Found problems: 85335
How many positive integers at most $420$ leave different remainders when divided by each of $5$, $6$, and $7$?
[i]Proposed by Milan Haiman.[/i]
Let $n\geq 1$ be a positive integer. We say that an integer $k$ is a [i]fan [/i]of $n$ if $0\leq k\leq n-1$ and there exist integers $x,y,z\in\mathbb{Z}$ such that
\begin{align*}
x^2+y^2+z^2 &\equiv 0 \pmod n;\\
xyz &\equiv k \pmod n.
\end{align*}
Let $f(n)$ be the number of fans of $n$. Determine $f(2020)$.
Calculate the value of $\sqrt{\dfrac{11^4+100^4+111^4}{2}}$ and answer in the form of an integer.
An equilateral triangle of side $n$ has been divided into little equilateral triangles of side $1$ in the usual way. We draw a path over the segments of this triangulation, in such a way that it visits exactly once each one of the $\frac{(n+1)(n+2)}{2}$ vertices. What is the minimum number of times the path can change its direction?
The figure below shows a valid path on a triangular board of side $4$, with exactly $9$ changes of direction.
[asy]
unitsize(30);
pair h = (1, 0);
pair v = dir(60);
pair d = dir(120);
for(int i = 0; i < 4; ++i)
{
draw(i*v -- i*v + (4 - i)*h);
draw(i*h -- i*h + (4 - i)*v);
draw((i + 1)*h -- (i + 1)*h + (i + 1)*d);
}
draw(h + v -- v -- (0, 0) -- 2*h -- 2*h + v -- h + 2*v -- 2*v -- 4*v -- 3*h + v -- 3*h -- 4*h, linewidth(2));
draw(3*h -- 4*h, EndArrow);
fill(circle(h + v, 0.1));
[/asy]
[i]Proposed by Oriol Solé[/i]
Let $P(x)$ be a polynomial such that $P(x)=x^3-2x^2+bx+c$. Roots of $P(x)$ belong to interval $(0,1)$. Prove that $8b+9c \leq 8$. When does equality hold?
The points $K$ and $L$ on the side $BC$ of a triangle $\triangle{ABC}$ are such that $\widehat{BAK}=\widehat{CAL}=90^\circ$. Prove that the midpoint of the altitude drawn from $A$, the midpoint of $KL$ and the circumcentre of $\triangle{ABC}$ are collinear.
[i](A. Akopyan, S. Boev, P. Kozhevnikov)[/i]
DK plays the following game in a simple graph: in each round, he does one of the two operations:
([i]i[/i]) Choose a vertex of odd degree and delete it. Before doing that, DK changes the relation between every two
neighbors of the chosen vertex (that is, if they were connected by an edge, then remove this edge, and, if this edge did not exist, then put this edge on the graph).
([i]ii[/i]) Choose a vertex of even degree and change the relation between every two neighbors of it (note that the chosen vertex is not deleted).
DK plays this game until there are no more edges on the graph. Show that the number of remaining vertices does not depend on the chosen operations.
Convex quadrilateral $ABCD$ has side-lengths $AB=7$, $BC=9$, $CD=15$, and there exists a circle, lying inside the quadrilateral and having center $I$, that is tangent to all four sides of the quadrilateral. Points $M$ and $N$ are the midpoints of $AC$ and $BD$ respectively. It can be proven that point $I$ always lies on segment $MN$. Supposing further that $I$ is the midpoint of $MN$, the area of quadrilateral $ABCD$ may be expressed as $p\sqrt q$, where $p$ and $q$ are positive integers and $q$ is not divisible by the square of any prime. Compute $p\cdot q$.
What is the smallest positive odd integer having the same number of positive divisors as $360$?
A function $f:\{ 1,2,3,\cdots ,2016\}\rightarrow \{ 1,2,3,\cdots , 2016\}$ is called [i]good[/i] if the function $g(n)=|f(n)-n|$ is injective. Furthermore, a good function $f$ is called [i]excellent[/i] if there exists another good function $f'$ such that $f(n)-f'(n)$ is nonzero for exactly one value of $n$. Let $N$ be the number of good functions that are not excellent. Find the remainder when $N$ is divided by $1000$.
[i]Proposed by Nathan Ramesh
Let $n$ be a fixed positive integer. Find the maximum possible value of \[ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, \] where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.
