Found problems: 85335
A convex quadrilateral $ABCD$ with $AC \neq BD$ is inscribed in a circle with center $O$. Let $E$ be the intersection of diagonals $AC$ and $BD$. If $P$ is a point inside $ABCD$ such that $\angle PAB+\angle PCB=\angle PBC+\angle PDC=90^\circ$, prove that $O$, $P$ and $E$ are collinear.
In a plane there are four fixed points $A, B, C, D$, no $3$ collinear. Construct a square with sides $a, b, c, d$ such that $A \in a$, $B \in b$, $C \in c$, $D \in d$.
Prove that for any integer $n$ one can find integers $a$ and $b$ such that
\[n=\left[ a\sqrt{2}\right]+\left[ b\sqrt{3}\right] \]
Let $ABC$ be an acute triangle. Let $D$ be the reflection of point $B$ with respect to the line $AC$. Let $E$ be the reflection of point $C$ with respect to the line $AB$. Let $\Gamma_1$ be the circle that passes through $A, B$, and $D$. Let $\Gamma_2$ be the circle that passes through $A, C$, and $E$. Let $P$ be the intersection of $\Gamma_1$ and $\Gamma_2$ , other than $A$. Let $\Gamma$ be the circle that passes through $A, B$, and $C$. Show that the center of $\Gamma$ lies on line $AP$.
Regular $n$-gon is divided to triangles using $n-3$ diagonals of which none of them have common points with another inside polygon. How much among this triangles can there be the most not congruent?
[i]Proposed by Dusan Djukic[/i]
Let $n \in \mathbb{N}$ and $S_{n}$ the set of all permutations of $\{1,2,3,...,n\}$. For every permutation $\sigma \in S_{n}$ denote $I(\sigma) := \{ i: \sigma (i) \le i \}$.
Compute the sum $\sum_ {\sigma \in S_{n}} \frac{1}{|I(\sigma )|} \sum_ {i \in I(\sigma)} (i+ \sigma(i))$.
Find all couples of positive integers $m$ and $n$ such that
$$n!+5=m^3$$
We will say that two sets of distinct numbers are $\textit{linked}$ to each other if between any two numbers of each set lies at least one number of the other set. Is it possible to fill the cells of a $100 \times 200$ rectangle with distinct numbers so that any two rows of the rectangle are linked to one another, and any two columns of the rectangle are linked to one another?
Prove that there are are no positive integers $x$ and $y$ such that $x^5+y^5+1=(x+2)^5+(y-3)^5$.
[hide="Note"]
The restriction $x,y$ are positive isn't necessary.[/hide]
Find four different three-digit decimal numbers starting with the same digit, such that their sum is divisible by three of them.
It is known that from segments of lengths $a$, $b$ and $c$, a triangle can be formed. Could it happen that from segments of lengths $$\sqrt{a^2 + \frac{2}{3} bc},\quad \sqrt{b^2 + \frac{2}{3} ca}\quad \text{and} \quad \sqrt{c^2 + \frac{2}{3} ab},$$ a right-angled triangle can be formed?
Find all perfect squares $n$ such that if the positive integer $a\ge 15$ is some divisor $n$ then $a+15$ is a prime power.
[i]Proposed by Saudi Arabia[/i]
Determine the minimum value of $f (x)$ where
f (x) = (3 sin x - 4 cos x - 10)(3 sin x + 4 cos x - 10).
Given is a natural number $n > 5$. On a circular strip of paper is written a sequence of zeros and ones. For each sequence $w$ of $n$ zeros and ones we count the number of ways to cut out a fragment from the strip on which is written $w$. It turned out that the largest number $M$ is achieved for the sequence $11 00...0$ ($n-2$ zeros) and the smallest - for the sequence $00...011$ ($n-2$ zeros). Prove that there is another sequence of $n$ zeros and ones that occurs exactly $M$ times.
Two right circular cylinders have the same volume. The radius of the second cylinder is $10\%$ more than the radius of the first. What is the relationship between the heights of the two cylinders?
