Found problems: 85335
Ukraine Correspondence MO - geometry, 2004.10.
In an isosceles triangle $ABC$ ($AB = AC$), the bisector of the angle $B$ intersects $AC$ at point $D$ such that $BC = BD + AD$. Find $\angle A$.
2022 Iran MO (2nd round), 3
Take a $n \times n$ chess page.Determine the $n$ such that we can put the numbers $1,2,3, \ldots ,n$ in the squares of the page such that we know the following two conditions are true:
a) for each row we know all the numbers $1,2,3, \ldots ,n$ have appeared on it and the numbers that are in the black squares of that row have the same sum as the sum of the numbers in the white squares of that row.
b) for each column we know all the numbers $1,2,3, \ldots ,n$ have appeared on it and the numbers that are in the black squares in that column have the same sum as the sum of the numbers in the white squares of that column.
2024 Korea Summer Program Practice Test, 6
Does there exist a real sequence $\{a_n\}_{n=1}^\infty$ such that
$$a_na_{n+1}\ge a_{n+2}^2 +1$$
for all $n\ge 1$?
2014 All-Russian Olympiad, 4
Given a triangle $ABC$ with $AB>BC$, let $ \Omega $ be the circumcircle. Let $M$, $N$ lie on the sides $AB$, $BC$ respectively, such that $AM=CN$. Let $K$ be the intersection of $MN$ and $AC$. Let $P$ be the incentre of the triangle $AMK$ and $Q$ be the $K$-excentre of the triangle $CNK$. If $R$ is midpoint of the arc $ABC$ of $ \Omega $ then prove that $RP=RQ$.
[i]M. Kungodjin[/i]
2012 Serbia Team Selection Test, 2
Let $\sigma(x)$ denote the sum of divisors of natural number $x$, including $1$ and $x$. For every $n\in \mathbb{N}$ define $f(n)$ as number of natural numbers $m, m\leq n$, for which $\sigma(m)$ is odd number. Prove that there are infinitely many natural numbers $n$, such that $f(n)|n$.
2016 Novosibirsk Oral Olympiad in Geometry, 2
Bisector of one angle of triangle $ABC$ is equal to the bisector of its external angle at the same vertex (see figure). Find the difference between the other two angles of the triangle.
[img]https://cdn.artofproblemsolving.com/attachments/c/3/d2efeb65544c45a15acccab8db05c8314eb5f2.png[/img]
2020 Czech and Slovak Olympiad III A, 5
Given an isosceles triangle $ABC$ with base $BC$. Inside the side $BC$ is given a point $D$. Let $E, F$ be respectively points on the sides $AB, AC$ that $|\angle BED | = |\angle DF C| > 90^o$ . Prove that the circles circumscribed around the triangles $ABF$ and $AEC$ intersect on the line $AD$ at a point different from point $A$.
(Patrik Bak, Michal Rolínek)
2019 Danube Mathematical Competition, 2
Let be a natural number $ n, $ and $ n $ real numbers $ a_1,a_2,\ldots ,a_n. $ Prove that there exists a real number $ a $ such that $ a+a_1,a+a_2,\ldots ,a+a_n $ are all irrational.
2011 India IMO Training Camp, 3
Let $T$ be a non-empty finite subset of positive integers $\ge 1$. A subset $S$ of $T$ is called [b]good [/b] if for every integer $t\in T$ there exists an $s$ in $S$ such that $gcd(t,s) >1$. Let
\[A={(X,Y)\mid X\subseteq T,Y\subseteq T,gcd(x,y)=1 \text{for all} x\in X, y\in Y}\]
Prove that :
$a)$ If $X_0$ is not [b]good[/b] then the number of pairs $(X_0,Y)$ in $A$ is [b]even[/b].
$b)$ the number of good subsets of $T$ is [b]odd[/b].
2000 Harvard-MIT Mathematics Tournament, 3
A twelve foot tree casts a five foot shadow. How long is Henry’s shadow (at the same time of day) if he is five and a half feet tall?
2017 AIME Problems, 3
For a positive integer $n$, let $d_n$ be the units digit of $1 + 2 + \dots + n$. Find the remainder when
\[\sum_{n=1}^{2017} d_n\]
is divided by $1000$.
2010 Saudi Arabia BMO TST, 4
Find all primes $p, q$ satisfying the equation $2p^q - q^p = 7.$
2001 Baltic Way, 11
The real-valued function $f$ is defined for all positive integers. For any integers $a>1, b>1$ with $d=\gcd (a, b)$, we have
\[f(ab)=f(d)\left(f\left(\frac{a}{d}\right)+f\left(\frac{b}{d}\right)\right) \]
Determine all possible values of $f(2001)$.
2013 National Olympiad First Round, 26
What is the maximum number of primes that divide both the numbers $n^3+2$ and $(n+1)^3+2$ where $n$ is a positive integer?
