Found problems: 85335
Let a triangle $ABC$ inscribed in a circle $\Gamma$ with center $O$. Let $I$ the incenter of triangle $ABC$ and $D, E, F$ the contact points of the incircle with sides $BC, AC, AB$ of triangle $ABC$ respectively . Let also $S$ the foot of the perpendicular line from $D$ to the line $EF$.Prove that line $SI$ passes from the antidiametric point $N$ of $A$ in the circle $\Gamma$.( $AN$ is a diametre of the circle $\Gamma$).
Regular hexagon $NOSAME$ with side length $1$ and square $UDON$ are drawn in the plane such that $UDON$ lies outside of $NOSAME$. Compute $[SAND] + [SEND]$, the sum of the areas of quadrilaterals $SAND$ and $SEND$.
The sets $A$ and $B$ are subsets of the positive integers. The sum of any two distinct elements of $A$ is an element of $B$. The quotient of any two distinct elements of $B$ (where we divide the largest by the smallest of the two) is an element of $A$. Determine the maximum number of elements in $A\cup B$.
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.
An acute triangle $ABC$ with $AB<AC<BC$ is inscribed in a circle $c(O,R)$. The circle $c_1(A,AC)$ intersects the circle $c$ at point $D$ and intersects $CB$ at $E$. If the line $AE$ intersects $c$ at $F$ and $G$ lies in $BC$ such that $EB=BG$, prove that $F,E,D,G$ are concyclic.
Let $G,A_1,A_2,A_3,A_4,B_1,B_2,B_3,B_4,B_5$ be ten points on a circle such that $GA_1A_2A_3A_4$ is a regular pentagon and $GB-1B_2B_3B_4B_5$ is a regular hexagon, and $B_1$ lies on minor arc $GA_1$. Let $B_5B_3$ intersect $B_1A_2$ at $G_1$, and let $B_5A_3$ intersect $GB_3$ at $G_2$. Determine the degree measure of $\angle GG2G_1$.
Pentagon \(ABCDE\) is inscribed in a circle. Its diagonals \(AC\) and \(BD\) intersect at \(F\). The bisectors of \(\angle BAC\) and \(\angle CDB\) intersect at \(G\). Let \(AG\) intersect \(BD\) at \(H\), let \(DG\) intersect \(AC\) at \(I\), and let \(EG\) intersect \(AD\) at \(J\). If \(FHGI\) is cyclic and \[JA \cdot FC \cdot GH = JD \cdot FB \cdot GI,\] prove that \(G\), \(F\) and \(E\) are collinear.
In space $4$ points are given. How many planes equidistant from these points are there? Consider separately
(a) the generic case (the points given do not lie on a single plane) and
(b) the degenerate cases.
If $a_k >0$ [ $k=$1, 2, $\cdots$, $n$], then prove the following inequality $$\left (\sum \limits_{k=1}^n a_k^5 \right )^4 \geq \frac{1}{n} \left (\frac{2}{n-1} \right )^5 \left (\sum \limits_{1\leq i<j\leq n} a_i^2a_j^2 \right )^5$$
Let $x, y, z$ be real numbers such that $$\displaystyle{\begin{cases} x^2+y^2+z^2+(x+y+z)^2=9 \\ xyz \leq \frac{15}{32} \end{cases}} $$
Find the maximum possible value of $x.$
An $n$-sum of type $1$ is a finite sequence of positive integers $a_1,a_2,\ldots,a_r$, such that:
$(1)$ $a_1+a_2+\ldots+a_r=n$;
$(2)$ $a_1>a_2+a_3,a_2>a_3+a_4,\ldots, a_{r-2}>a_{r-1}+a_r$, and $a_{r-1}>a_r$. For example, there are five $7$-sums of type $1$, namely: $7$; $6,1$; $5,2$; $4,3$; $4,2,1$. An $n$-sum of type $2$ is a finite sequence of positive integers $b_1,b_2,\ldots,b_s$ such that:
$(1)$ $b_1+b_2+\ldots+b_s=n$;
$(2)$ $b_1\ge b_2\ge\ldots\ge b_s$;
$(3)$ each $b_i$ is in the sequence $1,2,4,\ldots,g_j,\ldots$ defined by $g_1=1$, $g_2=2$, $g_j=g_{j-1}+g_{j-2}+1$; and
$(4)$ if $b_1=g_k$, then $1,2,4,\ldots,g_k$ is a subsequence. For example, there are five $7$-sums of type $2$, namely: $4,2,1$; $2,2,2,1$; $2,2,1,1,1$; $2,1,1,1,1,1$; $1,1,1,1,1,1,1$. Prove that for $n\ge1$ the number of type $1$ and type $2$ $n$-sums is the same.
