Found problems: 85335
Ukrainian From Tasks to Tasks - geometry, 2011.3
Let $O$ be the center of the circumcircle, and $AD$ be the angle bisector of the acute triangle $ABC$. The perpendicular drawn from point $D$ on the line $AO$ intersects the line $AC$ at the point $P$. Prove that $AP = AB$.
2021 Saint Petersburg Mathematical Olympiad, 7
A square is cut into red and blue rectangles. The sum of areas of red triangles is equal to the sum of areas of the blue ones. For each blue rectangle, we write the ratio of the length of its vertical side to the length of its horizontal one and for each red rectangle, the ratio of the length of its horizontal side to the length of its vertical side. Find the smallest possible value of the sum of all the written numbers.
Putnam 1938, A7
Do either $(1)$ or $(2)$
$(1)$ $S$ is a thin spherical shell of constant thickness and density with total mass $M$ and center $O.$ $P$ is a point outside $S.$ Prove that the gravitational attraction of $S$ at $P$ is the same as the gravitational attraction of a point mass $M$ at $O.$
$(2)$ $K$ is the surface $z = xy$ in Euclidean $3-$space. Find all straight lines lying in $S$. Draw a diagram to illustrate them.
2013 AMC 12/AHSME, 8
Line $\ell_1$ has equation $3x-2y=1$ and goes through $A=(-1,-2)$. Line $\ell_2$ has equation $y=1$ and meets line $\ell_1$ at point $B$. Line $\ell_3$ has positive slope, goes through point $A$, and meets $\ell_2$ at point $C$. The area of $\triangle ABC$ is $3$. What is the slope of $\ell_3$?
$ \textbf{(A)}\ \frac{2}{3}\qquad\textbf{(B)}\ \frac{3}{4}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{4}{3}\qquad\textbf{(E)}\ \frac{3}{2} $
2013 Iran MO (2nd Round), 3
Let $\{a_n\}_{n=1}^{\infty}$ be a sequence of positive integers for which
\[ a_{n+2} = \left[\frac{2a_n}{a_{n+1}}\right]+\left[\frac{2a_{n+1}}{a_n}\right]. \]
Prove that there exists a positive integer $m$ such that $a_m=4$ and $a_{m+1} \in\{3,4\}$.
[b]Note.[/b] $[x]$ is the greatest integer not exceeding $x$.
2003 National High School Mathematics League, 15
A circle $O$ with radius of $R$ is drawn on a piece of paper. $A$ is a fixed point inside circle $O$, and $OA=a$. Fold the paper, so that a point $A'$ on the circle is coincident with $A$. For all such foldings, a kink mark is remained. Find the set of points on a certain kink mark.
2016 IFYM, Sozopol, 5
Prove that for an arbitrary $\Delta ABC$ the following inequality holds:
$\frac{l_a}{m_a}+\frac{l_b}{m_b}+\frac{l_c}{m_c} >1$,
Where $l_a,l_b,l_c$ and $m_a,m_b,m_c$ are the lengths of the bisectors and medians through $A$, $B$, and $C$.
2022 Romania Team Selection Test, 2
Let $ABC$ be an acute triangle and let $B'$ and $C'$ be the feet of the heights $B$ and $C$ of triangle $ABC$ respectively. Let $B_A'$ and $B_C'$ be reflections of $B'$ with respect to the lines $BC$ and $AB$, respectively. The circle $BB_A'B_C'$, centered in $O_B$, intersects the line $AB$ in $X_B$ for the second time.
The points $C_A', C_B', O_C, X_C$ are defined analogously, by replacing the pair $(B, B')$ with the pair $(C, C')$. Show that $O_BX_B$ and $O_CX_C$ are parallel.
1999 National Olympiad First Round, 15
2 squares are painted in blue and 2 squares are painted in red on a $ 3\times 3$ board in such a way that two square with same color is neither at same row nor at same column. In how many different ways can these four squares be painted?
$\textbf{(A)}\ 198 \qquad\textbf{(B)}\ 288 \qquad\textbf{(C)}\ 396 \qquad\textbf{(D)}\ 576 \qquad\textbf{(E)}\ 792$
1976 IMO Longlists, 41
Determine the greatest number, who is the product of some positive integers, and the sum of these numbers is $1976.$
2001 Portugal MO, 2
The trapezium $[ABCD]$ has bases $[AB]$ and $[CD]$ (with $[AB]$ being the largest base). Knowing that $BC = 2 DA$ and that $\angle DAB + \angle ABC =120^o$ , determines the measure of $\angle DAB$.
