Found problems: 85335
Squares $ABCD$ and $BEFG$ are located as shown in the figure. It turned out that points $A, G$ and $E$ lie on the same straight line. Prove that then the points $D, F$ and $E$ also lie on the same line.
[img]https://cdn.artofproblemsolving.com/attachments/4/2/9faf29a399d3a622c84f5d4a3cfcf5e99539c0.png[/img]
There are rhombus $ABCD$ and circle $\Gamma_B$, which is centred at $B$ and has radius $BC$, and circle $\Gamma_C$, which is centred at $C$ and has radius $BC$. Circles $\Gamma_B$ and $\Gamma_C$ intersect at point $E$. The line $ED$ intersects $\Gamma_B$ at point $F$. Find all possible values of $\angle AFB$.
In how many regions can four straight lines divide the plane? List all possible cases.
Let $\mathcal{F}$ be a family of $4$-element subsets of a set of size $5^m$, where $m$ is a fixed positive integer. If the intersection of any two sets in $\mathcal{F}$ does not have size exactly $2$, find the maximal value of $|\mathcal{F}|$.
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $m,n\in \mathbb{N}$: \[f(f(m)+f(n))=m+n.\]
Define a sequence $(a_n)$ for $n\ge1$ by $a_1=2$ and $a_{n+1}=a_n^{1+n^{-3/2}}$. Is $(a_n)$ convergent (i.e. $\lim_{n\to\infty}a_n<\infty$)?
The circles $\gamma$ and $\delta$ are internally tangent to the circle $\omega$ at $A$ and $B$. From $A$, draw two tangent lines $\ell_1, \ell_2$ to $\delta$, . From $B$ draw two tangent lines $t_1, t_2$ to $\gamma$ . Let $\ell_1$ intersect $t_1$ at $X$ and $\ell_2$ intersect $t_2$ at $Y$ . Prove that the quadrilateral $AX BY$ is cyclic.
Nguyen Van Linh, High School of Natural Sciences, Hanoi National University
Using the fact that
$$\sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6},$$
compute
$$\int_0^1(\ln x)\ln(1-x)dx.$$
An [I]automorphism[/i] of a group \(\left(G,*\right)\) is a bijective function \(f:G\to G\) satisfying \(f(x*y)=f(x)*f(y)\) for all \(x,y\in G\).
Find a group \((G,*)\) with fewer than \((201.6)^2=40642.56\) unique elements and exactly \(2016^2\) unique automorphisms.
[i]Proposed by the ICMC Problem Committee[/i]
Let $g(x) = ax^2 + bx + c$ a quadratic function with real coefficients such that the equation $g(g(x)) = x$ has four distinct real roots. Prove that there isn't a function $f$: $R--R$ such that $f(f(x)) = g(x)$ for all $x$ real
Let $k$ and $d$ be integers such that $k>1$ and $0\leq d<9$. Prove that there exists some integer $n$ such that the $k$th digit from the right of $2^n$ is $d$.
[i](tatari/nightmare)[/i]
Quadratic equation $ x^2\plus{}ax\plus{}b\plus{}1\equal{}0$ have 2 positive integer roots, for integers $ a,b$. Show that $ a^2\plus{}b^2$ is not a prime.
Consider a set of $6$ fixed points in the plane, with no three collinear. Between some pairs of these points, we may draw one arrow from one point to the other. How many possible configurations of arrows are there such that if there is an arrow from point $A$ to point $B$ and an arrow from $B$ to $C,$ then there is an arrow from $A$ to $C?$ Your score will be $16-\frac{1}{800}|\textbf{Your Answer}-\textbf{Actual Answer}|$ rounded to the nearest integer or zero, whichever is higher.
