Found problems: 85335
Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$.
[i]Proposed by Evan Chen, Taiwan[/i]
Find the number of permutations $( p_1, p_2, p_3 , p_4 , p_5 , p_6)$ of $1, 2 ,3,4,5,6$ such that for any $k, 1 \leq k \leq 5$, $(p_1, \ldots, p_k)$ does not form a permutation of $1 , 2, \ldots, k$.
We say that a positive integer is [i]quad-divi[/i] if it is divisible by the sum of the squares of its digits, and also none of its digits is equal to zero.
a) Find a quad-divi number such that the sum of its digits is $24$.
b) Find a quad-divi number such that the sum of its digits is $1001$.
Let $s_k$ denote the sum of the $\textit{k}$th powers of the roots of the polynomial $x^3-5x^2+8x-13$. In particular, $s_0=3$, $s_1=5$, and $s_2=9$. Let $a$, $b$, and $c$ be real numbers such that $s_{k+1} = a \, s_k + b \, s_{k-1} + c \, s_{k-2}$ for $k = 2$, $3$, $....$ What is $a+b+c$?
$\textbf{(A)} \; -6 \qquad \textbf{(B)} \; 0 \qquad \textbf{(C)} \; 6 \qquad \textbf{(D)} \; 10 \qquad \textbf{(E)} \; 26$
There are $42$ students taking part in the Team Selection Test. It is known that every student knows exactly $20$ other students. Show that we can divide the students into $2$ groups or $21$ groups such that the number of students in each group is equal and every two students in the same group know each other.
How many positive integers have exactly three proper divisors, each of which is less than 50?
Let $n \in N$. Find the maximum number of irreducible fractions a/b (i.e., $gcd(a, b) = 1$) which lie in the interval $(0,1/n)$.
What integers $a, b, c$ satisfy $a^2 + b^2 + c^2 + 3 < ab + 3b + 2c$ ?
Let $ F $ be the primitive of a continuous function $ f:\mathbb{R}\longrightarrow (0,\infty ), $ with $ F(0)=0. $
Determine for which values of $ \lambda \in (0,1) $ the function $ \left( F^{-1}\circ \lambda F \right)/\text{id.} $ has limit at $ 0, $ and calculate it.
Find the number of all infinite sequences $a_1$, $a_2$, ... of positive integers such that $a_n+a_{n+1}=2a_{n+2}a_{n+3}+2005$ for all positive integers $n$.
Given the circumradius $R$ of a triangle, a side length $c$, and the ratio $a/b$ of the other two side lengths, determine all three sides and angles of this triangle.
The sum of two numbers is $ S$. Suppose 3 is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?
$ \textbf{(A)} \ 2S \plus{} 3 \qquad \textbf{(B)} \ 3S \plus{} 2 \qquad \textbf{(C)} \ 3S \plus{} 6 \qquad \textbf{(D)} \ 2S \plus{} 6 \qquad \textbf{(E)} \ 2S \plus{} 12$
Compute the square of the distance between the incenter (center of the inscribed circle) and circumcenter (center of the circumscribed circle) of a 30-60-90 right triangle with hypotenuse of length 2.
In the plane of traingle $ABC$ we consider a variable point $M$ which moves on line $MA$ towards $A$. Halfway there, it stops and starts moving in a straight line line towards $B$. Halfway there, it stops and starts moving in a straight line towards $C$, and halfway there it stops and starts moving in a straight line towards $A$, and so on. Show that $M$ will get as close as we want to the vertices of a fixed triangle with area $\text{area}(ABC)/7$.
Determine all functions $f : N_0 \to R$ satisfying $f (x+y)+ f (x-y)= f (3x)$ for all $x,y$.
In $\triangle ABC$, $AC>BC$, $CM$ is the median, and $CH$ is the altitude emanating from $C$, as shown in the figure on the right. Determine the measure of $\angle MCH$ if $\angle ACM$ and $\angle BCH$ each have measure $17^\circ$.
[asy]
size(200);
defaultpen(linewidth(0.8));
pair A=origin,B=(10,0),C=(7,5),M=(5,0),H=(7,0);
draw(A--C--B--cycle^^H--C--M);
label("$A$",A,NW);
label("$B$",B,NE);
label("$C$",C,NE);
label("$M$",M,NW);
label("$H$",H,NE);
[/asy]
An $8 \times 8 \times 8$ cube is painted red on $3$ faces and blue on $3$ faces such that no corner is surrounded by three faces of the same color. The cube is then cut into $512$ unit cubes. How many of these cubes contain both red and blue paint on at least one of their faces?
[i]Author: Ray Li[/i]
[hide="Clarification"]The problem asks for the number of cubes that contain red paint on at least one face and blue paint on at least one other face, not for the number of cubes that have both colors of paint on at least one face (which can't even happen.)[/hide]
Let $ABC$ be an equilateral triangle with side lenght is $1$ $cm$.Let $D \in [AB]$ is a point. Perpendiculars from $D$ to $[AC]$ and $[BC]$ intersects with $[AC]$ and $[BC]$ at points $E$ and $F$ respectively. Perpendiculars from $E$ and $F$ to $[AB]$ intersects with $[AB]$ at points $E_1$ and $F_1$. Prove that
$$[E_1F_1]=\frac{3}{4}$$
Find the complex numbers $ a,b $ having the properties that $ |a|=|b|=1=\bar{a} +\bar{b} -ab. $
Let $ABC$ be an acute-angled triangle and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$ to the sides $AC$ and $AB$ respectively. Suppose $AD$ is the diameter of the circle $ABC$. Let $M$ be the midpoint of $BC$. Let $K$ be the imsimilicenter of the incircles of the triangles $BMF$ and $CME$. Prove that the points $K, M, D$ are collinear.
[i]Proposed by Bilegdembrel Bat-Amgalan.[/i]
Solve the equation
$ 2(x^2\minus{}3x\plus{}2)\equal{}3 \sqrt{x^3\plus{}8}$, where $ x\in R$
When Pablo turns $15$, he throws a party inviting $43$ friends. He presents them with a cake n the form of a regular $15$-sided polygon and on it $15$ candles. The candles are arranged so that between candles and vertices there are never three aligned (any three candles are not aligned, nor are any two candles with a vertex of the polygon, nor are any two vertices of the polygon with a candle). Then Pablo divides the cake into triangular pieces, by means of cuts that join candles to each other or candles and vertices, but also do not intersect with others already made. Why, by doing this, Paul was able to distribute a piece to each of his guests but he was left without eating?
Let $O$ be the circumcenter of triangle $ABC,$ and $D$ be an arbitrary point on the circumcircle of $ABC.$ Let points $X, Y$ and $Z$ be the orthogonal projections of point $D$ onto lines $OA, OB$ and $OC,$ respectively. Prove that the incenter of triangle $XYZ$ is on the Simson-Wallace line of triangle $ABC$ corresponding to point $D.$
Find all integers $a$ and $b$ satisfying
\begin{align*}
a^6 + 1 & \mid b^{11} - 2023b^3 + 40b, \qquad \text{and}\\
a^4 - 1 & \mid b^{10} - 2023b^2 - 41.
\end{align*}
Solve equation
\[2^{\tfrac{1}{2}-2|x|} = \left| {\tan x + \frac{1}{2}} \right| + \left| {\tan x - \frac{1}{2}} \right|\]