2017 China Western Mathematical Olympiad, 4
Let $n$ and $k$ be given integers such that $n\ge k\ge 2$. Alice and Bob play a game on an $n$ by $n$ table with white cells. They take turns to pick a white cell and color it black. Alice moves first. The game ends as soon as there is at least one black cell in every $k$ by $k$ square after a player moves, who is declared the winner of the game. Who has the winning strategy?
2023 Bulgarian Autumn Math Competition, 12.3
Solve in positive integers the equation $$m^{\frac{1}{n}}+n^{\frac{1}{m}}=2+\frac{2}{mn(m+n)^{\frac{1}{m}+\frac{1}{n}}}.$$
2016 Balkan MO Shortlist, A4
The positive real numbers $a, b, c$ satisfy the equality $a + b + c = 1$. For every natural number $n$ find the minimal possible value of the expression $$E=\frac{a^{-n}+b}{1-a}+\frac{b^{-n}+c}{1-b}+\frac{c^{-n}+a}{1-c}$$
2009 Oral Moscow Geometry Olympiad, 2
Trapezium $ABCD$ and parallelogram $MBDK$ are located so that the sides of the parallelogram are parallel to the diagonals of the trapezoid (see fig.). Prove that the area of the gray part is equal to the sum of the areas of the black part.
(Yu. Blinkov)
[img]https://cdn.artofproblemsolving.com/attachments/b/9/cfff83b1b85aea16b603995d4f3d327256b60b.png[/img]
1966 Spain Mathematical Olympiad, 6
They tell us that a married couple has $5$ children. Calculate the probability that among them there are at least two men and at least one woman. Probability of being born male is considered $1/2$.
LMT Team Rounds 2021+, 15
There are $28$ students who have to be separated into two groups such that the number of students in each group
is a multiple of $4$. The number of ways to split them into the groups can be written as
$$\sum_{k \ge 0} 2^k a_k = a_0 +2a_1 +4a_2 +...$$
where each $a_i$ is either $0$ or $1$. Find the value of
$$\sum_{k \ge 0} ka_k = 0+ a_1 +2a_2 +3a3_ +....$$
2022 Balkan MO Shortlist, N1
Let $n{}$ be a positive integer. What is the smallest sum of digits that $5^n + 6^n + 2022^n$ can take?
2015 Purple Comet Problems, 13
Given that x, y, and z are real numbers satisfying $x+y +z = 10$ and $x^2 +y^2 +z^2 = 50$, find the maximum possible value of $(x + 2y + 3z)^2 + (y + 2z + 3x)^2 + (z + 2x + 3y)^2$.
1998 Cono Sur Olympiad, 1
We have $98$ cards, in each one we will write one of the numbers: $1, 2, 3, 4,...., 97, 98$.
We can order the $98$ cards, in a sequence such that two consecutive numbers $X$ and $Y$ and the number $X - Y$ is greater than $48$, determine how and how many ways we can make this sequence!!
2019 Belarus Team Selection Test, 6.2
The numbers $1,2,\ldots,49,50$ are written on the blackboard. Ann performs the following operation: she chooses three arbitrary numbers $a,b,c$ from the board, replaces them by their sum $a+b+c$ and writes $(a+b)(b+c)(c+a)$ to her notebook. Ann performs such operations until only two numbers remain on the board (in total 24 operations). Then she calculates the sum of all $24$ numbers written in the notebook. Let $A$ and $B$ be the maximum and the minimum possible sums that Ann san obtain.
Find the value of $\frac{A}{B}$.
[i](I. Voronovich)[/i]
2014 Chile TST Ibero, 1
Consider a function $f: \mathbb{R} \to \mathbb{R}$ satisfying for all $x \in \mathbb{R}$:
\[
f(x+1) = \frac{1}{2} + \sqrt{f(x) - f(x)^2}.
\]
Prove that there exists a $b > 0$ such that $f(x + b) = f(x)$ for all $x \in \mathbb{R}$.
LMT Guts Rounds, 34
A [i]prime power[/i] is an integer of the form $p^k,$ where $p$ is a prime and $k$ is a nonnegative integer. How many prime powers are there less than or equal to $10^6?$ Your score will be $16-80|\frac{\textbf{Your Answer}}{\textbf{Actual Answer}}-1|$ rounded to the nearest integer or $0,$ whichever is higher.
2008 SDMO (Middle School), 5
For a positive integer $n$, let $f\left(n\right)$ be the sum of the first $n$ terms of the sequence $$0,1,1,2,2,3,3,4,4,\ldots,r,r,r+1,r+1,\ldots.$$ For example, $f\left(5\right)=0+1+1+2+2=6$.
(a) Find a formula for $f\left(n\right)$.
(b) Prove that $f\left(s+t\right)-f\left(s-t\right)=st$ for all positive integers $s$ and $t$, where $s>t$.
2016 IMO Shortlist, A2
Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that
\[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]