$m$ boys and $n$ girls ($m>n$) sat across a round table, supervised by a teacher, and they did a game, which went like this. At first, the teacher pointed a boy to start the game. The chosen boy put a coin on the table. Then, consecutively in a clockwise order, everyone did his turn. If the next person is a boy, he will put a coin to the existing pile of coins. If the next person is a girl, she will take a coin from the existing pile of coins. If there is no coin on the table, the game ends. Notice that depending on the chosen boy, the game could end early, or it could go for a full turn. If the teacher wants the game to go for at least a full turn, how many possible boys could be chosen?
[i]Hendrata Dharmawan, Boston, USA[/i]
The segments connecting the interior point of a convex non-sided $n$-gon with its vertices divide the $n$-gon into $n$ congruent triangles. For what is the smallest $n$ that is possible?
In any cell of an $n \times n$ table a number is written such that all the rows are distinct. Prove that we can remove a column such that the rows in the new table are still distinct.
Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, $CA = 15$. Let $H$ be the orthocenter of $ABC$. Find the distance between the circumcenters of triangles $AHB$ and $AHC$.
Let $n$ be a natural number larger than $2$. Vanessa has $n$ piles of jade stones, and all the piles have a different number of stones. Vanessa can distribute the stones from any pile by the other piles and stay with $n-1$ piles with the same number of stones. She also can distribute the stones from any two piles by the other piles and stay with $n-2$ piles with the same number of stones. Find the smallest possible number of jade's stones that the pile with the largest number of stones can have.
Given an acute triangle $ABC$, point $D$ is chosen on the side $AB$ and a point $E$ is chosen on the extension of $BC$ beyond $C$. It became known that the line through $E$ parallel to $AB$ is tangent to the circumcircle of $\triangle ADC$. Prove that one of the tangents from $E$ to the circumcircle of $\triangle BCD$ cuts the angle $\angle ABE$ in such a way that a triangle similar to $\triangle ABC$ is formed.
Find all functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that $$f(x)f\left(yf(x)-1\right)=x^2f(y)-f(x),$$for all $x,y \in \mathbb{R}$.
For an arbitrary natural $n$, prove the equality
$$\sin \frac{\pi}{2n}\sin \frac{3\pi}{2n}\sin \frac{5\pi}{2n}...\sin \frac{n'\pi}{2n}=2^{\dfrac{1-n}{2}}$$
where $n'$ is the largest odd number not exceeding $n$.
Let $x,y$ be complex numbers such that $\dfrac{x^2+y^2}{x+y}=4$ and $\dfrac{x^4+y^4}{x^3+y^3}=2$. Find all possible values of $\dfrac{x^6+y^6}{x^5+y^5}$.
Consider the sequence given by $a_0 = 3$ and such that for $i \ge 1$, we have $ai = 2^{a_{i-1}} + 1$. Let $m$ be the smallest integer such that $a^3_3$ divides $a_m$. Let $m'$ the smallest integer such that $a^3_m$ divides $a_{m'}$ . Find the value of $m'$.
Let $ABC$ be a triangle with $m(\widehat A) = 70^\circ$ and the incenter $I$. If $|BC|=|AC|+|AI|$, then what is $m(\widehat B)$?
$
\textbf{(A)}\ 35^\circ
\qquad\textbf{(B)}\ 36^\circ
\qquad\textbf{(C)}\ 42^\circ
\qquad\textbf{(D)}\ 45^\circ
\qquad\textbf{(E)}\ \text{None of above}
$
Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$.
Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$.
[i]Alternative formulation.[/i] Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines $AN$ and $CM$ intersect at a point $Q$.
Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$.
Sale prices at the Ajax Outlet Store are $ 50 \%$ below original prices. On Saturdays an additional discount of $ 20 \%$ off the sale price is given. What is the Saturday price of a coat whose original price is $ \$180$?
\[ \textbf{(A)}\ \$54 \qquad
\textbf{(B)}\ \$72 \qquad
\textbf{(C)}\ \$90 \qquad
\textbf{(D)}\ \$108 \qquad
\textbf{(E)}\ \$110
\]
Let $ \lambda $ be the Lebesgue measure in the plane, let $ u,v\in\mathbb{R}^2, $ let $ A\subset\mathbb{R}^2 $ such that $ \lambda (A)>0 $ and let be the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined as
$$ f(t)=\int_A \chi_{A+tu}\cdot\chi_{A+tv}\cdot d\lambda $$
[b]a)[/b] Show that $ f $ is continuous.
[b]b)[/b] Prove that any Lebesgue measurable subset of the plane that has nonzero Lebesgue measure contains the vertices of an equilateral triangle.