$\textbf{(A) }\text{The second height is 10\% less than the first.}$
$\textbf{(B) }\text{The first height is 10\% more than the second.}$
$\textbf{(C) }\text{The second height is 21\% less than the first.}$
$\textbf{(D) }\text{The first height is 21\% more than the second.}$
$\textbf{(E) }\text{The second height is 80\% of the first.}$
Prove that for all $x_1, x_2,\ldots , x_n \in \mathbb R$ the following inequality holds:
\[\sum_{n \geq i >j \geq 1} \cos^2(x_i - x_j ) \geq \frac{n(n-2)}{4}\]
Find all monic polynomials $f(x)$ in $\mathbb Z[x]$ such that $f(\mathbb Z)$ is closed under multiplication.
[i]By Mohsen Jamali[/i]
Let $AK$, $BL$ and $CM$ be the angle bisectors of a triangle $ABC$, with $K$ on $BC$. Let $P$ and $Q$ be the points on the lines $BL$ and $CM$ respectively such that $AP = PK$ and $AQ = QK$. Prove that $\angle PAQ = 90^o -\frac12 \angle B AC.$
(I Sharygin)
Is there a square with side lenght $\ell < 1$ that can completely cover any rectangle of diagonal $1$?
Noah has to fit 8 species of animals into 4 cages of the Arc. He planes to put two species of animal in each cage. It turns out that, for each species of animal, there are at most 3 other species with which it cannot share a cage. Prove that there is a way to assign the animals to the cages so that each species shares a cage with a compatible species.
The function $\psi : \mathbb{N}\rightarrow\mathbb{N}$ is defined by $\psi (n)=\sum_{k=1}^n\gcd (k,n)$.
$(a)$ Prove that $\psi (mn)=\psi (m)\psi (n)$ for every two coprime $m,n \in \mathbb{N}$.
$(b)$ Prove that for each $a\in\mathbb{N}$ the equation $\psi (x)=ax$ has a solution.
[i]20 problems for 25 minutes.[/i]
[b]p1.[/b] Ben was trying to solve for $x$ in the equation $6 + x = 1$. Unfortunately, he was reading upside-down and misread the equation as $1 = x + 9$. What is the positive difference between Ben's answer and the correct answer?
[b]p2.[/b] Anjali and Meili each have a chocolate bar shaped like a rectangular box. Meili's bar is four times as long as Anjali's, while Anjali's is three times as wide and twice as thick as Meili's. What is the ratio of the volume of Anjali's chocolate to the volume of Meili's chocolate?
[b]p3.[/b] For any two nonnegative integers $m, n$, not both zero, define $m?n = m^n + n^m$. Compute the value of $((2?0)?1)?7$.
[b]p4.[/b] Eliza is making an in-scale model of the Phillips Exeter Academy library, and her prototype is a cube with side length $6$ inches. The real library is shaped like a cube with side length $120$ feet, and it contains an entrance chamber in the front. If the chamber in Eliza's model is $0.8$ inches wide, how wide is the real chamber, in feet?
[b]p5.[/b] One day, Isaac begins sailing from Marseille to New York City. On the exact same day, Evan begins sailing from New York City to Marseille along the exact same route as Isaac. If Marseille and New York are exactly $3000$ miles apart, and Evan sails exactly 40 miles per day, how many miles must Isaac sail each day to meet Evan's ship in $30$ days?
[b]p6.[/b] The conversion from Celsius temperature C to Fahrenheit temperature F is: $$F = 1.8C + 32.$$ If the lowest temperature at Exeter one day was $20^o$ F, and the next day the lowest temperature was $5^o$ C higher, what would be the lowest temperature that day, in degrees Fahrenheit?
[b]p7.[/b] In a school, $60\%$ of the students are boys and $40\%$ are girls. Given that $40\%$ of the boys like math and $50\%$ of the people who like math are girls, what percentage of girls like math?