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 1
\qquad\textbf{(D)}\ 0
\qquad\textbf{(E)}\ \text{None of above}
$
Kyiv City MO 1984-93 - geometry, 1991.7.5
Inside the rectangle $ABCD$ is taken a point $M$ such that $\angle BMC + \angle AMD = 180^o$. Determine the sum of the angles $BCM$ and $DAM$.
2015 Princeton University Math Competition, A8
The incircle of acute triangle $ABC$ touches $BC, AC$, and $AB$ at points $D, E$, and $F$, respectively. Let $P$ be the second intersection of line $AD$ and the incircle. The line through $P$ tangent to the incircle intersects $AB$ and $AC$ at points $M$ and $N$, respectively. Given that $\overline{AB} = 8, \overline{AC} = 10$, and $\overline{AN} = 4$, let $\overline{AM} = \tfrac{a}{b}$ where $a$ and $b$ are positive coprime integers. What is $a + b$?
2014 Saudi Arabia GMO TST, 3
Turki has divided a square into finitely many white and green rectangles, each with sides parallel to the sides of the square. Within each white rectangle, he writes down its width divided by its height. Within each green rectangle, he writes down its height divided by its width. Finally, he calculates $S$, the sum of these numbers. If the total area of white rectangles equals the total area of green rectangles, determine the minimum possible value of $S$.
2021 Peru Cono Sur TST., P6
Prove that there are no positive integers $a_1, a_2, \ldots , a_{2021}$ (not necessarily distinct) such that for $k = 1, 2, 3, \ldots , 2021$ the number of elements in the set
$$A_k = \{ j \in \mathbb{N} : 1 \le j \le 2021 \text{ and } a_j|k \}$$
be exactly $a_k$.
2016 Indonesia TST, 2
Given a convex polygon with $n$ sides and perimeter $S$, which has an incircle $\omega$ with radius $R$. A regular polygon with $n$ sides, whose vertices lie on $\omega$, has a perimeter $s$. Determine whether the following inequality holds:
\[ S \ge \frac{2sRn}{\sqrt{4n^2R^2-s^2}}. \]
2002 Iran Team Selection Test, 10
Suppose from $(m+2)\times(n+2)$ rectangle we cut $4$, $1\times1$ corners. Now on first and last row first and last columns we write $2(m+n)$ real numbers. Prove we can fill the interior $m\times n$ rectangle with real numbers that every number is average of it's $4$ neighbors.
2006 Austrian-Polish Competition, 9
We have an 8x8 chessboard with 64 squares. Then we have 3x1 dominoes which cover exactly 3 squares. Such dominoes can only be moved parallel to the borders of the chessboard and also only if the passing squares are free. If no dominoes can be moved, then the position is called stable.
a. Find the smalles number of covered squares neccessary for a stable position.
b. Prove: There exist a stable position with only one square uncovered.
c. Find all Squares which are uncoverd in at least one position of b).
1961 Miklós Schweitzer, 6
[b]6.[/b] Consider a sequence $\{ a_n \}_{n=1}^{\infty}$ such that, for any convergent subsequence $\{ a_{n_k} \}$ of $\{a_n\}$, the sequence $\{ a_{n_k +1} \}$ also is convergent and has the same limit as $\{ a_{n_k}\}$. Prove that the sequence $\{ a_n \}$ is either convergent of has infinitely many accumulation points the set of which is dense in itself. Give an example for the second case. (A sequence $ x_n \to \infty $ or $-\infty$ is considered to be convergente, too)
[b](S. 13)[/b]
2020 Harvard-MIT Mathematics Tournament, 8
Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let the internal angle bisector of $\angle BAC$ intersect $BC$ and $\Gamma$ at $E$ and $N$, respectively. Let $A'$ be the antipode of $A$ on $\Gamma$ and let $V$ be the point where $AA'$ intersects $BC$. Given that $EV=6$, $VA'=7$, and $A'N=9$, compute the radius of $\Gamma$.
[i]Proposed by James Lin.[/i]
2006 Purple Comet Problems, 5
The sizes of the freshmen class and the sophomore class are in the ratio $5:4$. The sizes of the sophomore class and the junior class are in the ratio $7:8$. The sizes of the junior class and the senior class are in the ratio $9:7$. If these four classes together have a total of $2158$ students, how many of the students are freshmen?
2022 Rioplatense Mathematical Olympiad, 3
Let $ABC$ be a triangle with $AB<AC$. There are two points $X$ and $Y$ on the angle bisector of $B\widehat AC$ such that $X$ is between $A$ and $Y$ and $BX$ is parallel to $CY$. Let $Z$ be the reflection of $X$ with respect to $BC$. Line $YZ$ cuts line $BC$ at point $P$. If line $BY$ cuts line $CX$ at point $K$, prove that $KA=KP$.