Found problems: 85335
Let $a_1,a_2,\dots,a_{2018}$ be a strictly increasing sequence of positive integers such that $$a_1+a_2+\cdots+a_{2018}=2018^{2018}.$$
What is the remainder when $a_1^3+a_2^3+\cdots+a_{2018}^3$ is divided by $6$?
$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$
We call an $n\times n$ matrix [i]well groomed[/i] if it only contains elements $0$ and $1$, and it does not contain the submatrix $\begin{pmatrix}
1& 0\\
0 & 1
\end{pmatrix}.$ Show that there exists a constant $c>0$ such that every well groomed, $n\times n$ matrix contains a submatrix of size at least $cn\times cn$ such that all of the elements of the submatrix are equal. (A well groomed matrix may contain the submatrix $\begin{pmatrix}
0& 1\\
1 & 0
\end{pmatrix}.$ )
We are given a triangle $ ABC$ such that $ AC \equal{} BC$. There is a point $ D$ lying on the segment $ AB$, and $ AD < DB$. The point $ E$ is symmetrical to $ A$ with respect to $ CD$. Prove that:
\[\frac {AC}{CD} \equal{} \frac {BE}{BD \minus{} AD}\]
In convex quadrilateral $ABCD$, $\angle CAB = \angle BCD$. $P$ lies on line $BC$ such that $AP = PC$, $Q$ lies on line $AP$ such that $AC$ and $DQ$ are parallel, $R$ is the point of intersection of lines $AB$ and $CD$, and $S$ is the point of intersection of lines $AC$ and $QR$. Line $AD$ meets the circumcircle of $AQS$ again at $T$. Prove that $AB$ and $QT$ are parallel.
Let $ x,y,z $ be real numbers. Prove that the following conditions are equivalent:
(i) $ x,y,z $ are positive numbers and $ \dfrac 1x + \dfrac 1y + \dfrac 1z \leq 1 $;
(ii) $ a^2x+b^2y+c^2z>d^2 $ holds for every quadrilateral with sides $ a,b,c,d $.
For each positive integer $k$, let $n_k$ be the smallest positive integer such that there exists a finite set $A$ of integers satisfy the following properties:
[list]
[*]For every $a\in A$, there exists $x,y\in A$ (not necessary distinct) that
$$n_k\mid a-x-y$$[/*]
[*]There's no subset $B$ of $A$ that $|B|\leq k$ and $$n_k\mid \sum_{b\in B}{b}.$$
[/list]
Show that for all positive integers $k\geq 3$, we've $$n_k<\Big( \frac{13}{8}\Big)^{k+2}.$$
In convex pentagon $ ABCDE$, denote by
$ AD\cap BE = F,BE\cap CA = G,CA\cap DB = H,DB\cap EC = I,EC\cap AD = J; AI\cap BE = A',BJ%Error. "capCA" is a bad command.
= B',CF%Error. "capDB" is a bad command.
= C',DG\cap EC = D',EH\cap AD = E'.$
Prove that $ \frac {AB'}{B'C}\cdot\frac {CD'}{D'E}\cdot\frac {EA'}{A'B}\cdot\frac {BC'}{C'D}\cdot\frac {DE'}{E'A} = 1$.
prove that for almost every real number $\alpha \in [0,1]$ there exists natural number $n_{\alpha} \in \mathbb N$ such that the inequality
$|\alpha-\frac{p}{q}|\le \frac{1}{q^n}$
for natural $n\ge n_{\alpha}$ and rational $\frac{p}{q}$ has no answers.
Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$
[b]p1.[/b] Compute the greatest integer less than or equal to $$\frac{10 + 12 + 14 + 16 + 18 + 20}{21}$$
[b]p2.[/b] Let$ A = 1$.$B = 2$, $C = 3$, $...$, $Z = 26$. Find $A + B +M + C$.
[b]p3.[/b] In Mr. M's farm, there are $10$ cows, $8$ chickens, and $4$ spiders. How many legs are there (including Mr. M's legs)?
[b]p4.[/b] The area of an equilateral triangle with perimeter $18$ inches can be expressed in the form $a\sqrt{b}{c}$ , where $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime. Find $a + b + c$.
[b]p5.[/b] Let $f$ be a linear function so $f(x) = ax + b$ for some $a$ and $b$. If $f(1) = 2017$ and $f(2) = 2018$, what is $f(2019)$?
[b]p6.[/b] How many integers $m$ satisfy $4 < m^2 \le 216$?
[b]p7.[/b] Allen and Michael Phelps compete at the Olympics for swimming. Allen swims $\frac98$ the distance Phelps swims, but Allen swims in $\frac59$ of Phelps's time. If Phelps swims at a rate of $3$ kilometers per hour, what is Allen's rate of swimming? The answer can be expressed as $m/n$ for relatively prime positive integers $m, n$. Find $m + n$.
[b]p8.[/b] Let $X$ be the number of distinct arrangements of the letters in "POONAM," $Y$ be the number of distinct arrangements of the letters in "ALLEN" and $Z$ be the number of distinct arrangements of the letters in "NITHIN." Evaluate $\frac{X+Z}{Y}$ :
[b]p9.[/b] Two overlapping circles, both of radius $9$ cm, have centers that are $9$ cm apart. The combined area of the two circles can be expressed as $\frac{a\pi+b\sqrt{c}+d}{e}$ where $c$ is not divisible by the square of any prime and the fraction is simplified. Find $a + b + c + d + e$.
[b]p10.[/b] In the Boxborough-Acton Regional High School (BARHS), $99$ people take Korean, $55$ people take Maori, and $27$ people take Pig Latin. $4$ people take both Korean and Maori, $6$ people take both Korean and Pig Latin, and $5$ people take both Maori and Pig Latin. $1$ especially ambitious person takes all three languages, and and $100$ people do not take a language. If BARHS does not o er any other languages, how many students attend BARHS?
