Found problems: 85335
The orthocenter $ H $ of a triangle $ ABC $ is distinct from its vertices and its circumcenter $ O. $ $ M,N,P $ are the circumcenters of the triangles $ HBC,HCA, $ respectively, $ HAB. $ Prove that $ AM,BN,CP $ and $ OH $ are concurrent.
An integer sequence satisfies $a_{n+1}={a_n}^3 +1999$. Show that it contains at most one square.
Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$. Prove that
\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]
[i]Proposed by Angelo Di Pasquale, Australia[/i]
Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied:
[list=1]
[*] Each number in the table is congruent to $1$ modulo $n$.
[*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$.
[/list]
Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.
In each cell of a chessboard with $2$ rows and $2019$ columns a real number is written so that:
[LIST]
[*] There are no two numbers written in the first row that are equal to each other.[/*]
[*] The numbers written in the second row coincide with (in some another order) the numbers written in the first row.[/*]
[*] The two numbers written in each column are different and they add up to a rational number.[/*]
[/LIST]
Determine the maximum quantity of irrational numbers that can be in the chessboard.
Let $ABC$ be a triangle with $AB=24$, $BC=30$, and $AC=36$. Point $M$ lies on $BC$ such that $BM=12$, and point $N$ lies on $AC$ such that $CN=20$. Let $X$ be the intersection of $AM$ and $BN$ and let line $CX$ intersect $AB$ at point $L$. Compute
$$\frac{AX}{XM}+\frac{BX}{XN}+\frac{CX}{XL}.$$
Show that
\[
1 - \frac{1}{k} \leq n\left(\sqrt[n]{k}-1\right) \leq k - 1
\]
for all positive integers $n$ and positive reals $k$.
For constant $a$, find the differentiable function $f(x)$ satisfying $\int_0^x (e^{-x}-ae^{-t})f(t)dt=0$.
Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.
Let $x, y, z$ be real numbers that are greater than or equal to $0$ and less than or equal to $\frac{1}{2}$
[list]
[*] (a) Determine the minimum possible value of \[x+y+z-xy-yz-zx\] and determine all triples $(x,y,z)$ for which this minimum is obtained.
[*] (b) Determine the maximum possible value of \[x+y+z-xy-yz-zx\] and determine all triples $(x,y,z)$ for which this maximum is obtained.[/list]
Determine all pairs $(a,b)$ of non-negative integers such that
$$ \frac{a+b}{2}-\sqrt{ab}=1.$$
Distinct points $B,B',C,C'$ lie on an arbitrary line $\ell$. $A$ is a point not lying on $\ell$. A line passing through $B$ and parallel to $AB'$ intersects with $AC$ in $E$ and a line passing through $C$ and parallel to $AC'$ intersects with $AB$ in $F$. Let $X$ be the intersection point of the circumcircles of $\triangle{ABC}$ and $\triangle{AB'C'}$($A \neq X$). Prove that $EF \parallel AX$.
Suppose that $ \tau$ is a metrizable topology on a set $ X$ of cardinality less than or equal to continuum. Prove that there exists a separable and metrizable topology on $ X$ that is coarser that $ \tau$.
[i]L. Juhasz[/i]
The vertices of $\triangle ABC$ are $A = (0,0)$, $B = (0,420)$, and $C = (560,0)$. The six faces of a die are labeled with two $A$'s, two $B$'s, and two $C$'s. Point $P_1 = (k,m)$ is chosen in the interior of $\triangle ABC$, and points $P_2$, $P_3$, $P_4, \dots$ are generated by rolling the die repeatedly and applying the rule: If the die shows label $L$, where $L \in \{A, B, C\}$, and $P_n$ is the most recently obtained point, then $P_{n + 1}$ is the midpoint of $\overline{P_n L}$. Given that $P_7 = (14,92)$, what is $k + m$?
In a rectangular coordinate system we call a horizontal line parallel to the $x$ -axis triangular if it intersects the curve with equation \[y = x^4 + px^3 + qx^2 + rx + s\] in the points $A,B,C$ and $D$ (from left to right) such that the segments $AB, AC$ and $AD$ are the sides of a triangle. Prove that the lines parallel to the $x$ - axis intersecting the curve in four distinct points are all triangular or none of them is triangular.
Kevin is trying to solve an economics question which has six steps. At each step, he has a probability $p$ of making a sign error. Let $q$ be the probability that Kevin makes an even number of sign errors (thus answering the question correctly!). For how many values of $0 \le p \le 1$ is it true that $p+q=1$?
[i]Proposed by Evan Chen[/i]
At a meeting of $ 12k$ people, each person exchanges greetings with exactly $ 3k\plus{}6$ others. For any two people, the number who exchange greetings with both is the same. How many people are at the meeting?
Compute the following: $$\sum^{99}_{x=0} (x^2 + 1)^{-1} \,\,\, (mod \,\,\,199)$$ where $x^{-1}$ is the value $0 \le y \le 199$ such that $xy - 1$ is divisible by $199$.
Given:
\[ S = 1 + \frac{1}{1 + \frac{1}{3}} + \frac{1}{1 + \frac{1}{3} + \frac{1} {6}} + \cdots + \frac{1}{1 + \frac{1}{3} + \frac{1}{6} + \cdots + \frac{1} {1993006}} \]
where the denominators contain partial sums of the sequence of reciprocals of triangular numbers (i.e. $k=\frac{n(n+1)}{2}$ for $n = 1$, $2$, $\ldots$,$1996$). Prove that $S>1001$.
If $x \in \mathbb{R}\backslash \left \{\frac{k\pi}{2}\ |\ k\in \mathbb{Z} \right \}$, then $$\left (\sum \limits_{0\leq j<k\leq n} \sin (2(j+k)x)\right )^2 + \left (\sum \limits_{0\leq j<k\leq n} \cos (2(j+k)x)\right )^2 = \frac{\sin ^2 nx \sin ^2 (n+1)x}{\sin ^2x \sin^22x}$$
Find all natural numbers $x,y>1$and primes $p$ that satisfy $$\frac{x^2-1}{y^2-1}=(p+1)^2. $$
Consider the set $S= \{(a + b)^7 - a^7 - b^7 : a,b \in Z\}$. Find the greatest common divisor of all members in $S$.
Let $D$ be the point on the base $BC$ of an isosceles $\vartriangle ABC$ triangle such that $\frac{\left| BD \right|}{\left| DC \right|}=\text{ }2$, and let $P$ be the point on the segment $\left[ AD \right]$ such that $\angle BAC=\angle BPD$. Prove that $\angle DPC=\frac{1}{2}\angle BAC$.
We choose 5 points $A_1, A_2, \ldots, A_5$ on a circumference of radius 1 and centre $O.$ $P$ is a point inside the circle. Denote $Q_i$ as the intersection of $A_iA_{i+2}$ and $A_{i+1}P$, where $A_7 = A_2$ and $A_6 = A_1.$ Let $OQ_i = d_i, i = 1,2, \ldots, 5.$ Find the product $\prod^5_{i=1} A_iQ_i$ in terms of $d_i.$
For a positive integer $n$, define $d(n)$ to be the sum of the digits of $n$ when written in binary (for example, $d(13)=1+1+0+1=3$). Let
\[
S=\sum_{k=1}^{2020}(-1)^{d(k)}k^3.
\]
Determine $S$ modulo $2020$.