Isosceles $\triangle{ABC}$ has interior point $O$ such that $AO = \sqrt{52}$, $BO = 3$, and $CO = 5$. Given that $\angle{ABC}=120^{\circ}$, find the length $AB$.
[i]Proposed by Powell Zhang[/i]
How many $\textit{odd}$ four-digit integers have the property that their digits, read left to right, are in strictly decreasing order?
Jasper and Rose are playing a game. Twenty-six $32$-ounce jugs are in a line, labeled Quart $\text{A}$ through Quart $\text{Z}$ from left to right. All twenty-six jugs are initially full. Jasper and Rose take turns making one of the following two moves:
[list]
[*] remove a positive integer number of ounces (possibly all) from the leftmost nonempty jug, or
[*] remove an [i]equal[/i] positive integer number of ounces from the two leftmost nonempty jugs, possibly
emptying them. Neither player may remove more ounces from a jug than it currently contains.
[/list]
Jasper plays first. A player’s score is the number of ounces they take from Quart $\text{Z}.$ If both players play to maximize their score, compute the maximum score that Jasper can guarantee.
Determine which integers $n > 1$ have the property that there exists an infinite sequence $a_1, a_2, a_3, \ldots$ of nonzero integers such that the equality \[a_k+2a_{2k}+\ldots+na_{nk}=0\]holds for every positive integer $k$.
$ 1990-1980+1970-1960+\cdots-20+10 = $
$ \text{(A)}\ -990\qquad\text{(B)}\ -10\qquad\text{(C)}\ 990\qquad\text{(D)}\ 1000\qquad\text{(E)}\ 1990 $
Let $ n\ge 2 $ be a perfect square and let be $ n $ natural numbers $ m_1,m_2,\ldots ,m_n. $ Prove that if the polynom
$$ X^2-\left( 1+ m_1^2+m_2^2+\cdots +m_n^2 \right) X+m_1m_2+m_2m_3+\cdots +m_{n-1}m_n +m_nm_1\in \mathbb{N} [X] $$
is reducible, then its two roots are perfect squares.
Find a positive integer $n$ with five non-zero different digits, which satisfies to be equal to the sum of all the three-digit numbers that can be formed using the digits of $n$.
Let $T_k = k - 1$ for $k = 1, 2, 3,4$ and
\[T_{2k-1} = T_{2k-2} + 2^{k-2}, T_{2k} = T_{2k-5} + 2^k \qquad (k \geq 3).\]
Show that for all $k$,
\[1 + T_{2n-1} = \left[ \frac{12}{7}2^{n-1} \right] \quad \text{and} \quad 1 + T_{2n} = \left[ \frac{17}{7}2^{n-1} \right],\]
where $[x]$ denotes the greatest integer not exceeding $x.$
$2020*N$ is a perfect cube. If $N$ can be expressed as $2^a*5^b*101^c,$ find the least possible value of $a+b+c$ such that $a,b,c$ are all positive integers and not necessarily distinct.
[i]Proposed by Ephram Chun[/i]
Given a $ 4\times 4$ squares table. How many ways that we can fill the table with $ \{0,1\}$ such that two neighbor squares (have one common side) have product which is equal to $ 0$?
$ABC$ is acute-angled triangle, $AA_1,CC_1$ are altitudes. $M$ is centroid. $M$ lies on circumcircle of $A_1BC_1$. Find all values of $\angle B$
Show that the equation $2x^2-3x=3y^2$ has infinitely many solutions in positive integers.
Solve the equation $[x]\{x\} = 1999x$, where $[x]$ denotes the largest integer less than or equal to $x$, and $\{x\} = x -[x] $
i) If $x = \left(1+\frac{1}{n}\right)^{n}$ and $y=\left(1+\frac{1}{n}\right)^{n+1}$, show that $y^{x}= x^{y}$.
ii) Show that, for all positive integers $n$, \[1^{2}-2^{2}+3^{2}-4^{2}+\cdots+(-1)^{n}(n-1)^{2}+(-1)^{n+1}n^{2}= (-1)^{n+1}(1+2+\cdots+n).\]