[b]p11.[/b] Let $H$ be a regular hexagon of side length $2$. Let $M$ be the circumcircle of $H$ and $N$ be the inscribed circle of $H$. Let $m, n$ be the area of $M$ and $N$ respectively. The quantity $m - n$ is in the form $\pi a$, where $a$ is an integer. Find $a$.
[b]p12.[/b] How many ordered quadruples of positive integers $(p, q, r, s)$ are there such that $p + q + r + s \le 12$?
[b]p13.[/b] Let $K = 2^{\left(1+ \frac{1}{3^2} \right)\left(1+ \frac{1}{3^4} \right)\left(1+ \frac{1}{3^8}\right)\left(1+ \frac{1}{3^{16}} \right)...}$. What is $K^8$?
[b]p14.[/b] Neetin, Neeton, Neethan, Neethine, and Neekhil are playing basketball. Neetin starts out with the ball. How many ways can they pass 5 times so that Neethan ends up with the ball?
[b]p15.[/b] In an octahedron with side lengths $3$, inscribe a sphere. Then inscribe a second sphere tangent to the first sphere and to $4$ faces of the octahedron. The radius of the second sphere can be expressed in the form $\frac{\sqrt{a}-\sqrt{b}}{c}$ , where the square of any prime factor of $c$ does not evenly divide into $b$. Compute $a + b + c$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let ABC be a scalene triangle and AM is the median relative to side BC. The diameter circumference AM intersects for the second time the side AB and AC at points P and Q, respectively, both different from A. Assuming that PQ is parallel to BC, determine the angle measurement <BAC.
Any solution without trigonometry?
\[\int_1^\infty \frac{\sec^{-1}(x^{2})-\sec^{-1}(\sqrt x)}{x\log(x)}\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
In $\Delta ABC$, $AC=kAB$, with $k>1$. The internal angle bisector of $\angle BAC$ meets $BC$ at $D$. The circle with $AC$ as diameter cuts the extension of $AD$ at $E$. Express $\dfrac{AD}{AE}$ in terms of $k$.
Prove that if $a,b,c \ge 1$ and $a+b+c=9$, then $\sqrt{ab+bc+ca} \le \sqrt{a} +\sqrt{b} + \sqrt{c}$
If you pick a random $3$-digit number, what is the probability that its hundreds digit is triple the ones digit?
Find all pairs of natural numbers $a,b$ , with $a\ne b$ , such that $a+b$ and $ab+1$ are powers of $2$.
Bassanio has three red coins, four yellow coins, and five blue coins. At any point, he may give Shylock any two coins of different colors in exchange for one coin of the other color; for example, he may give Shylock one red coin and one blue coin, and receive one yellow coin in return. Bassanio wishes to end with coins that are all the same color, and he wishes to do this while having as many coins as possible. How many coins will he end up with, and what color will they be?
Given $n\geq 2$, $a_1$, $a_2$, $\cdots$, $a_n\in\mathbb {R}$ satisfy
$$a_1\geqslant a_2\geqslant \cdots \geqslant a_n\geqslant 0,a_1+a_2+\cdots +a_n=n.$$
Find the minimum value of $a_1+a_1a_2+\cdots +a_1a_2\cdots a_n$.
Does there exist a sequence of positive integers $a_1,a_2,...$ such that every positive integer occurs exactly once and that the number $\tau (na_{n+1}^n+(n+1)a_n^{n+1})$ is divisible by $n$ for all positive integer.
Here $\tau (n)$ denotes the number of positive divisor of $n$.
Let $ABC$ be an acute triangle with $AB<AC<BC$ and let $D$ be a point on it's extension of $BC$ towards $C$. Circle $c_1$, with center $A$ and radius $AD$, intersects lines $AC,AB$ and $CB$ at points $E,F$, and $G$ respectively. Circumscribed circle $c_2$ of triangle $AFG$ intersects again lines $FE,BC,GE$ and $DF$ at points $J,H,H' $ and $J'$ respectively. Circumscribed circle $c_3$ of triangle $ADE$ intersects again lines $FE,BC,GE$ and $DF$ at points $I,K,K' $ and $I' $ respectively. Prove that the quadrilaterals $HIJK$ and $H'I'J'K '$ are cyclic and the centers of their circumscribed circles coincide.
by Evangelos Psychas, Greece
Let \(\mathcal E\) be an ellipse with foci \(F_1\) and \(F_2\), and let \(P\) be a point on \(\mathcal E\). Suppose lines \(PF_1\) and \(PF_2\) intersect \(\mathcal E\) again at distinct points \(A\) and \(B\), and the tangents to \(\mathcal E\) at \(A\) and \(B\) intersect at point \(Q\). Show that the midpoint of \(\overline{PQ}\) lies on the circumcircle of \(\triangle PF_1F_2\).
[i]Proposed by Karthik Vedula[/i]
The plane is divided in $1\times1$ squares. In each square there is a real number such that it is the arithmetic mean of the four adjacent squares (with a common side). In a square there is $2024.$ Is it possible for $2024^{2024}$ to be written in another square if all the numbers are: a) nonnegative integers; b) integers?
There are five points in the plane, no three of which are collinear. Show that some four of these points are the vertices of a convex quadrilateral.
The numbers $3, 5, 7, a,$ and $b$ have an average (arithmetic mean) of $15$. What is the average of $a$ and $b$?
$\textbf{(A) } 0 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 30 \qquad\textbf{(D) } 45 \qquad\textbf{(E) } 60$
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for any real numbers $x$ and $y$ the following is true:
$$x^2+y^2+2f(xy)=f(x+y)(f(x)+f(y))$$