Found problems: 85335
Let the sides $AD$ and $BC$ of the quadrilateral $ABCD$ (such that $AB$ is not parallel to $CD$) intersect at point $P$. Points $O_1$ and $O_2$ are circumcenters and points $H_1$ and $H_2$ are orthocenters of triangles $ABP$ and $CDP$, respectively. Denote the midpoints of segments $O_1H_1$ and $O_2H_2$ by $E_1$ and $E_2$, respectively. Prove that the perpendicular from $E_1$ on $CD$, the perpendicular from $E_2$ on $AB$ and the lines $H_1H_2$ are concurrent.
[i]Proposed by Eugene Bilopitov, Ukraine[/i]
Factorise $ 5^{1985}\minus{}1$ as a product of three integers, each greater than $ 5^{100}$.
A nonconstant polynomial $f$ with integral coefficients has the property that, for each prime $p$, there exist a prime $q$ and a positive integer $m$ such that $f(p) = q^m$. Prove that $f = X^n$ for some positive integer $n$.
[i]AMM Magazine[/i]
Given that $9\times10\times11\times\cdots\times15=32432400$, what is $1\times3\times5\times\cdots\times15$?
[i]2017 CCA Math Bonanza Team Round #1[/i]
Let $ ABC$ be a triangle with $ \angle A = 60^{\circ}$.Prove that if $ T$ is point of contact of Incircle And Nine-Point Circle, Then $ AT = r$,
$ r$ being inradius.
Let $a,b,c$ be positive real numbers with $a^2+b^2+c^2 \geq 3$. Prove that:
$$\frac{(a+1)(b+2)}{(b+1)(b+5)}+\frac{(b+1)(c+2)}{(c+1)(c+5)}+\frac{(c+1)(a+2)}{(a+1)(a+5)} \geq \frac{3}{2}.$$
Prove that exists a infinity of triplets $a, b, c\in\mathbb{R}$ satisfying simultaneously the relations $a+b+c=0$ and $a^4+b^4+c^4=50$.
Moldova National Math Olympiad 2010, 12th grade
A square is placed in the plane and a point $P$ is marked in this plane with invisible ink. A certain person can see this point through special glasses. One can draw a straight line and this person will say on which side of the line the point $P$ lies. If $P$ lies on the line, the person says so. What is the minimal number of questions one needs to find out if $P$ lies inside the square or not?
(Folklore)
Solve equation :
$\sqrt{x+\sqrt{4x+\sqrt{16x}+..+\sqrt{4^nx+3}}}-\sqrt{x}=1$
Find all pairs of natural numbers $ (a, b)$ such that $ 7^a \minus{} 3^b$ divides $ a^4 \plus{} b^2$.
[i]Author: Stephan Wagner, Austria[/i]
Triangle $ ABC$ has vertices $ A\equal{}(3,0)$, $ B\equal{}(0,3)$, and $ C$, where $ C$ is on the line $ x\plus{}y\equal{}7$. What is the area of $ \triangle ABC$?
$ \textbf{(A)}\ 6\qquad
\textbf{(B)}\ 8\qquad
\textbf{(C)}\ 10\qquad
\textbf{(D)}\ 12\qquad
\textbf{(E)}\ 14$
Sandwiches at Joe's Fast Food cost $ \$3$ each and sodas cost $ \$2$ each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas?
$ \textbf{(A) } 31\qquad \textbf{(B) } 32\qquad \textbf{(C) } 33\qquad \textbf{(D) } 34\qquad \textbf{(E) } 35$
For which positive integers $n$, one can find real numbers $x_1,x_2,\cdots ,x_n$ such that $$\dfrac{x_1^2+x_2^2+\cdots+x_n^2}{\left(x_1+2x_2+\cdots+nx_n\right)^2}=\dfrac{27}{4n(n+1)(2n+1)}$$ and $i\leq x_i\leq 2i$ for all $i=1,2,\cdots ,n$ ?