Let $ABC$ be a triangle with side lengths $AB=13$, $BC=15$, $CA=14$. Let $\ell$ be the line passing through $A$ parallel to $BC$. Define $H$ as the orthocenter of $\triangle ABC$, and extend $BH$ to intersect $AC$ at $E$ and $\ell$ at $G$. Similarly, extend $CH$ to intersect $AB$ at $F$ and $\ell$ at $D$. Let $M$ be the midpoint of $BC$, and let $AM$ intersect the circumcircle of $AEF$ again at $P$. The ratio $\frac{PD}{PG}$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Individual #15[/i]
A college student drove his compact car $ 120$ miles home for the weekend and averaged $ 30$ miles per gallon. On the return trip the student drove his parents' SUV and averaged only $ 20$ miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?
$ \textbf{(A)}\ 22 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 26 \qquad \textbf{(E)}\ 28$
Find the angles of the triangle $ABC$, if we know that its center $O$ of the circumscribed circle and the center $I_A$ of the exscribed circle (tangent to $BC$) are symmetric wrt $BC$.
(Bogdan Rublev)
Determine all triplets of positive integers $(p,m,n)$ such that $p$ is a prime, $m \neq n < 2p$ and $2 \nmid n$. Also, the following polynomial is reducible in $\mathbb{Z}[x]$
$$x^{2p} - 2px^m - p^2x^n - 1$$
Parallel lines $\ell_1$, $\ell_2$, $\ell_3$, $\ell_4$ are evenly spaced in the plane, in that order. Square $ABCD$ has the property that $A$ lies on $\ell_1$ and $C$ lies on $\ell_4$. Let $P$ be a uniformly random point in the interior of $ABCD$ and let $Q$ be a uniformly random point on the perimeter of $ABCD$. Given that the probability that $P$ lies between $\ell_2$ and $\ell_3$ is $\frac{53}{100}$ , the probability that $Q$ lies between $\ell_2$ and $\ell_3$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.
Let $\omega$ be the circumcircle of the triangle $ABC$ with $\angle B = 3\angle C$. The internal angle bisector of $\angle A$, intersects $\omega$ and $BC$ at $M$ and $D$, respectively. Point $E$ lies on the extension of the line $MC$ from $M$ such that $ME$ is equal to the radius of $\omega$. Prove that circumcircles of triangles $ACE$ and $BDM$ are tangent.
[i]Proposed by Mehran Talaei - Iran[/i]
by using the formula $\pi cot(\pi z)=\frac{1}{z}+\sum_{n=1}^{\infty}\frac{2z}{z^2-n^2}$ calculate values of $\zeta(2k)$ on terms of bernoli numbers and powers of $\pi$.
An isosceles triangle $ABC$ with base $AC$ is given. Point $H$ is the intersection of altitudes. On the sides $AB$ and $BC$, points $M$ and $K$ are selected, respectively, so that the angle $KMH$ is right. Prove that a right-angled triangle can be constructed from the segments $AK, CM$ and $MK$.
Find all $f$ functions from real numbers to itself such that for all real numbers $x,y$ the equation
\[f(f(y)+x^2+1)+2x=y+(f(x+1))^2\]
holds.
Let $ABC$ ($AB < AC$) be a triangle with circumcircle $k$. The tangent to $k$ at $A$ intersects the line $BC$ at $D$ and the point $E\neq A$ on $k$ is such that $DE$ is tangent to $k$. The point $X$ on line $BE$ is such that $B$ is between $E$ and $X$ and $DX = DA$ and the point $Y$ on the line $CX$ is such that $Y$ is between $C$ and $X$ and $DY = DA$. Prove that the lines $BC$ and $YE$ are perpendicular.
Let $a, b$ be positive real numbers, and let $x, y$ be complex numbers such that $|x| = a$ and $|y| = b$. Find the minimal and maximal value of
\[\left|\frac{x + y}{1 + x\overline{y}}\right|\]
If $x,y,z$ are positive integers and $z(xz+1)^2=(5z+2y)(2z+y)$, prove that $z$ is an odd perfect square.