final number be $1000$.