Let $ABC$ be a triangle with $\angle A=60^o$ and $T$ be a point such that $\angle ATB=\angle BTC=\angle ATC$. A circle passing through $B,C$ and $T$ meets $AB$ and $AC$ for the second time at points $K$ and $L$.Prove that the distances from $K$ and $L$ to $AT$ are equal.
Farmer John has $ 5$ cows, $ 4$ pigs, and $ 7$ horses. How many ways can he pair up the animals so that every pair consists of animals of different species? (Assume that all animals are distinguishable from each other.)
A chess knight has injured his leg and is limping. He alternates between a normal move and a short move where he moves to any diagonally neighbouring cell.
The limping knight moves on a $5 \times 6$ cell chessboard starting with a normal move. What is the largest number of moves he can make if he is starting from a cell of his own choice and is not allowed to visit any cell (including the initial cell) more than once?
Richard likes to solve problems from the IMO Shortlist. In 2013, Richard solves $5$ problems each Saturday and $7$ problems each Sunday. He has school on weekdays, so he ``only'' solves $2$, $1$, $2$, $1$, $2$ problems on each Monday, Tuesday, Wednesday, Thursday, and Friday, respectively -- with the exception of December 3, 2013, where he solved $60$ problems out of boredom. Altogether, how many problems does Richard solve in 2013?
[i]Proposed by Evan Chen[/i]
A twelve foot tree casts a five foot shadow. How long is Henry's shadow (at the same time of day) if he is five and a half feet tall?
A parabola $p$ is drawn on the coordinate plane — the graph of the equation $y =-x^2$, and a point $A$ is marked that does not lie on the parabola $p$. All possible parabolas $q$ of the form $y = x^2+ax+b$ are drawn through point $A$, intersecting $p$ at two points $X$ and $Y$ . Prove that all possible $XY$ lines pass through a fixed point in the plane.
[i]P.A.Kozhevnikov[/i]
We are writing either $0$ or $1$ to unit squares of an $8\times 8 $ chessboard. If the sum of numbers is even vertically, horizontally, or diagonally, what is the greatest possible value of the sum of the all numbers on the board?
$
\textbf{(A)}\ 32
\qquad\textbf{(B)}\ 48
\qquad\textbf{(C)}\ 52
\qquad\textbf{(D)}\ 56
\qquad\textbf{(E)}\ 64
$
Define $a*b=\frac{a-b}{1-ab}$. What is $(1*(2*(3*\cdots (n*(n+1))\cdots )))$?
Let $ABCD$ be a parallelogram in the plane. We draw two circles of radius $R$, one through the points $A$ and
$B$, the other through $B$ and $C$. Let $E$ be the other intersection point of the circles. We assume that $E$ is not a vertex of the parallelogram. Show that the circle passing through $A, D$, and $E$ also has radius $R$.
A sequence with first term $a_0$ is defined such that $a_{n+1}=2a_n^2-1$ for $n\geq0.$ Let $N$ denote the number of possible values of $a_0$ such that $a_0=a_{2020}.$ Find the number of factors of $N.$
[i]Proposed by Alex Li[/i]
A checkered table consists of $2012$ rows and $k > 2$ columns. A marker is placed in a cell of the left-most column. Two players move the marker in turns. During each move, the player moves the marker by $1$ cell to the right, up or down to a cell that had never been occupied by the marker before. The game is over when any of the players moves the marker to the right-most column. However, whether this player is to win or to lose, the players are advised only when the marker reaches the second column from the right. Can any player secure his win?
Let $\alpha$ be a real number in the interval $(0,1).$ Prove that there exists a sequence $(\varepsilon_n)_{n\geq 1}$ where each term is either $0$ or $1$ such that the sequence $(s_n)_{n\geq 1}$ \[s_n=\frac{\varepsilon_1}{n(n+1)}+\frac{\varepsilon_2}{(n+1)(n+2)}+...+\frac{\varepsilon_n}{(2n-1)2n}\]verifies the inequality \[0\leq \alpha-2ns_n\leq\frac{2}{n+1}\] for any $n\geq 2.$