Found problems: 85335
A non-isosceles triangle $\triangle ABC$ has its incircle tangent to $BC, CA, AB$ at points $D, E, F$.
Let the incenter be $I$. Say $AD$ hits the incircle again at $G$, at let the tangent to the incircle at $G$ hit $AC$ at $H$. Let $IH \cap AD = K$, and let the foot of the perpendicular from $I$ to $AD$ be $L$.
Prove that $IE \cdot IK= IC \cdot IL$.
Let $ABC$ be an acute-angled triangle with circumcircle $\omega$. A circle $\Gamma$ is internally tangent to $\omega$ at $A$ and also tangent to $BC$ at $D$. Let $AB$ and $AC$ intersect $\Gamma$ at $P$ and $Q$ respectively. Let $M$ and $N$ be points on line $BC$ such that $B$ is the midpoint of $DM$ and $C$ is the midpoint of $DN$. Lines $MP$ and $NQ$ meet at $K$ and intersect $\Gamma$ again at $I$ and $J$ respectively. The ray $KA$ meets the circumcircle of triangle $IJK$ again at $X\neq K$.
Prove that $\angle BXP = \angle CXQ$.
[i]Kian Moshiri, United Kingdom[/i]
The function $f$ is given by the table
\[\begin{array}{|c||c|c|c|c|c|}\hline x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 4 & 1 & 3 & 5 & 2 \\ \hline \end{array}\]
If $u_0=4$ and $u_{n+1}=f(u_n)$ for $n\geq 0$, find $u_{2002}$.
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$
Let $a_1, a_2, \ldots, a_n$ be positive numbers, $m_g = \sqrt[n]{(a_1a_2 \cdots a_n)}$ their geometric mean, and $m_a = \frac{(a_1 + a_2 + \cdots + a_n)}{n}$ their arithmetic mean. Prove that
\[(1 + m_g)^n \leq (1 + a_1) \cdots(1 + a_n) \leq (1 + m_a)^n.\]
In the interior of square $ABCD$ with side length $1$, a point $P$ is chosen such that the lines $\ell_1, \ell_2$ through $P$ parallel to $AC$ and $BD$, respectively, divide the square into four distinct regions, the smallest of which has area $\mathcal{R}$. The area of the region of all points $P$ for which $\mathcal{R} \geq \tfrac{1}{6}$ can be expressed as $\frac{a-b\sqrt{c}}{d}$ where $\gcd(a,b,d)=1$ and $c$ is not divisible by the square of any prime. Compute $a+b+c+d$.
[i]Proposed by Andrew Wen[/i]
Let $f$ and $g$ be linear functions such that $f(g(2021))-g(f(2021)) = 20$. Compute $f(g(2022))- g(f(2022))$.
(Note: A function h is linear if $h(x) = ax + b$ for all real numbers $x$.)
Let $H$ be the orthocenter of acute triangle $ABC$, let $F$ be the foot of the altitude from $C$ to $AB$, and let $P$ be the reflection of $H$ across $BC$. Suppose that the circumcircle of triangle $AFP$ intersects line $BC$ at two distinct points $X$ and $Y$. Prove that $C$ is the midpoint of $XY$.
Find all natural numbers which can be written in the form $\frac{(a+b+c)^2}{abc}$ , where $a,b,c \in N$.
Let $a,b, c$ be $3$ distinct numbers from $\{1, 2,3, 4, 5, 6\}$
Show that $7$ divides $abc + (7 - a)(7 - b)(7 - c)$
If the probability that the sum of three distinct integers between $16$ and $30$ (inclusive) is even can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, find $m + n$.
We call an even positive integer $n$ [i]nice[/i] if the set $\{1, 2, \dots, n\}$ can be partitioned into $\frac{n}{2}$ two-element subsets, such that the sum of the elements in each subset is a power of $3$. For example, $6$ is nice, because the set $\{1, 2, 3, 4, 5, 6\}$ can be partitioned into subsets $\{1, 2\}$, $\{3, 6\}$, $\{4, 5\}$. Find the number of nice positive integers which are smaller than $3^{2022}$.
In a right triangle with the length legs $b$ and $c$, and the length hypotenuse $a$, the ratio between the length of the hypotenuse and the length of the diameter of the inscribed circle does not exceed $1 + \sqrt2$. Determine the numerical value of the expression of $E =\frac{a}{b + c}+\frac{b}{c + a}+\frac{c}{a + b}$.