[b]p8.[/b] Adam and Victor go to an ice cream shop. There are four sizes available (kiddie, small, medium, large) and seventeen different flavors, including three that contain chocolate. If Victor insists on getting a size at least as large as Adam's, and Adam refuses to eat anything with chocolate, how many different ways are there for the two of them to order ice cream?
[b]p9.[/b] There are $10$ (not necessarily distinct) positive integers with arithmetic mean $10$. Determine the maximum possible range of the integers. (The range is defined to be the nonnegative difference between the largest and smallest number within a list of numbers.)
[b]p10.[/b] Find the sum of all distinct prime factors of $11! - 10! + 9!$.
[b]p11.[/b] Inside regular hexagon $ZUMING$, construct square $FENG$. What fraction of the area of the hexagon is occupied by rectangle $FUME$?
[b]p12.[/b] How many ordered pairs $(x, y)$ of nonnegative integers satisfy the equation $4^x \cdot 8^y = 16^{10}$?
[b]p13.[/b] In triangle $ABC$ with $BC = 5$, $CA = 13$, and $AB = 12$, Points $E$ and $F$ are chosen on sides $AC$ and $AB$, respectively, such that $EF \parallel BC$. Given that triangle $AEF$ and trapezoid $EFBC$ have the same perimeter, find the length of $EF$.
[b]p14.[/b] Find the number of two-digit positive integers with exactly $6$ positive divisors. (Note that $1$ and $n$ are both counted among the divisors of a number $n$.)
[b]p15.[/b] How many ways are there to put two identical red marbles, two identical green marbles, and two identical blue marbles in a row such that no red marble is next to a green marble?
[b]p16.[/b] Every day, Yannick submits $8$ more problems to the EMCC problem database than he did the previous day. Every day, Vinjai submits twice as many problems to the EMCC problem database as he did the previous day. If Yannick and Vinjai initially both submit one problem to the database on a Monday, on what day of the week will the total number of Vinjai's problems first exceed the total number of Yannick's problems?
[b]p17.[/b] The tiny island nation of Konistan is a cone with height twelve meters and base radius nine meters, with the base of the cone at sea level. If the sea level rises four meters, what is the surface area of Konistan that is still above water, in square meters?
[b]p18.[/b] Nicky likes to doodle. On a convex octagon, he starts from a random vertex and doodles a path, which consists of seven line segments between vertices. At each step, he chooses a vertex randomly among all unvisited vertices to visit, such that the path goes through all eight vertices and does not visit the same vertex twice. What is the probability that this path does not cross itself?
[b]p19.[/b] In a right-angled trapezoid $ABCD$, $\angle B = \angle C = 90^o$, $AB = 20$, $CD = 17$, and $BC = 37$. A line perpendicular to $DA$ intersects segment $BC$ and $DA$ at $P$ and $Q$ respectively and separates the trapezoid into two quadrilaterals with equal area. Determine the length of $BP$.
[b]p20.[/b] A sequence of integers $a_i$ is defined by $a_1 = 1$ and $a_{i+1} = 3i - 2a_i$ for all integers $i \ge 1$. Given that $a_{15} = 5476$, compute the sum $a_1 + a_2 + a_3 + ...+ a_{15}$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Find
$$\lim_{x\rightarrow0}\frac{\sin(x)-x}{x\cos(x)-x}.$$
Let $ABCD$ be a convex cyclic quadrilateral with $AD=BD$. The diagonals $AC$ and $BD$ intersect in $E$. Let the incenter of triangle $\triangle BCE$ be $I$. The circumcircle of triangle $\triangle BIE$ intersects side $AE$ in $N$.
Prove
\[ AN \cdot NC = CD \cdot BN. \]
Consider a container of the shape obtained by revolving a segment of parabola $x = 1 + y^2$ around the $y$-axis as shown below. The container is initially empty. Water is poured at a constant rate of $1\, \text{cm}^3$ into the container. Let $h(t)$ be the height of water inside container at time $t$. Find the time $t$ when the rate of change of $h(t)$ is maximum.