Found problems: 85335
In a triangle $ ABC$, the angle bisector at $ B$ intersects $ AC$ at $ D$ and the circumcircle again at $ E$. The circumcircle of the triangle $ DAE$ meets the segment $ AB$ again at $ F$. Prove that the triangles $ DBC$ and $ DBF$ are congruent.
Quadrilateral $ABCD$ is circumscribed. Its incircle touches sides $AB, BC, CD, DA$ in points $K, L, M, N$ respectively. Points $A', B', C', D'$ are the midpoints of segments $LM, MN, NK, KL$. Prove that the quadrilateral formed by lines $AA', BB', CC', DD'$ is cyclic.
Find the minimum value of $ f(a)=\int_{0}^{1}x|x-a|\ dx$.
Find all positive rational $(x,y)$ that satisfy the equation : $$yx^y=y+1$$
Does there exist any integer $a,b,c$ such that $a^2bc+2,ab^2c+2,abc^2+2$ are perfect squares?
Let $ABC$ be an equilateral triangle with side length $1$. Let $A_1$ and $A_2$ be the trisection points of $AB$ with $A_1$ closer to $A$, $B_1$ and $B_2$ be the trisection points of $BC$ with $B_1$ closer to $B$, and $C_1$ and $C_2$ be the trisection points of $CA$ with $C_1$ closer to $C$. Grogg has an orange equilateral triangle the size of triangle $A_1B_1C_1$. He puts the orange triangle over triangle $A_1B_1C_1$ and then rotates it about its center in the shortest direction until its vertices are over $A_2B_2C_2$. Find the area of the region that the orange triangle traveled over during its rotation.
Let $N$ be the number of consecutive 0's at the right end of the decimal representation of the product $1!\times2!\times3!\times4!\cdots99!\times100!.$ Find the remainder when $N$ is divided by 1000.
Given a triangle $ABC$, the line passing through the vertex $A$ symmetric to the median $AM$ wrt the line containing the bisector of the angle $\angle BAC$ intersects the circle circumscribed around the triangle $ABC$ at points $A$ and $K$. Let $L$ be the midpoint of the segment $AK$. Prove that $\angle BLC=2\angle BAC$.
What are the last two digits of $$2003^{{2005}^{{2007}^{2009}}}$$ , where $a^{b{^c}}$ means $a^{(b^c)}$?
Prove that for every positive integer $n$ and a non-negative real number $a$, the following inequality holds: $$n(n+1)a+2n \geqslant 4\sqrt{a}(\sqrt{1}+\sqrt{2}+\dots+\sqrt{n}).$$
Find all positive integers $a, b, c$ such that $ab+1$ and $c$ are coprimes and:
$$a(ba+1)(ca^2+ba+1)=2021^{2021}$$
Prove that the arithmetic progression $3,7,11,15,...$. contains infinitely many prime numbers.
In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on $20\%$ of her three-point shots and $30\%$ of her two-point shots. Shenille attempted $30$ shots. How many points did she score?
$ \textbf{(A)}\ 12\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 36 $
Given an acute triangle $ABC$ with altituties AD and BE. O circumcinter of $ABC$.If o lies on the segment DE then find the value of $sinAsinBcosC$
Determine whether the equation $x^2 +xy+y^2 = 2$ has a solution $(x,y)$ in rational numbers.
Let $ABC$ be an isosceles acute triangle ($AB = BC$). On the side $BC$ we mark a point $P$, such that $\angle PAC = 45^o$, and $Q$ is the point of intersection of the perpendicular bisector of the segment $AP$ with the side $AB$. Prove that $PQ \perp BC$.
Determine the integers $n\geqslant 2$ for which the equation $x^2-\hat{3}\cdot x+\hat{5}=\hat{0}$ has a unique solution in $(\mathbb{Z}_n,+,\cdot).$
Let $x_1, x_2, \ldots$ be a sequence of $0,1$, such that it satisfies the following three conditions:
1) $x_2=x_{100}=1$, $x_i=0$ for $1 \leq i \leq 100$ and $i \neq 2,100$;
2) $x_{2n-1}=x_{n-50}+1, x_{2n}=x_{n-50}$ for $51 \leq n \leq 100$;
3) $x_{2n}=x_{n-50}, x_{2n-1}=x_{n-50}+x_{n-100}$ for $n>100$.
Show that the sequence is periodic.
How many ordered integer pairs $(x,y)$ ($0\leq x,y < 31$) are there satisfying $(x^2-18)^2\equiv y^2 (\mod 31)$?
$ \textbf{(A)}\ 59
\qquad\textbf{(B)}\ 60
\qquad\textbf{(C)}\ 61
\qquad\textbf{(D)}\ 62
\qquad\textbf{(E)}\ \text{None}
$
Let $n{}$ be a non-negative integer and consider the standard power expansion of the following polynomial \[\sum_{k=0}^n\binom{n}{k}^2(X+1)^{2k}(X-1)^{2(n-k)}=\sum_{k=0}^{2n}a_kX^k.\]The coefficients $a_{2k+1}$ all vanish since the polynomial is invariant under the change $X\mapsto -X.$ Prove that the coefficients $a_{2k}$ are all positive.
Let $ a, b \in \mathbb{N}$ with $ 1 \leq a \leq b,$ and $ M \equal{} \left[\frac {a \plus{} b}{2} \right].$ Define a function $ f: \mathbb{Z} \mapsto \mathbb{Z}$ by
\[ f(n) \equal{} \begin{cases} n \plus{} a, & \text{if } n \leq M, \\
n \minus{} b, & \text{if } n >M. \end{cases}
\]
Let $ f^1(n) \equal{} f(n),$ $ f_{i \plus{} 1}(n) \equal{} f(f^i(n)),$ $ i \equal{} 1, 2, \ldots$ Find the smallest natural number $ k$ such that $ f^k(0) \equal{} 0.$
Let $p$ and $q=4p+1$ be prime numbers. Determine the least power $i$ of 2 for which $2^i\equiv 1\,(mod\, q)$.
An $m \times n$ grid of squares (with $m$ rows and $n$ columns) has some of its squares colored blue. The grid is called [i]fish-friendly[/i] if a fish can swim from the left edge of the grid to the right edge of the grid only moving through blue squares. In other words, there is a sequence of blue squares, each horizontally or vertically adjacent to the previous square, starting in the first column and ending in the last column.
Prove that the number of fish-friendly $42 \times 49$ grids is at least $2^{2022}$.
If P is a point inside a convex quadrilateral $ABCD$, let the angle bisectors of $\angle APB$, $\angle BPC$, $\angle CPD$ and $\angle DPA$ meet $AB$, $BC$, $CD$ and $DA$ at $K$, $L$, $M$ and $N$ respectively.
(a) Find a point $P$ such that $KLMN$ is a parallelogram.
(b) Find the locus of all such points $P$.
(S Tokarev)
Find all pairs $(x,y)$ of positive integers such that $x(x+y)=y^2+1.$