Given that $n$ is a natural number such that the leftmost digits in the decimal representations of $2^n$ and $3^n$ are the same, find all possible values of the leftmost digit.
Point $O$ is a center of circumcircle of acute triangle $ABC$, bisector of angle $BAC$ cuts side $BC$ in point $D$. Let $M$ be a point such that, $MC \perp BC$ and $MA \perp AD$. Lines $BM$ and $OA$ intersect in point $P$. Show that circle of center in point $P$ passing through a point $A$ is tangent to line $BC$.
Find all integer solutions $x$,$y$ of the equation $(x+y^2)(x^2+y)=(x+y)^3$.
A set of three nonnegative integers $\{x,y,z\}$ with $x < y < z$ is called [i]historic[/i] if $\{z-y,y-x\} = \{1776,2001\}$. Show that the set of all nonnegative integers can be written as the union of pairwise disjoint historic sets.
The natural number $n$ is said to be good if there are natural numbers $a$ and $b$ that satisfy $a + b = n$ and $ab | n^2 + n + 1$.
(a) Show that there are infinitely many good numbers.
(b) Show that if $n$ is a good number, then $7 \nmid n$.
Evaluate
\[\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}\sum_{n=0}^{\infty}\frac{1}{k2^n+1}.\]
A [i]Benelux n-square[/i] (with $n\geq 2$) is an $n\times n$ grid consisting of $n^2$ cells, each of them containing a positive integer, satisfying the following conditions:
$\bullet$ the $n^2$ positive integers are pairwise distinct.
$\bullet$ if for each row and each column we compute the greatest common divisor of the $n$ numbers in that row/column, then we obtain $2n$ different outcomes.
(a) Prove that, in each Benelux n-square (with $n \geq 2$), there exists a cell containing a number which is at least $2n^2.$
(b) Call a Benelux n-square [i]minimal[/i] if all $n^2$ numbers in the cells are at most $2n^2.$ Determine all $n\geq 2$ for which there exists a minimal Benelux n-square.
Find all reals $z$ such that $z^4 - z^3 - 2z^2 - 3z - 1= 0$.
There are three circles $w_1, w_2, w_3$. Let $w_{i+1} \cap w_{i+2} = A_i, B_i$, where $A_i$ lies insides of $w_i$. Let $\gamma$ be the circle that is inside $w_1,w_2,w_3$ and tangent to the three said circles at $T_1, T_2, T_3$. Let $T_iA_{i+1}T_{i+2}$'s circumcircle and $T_iA_{i+2}T_{i+1}$'s circumcircle meet at $S_i$. Prove that the circumcircles of $A_iB_iS_i$ meet at two points. ($1 \le i \le 3$, indices taken modulo $3$)
If one of $A_i,B_i,S_i$ are collinear - the intersections of the other two circles lie on this line. Prove this as well.
A square with side length $3$ is inscribed in an isosceles triangle with one side of the square along the base of the triangle. A square with side length $2$ has two vertices on the other square and the other two on sides of the triangle, as shown. What is the area of the triangle?
[asy]
//diagram by kante314
draw((0,0)--(8,0)--(4,8)--cycle, linewidth(1.5));
draw((2,0)--(2,4)--(6,4)--(6,0)--cycle, linewidth(1.5));
draw((3,4)--(3,6)--(5,6)--(5,4)--cycle, linewidth(1.5));
[/asy]
$(\textbf{A})\: 19\frac14\qquad(\textbf{B}) \: 20\frac14\qquad(\textbf{C}) \: 21 \frac34\qquad(\textbf{D}) \: 22\frac12\qquad(\textbf{E}) \: 23\frac34$
Sequence $\{ a_n \}$ satisfies: $a_1=3$, $a_2=7$, $a_n^2+5=a_{n-1}a_{n+1}$, $n \geq 2$. If $a_n+(-1)^n$ is prime, prove that there exists a nonnegative integer $m$ such that $n=3^m$.
A convex polyhedron has two triangle and three quadrilateral faces. Connect every vertex of one of the triangle faces with the intersection point of the diagonals in the quadrilateral face opposite to it. Show that the resulting three lines are concurrent.
A [i]domino[/i] is a $2 \times 1$ or $1 \times 2$ tile. Determine in how many ways exactly $n^2$ dominoes can be placed without overlapping on a $2n \times 2n$ chessboard so that every $2 \times 2$ square contains at least two uncovered unit squares which lie in the same row or column.