Found problems: 85335
2010 Contests, 2
In an acute-angled triangle $ABC$, $M$ is the midpoint of side $BC$, and $D, E$ and $F$ the feet of the altitudes from $A, B$ and $C$, respectively. Let $H$ be the orthocenter of $\Delta ABC$, $S$ the midpoint of $AH$, and $G$ the intersection of $FE$ and $AH$. If $N$ is the intersection of the median $AM$ and the circumcircle of $\Delta BCH$, prove that $\angle HMA = \angle GNS$.
[i]Proposed by Marko Djikic[/i]
2022 Saudi Arabia JBMO TST, 4
Determine the smallest positive integer $a$ for which there exist a prime number $p$ and a positive integer $b \ge 2$ such that $$\frac{a^p -a}{p}=b^2.$$
1998 IMC, 2
Consider the following statement: for any permutation $\pi_1\not=\mathbb{I}$ of $\{1,2,...,n\}$ there is a permutation $\pi_2$ such that any permutation on these numbers can be obtained by a finite compostion of $\pi_1$ and $\pi_2$.
(a) Prove the statement for $n=3$ and $n=5$.
(b) Disprove the statement for $n=4$.
2010 Thailand Mathematical Olympiad, 4
For $i = 1, 2$ let $\vartriangle A_iB_iC_i$ be a triangle with side lengths $a_i, b_i, c_i$ and altitude lengths $p_i, q_i, r_i$.
Define $a_3 =\sqrt{a_1^2 + a_2^2}, b_3 =\sqrt{b_1^2 + b_2^2}$ , and $c_3 =\sqrt{c_1^2 + c_2^2}$.
Prove that $a_3, b_3, c_3$ are side lengths of a triangle, and if $p_3, q_3, r_3$ are the lengths of altitudes of this triangle,
then $p_3^2 \ge p_1^2 +p_2^2$, $q_3^2 \ge q_1^2 +q_2^2$ , and $r_3^2 \ge r_1^2 +r_2^2$
1954 Miklós Schweitzer, 7
[b]7.[/b] Find the finite groups having only one proper maximal subgroup. [b](A.12)[/b]
2018 Czech and Slovak Olympiad III A, 4
Let $a,b,c$ be integers which are lengths of sides of a triangle, $\gcd(a,b,c)=1$ and all the values $$\frac{a^2+b^2-c^2}{a+b-c},\quad\frac{b^2+c^2-a^2}{b+c-a},\quad\frac{c^2+a^2-b^2}{c+a-b}$$
are integers as well. Show that $(a+b-c)(b+c-a)(c+a-b)$ or $2(a+b-c)(b+c-a)(c+a-b)$ is a perfect square.
1987 Bundeswettbewerb Mathematik, 3
Let $(a_n)_{n\ge 1}$ and $(b_n)_{n\ge 1}$ be two sequences of natural numbers such that $a_{n+1} = na_n + 1, b_{n+1} = nb_n - 1$ for every $n\ge 1$. Show that these two sequences can have only a finite number of terms in common.
2024 Iranian Geometry Olympiad, 2
Point $P$ lies on the side $CD$ of the cyclic quadrilateral $ABCD$ such that $\angle CBP = 90^{\circ}$. Let $K$ be the intersection of $AC,BP$ such that $AK = AP = AD$. $H$ is the projection of $B$ onto the line $AC$. Prove that $\angle APH = 90^{\circ}$.
[i]Proposed by Iman Maghsoudi - Iran[/i]
2007 Indonesia TST, 2
Let $a > 3$ be an odd integer. Show that for every positive integer $n$ the number $a^{2^n}- 1$ has at least $n + 1$ distinct prime divisors.
2016 Romania Team Selection Tests, 3
Let $n$ be a positive integer, and let $a_1,a_2,..,a_n$ be pairwise distinct positive integers. Show that $$\sum_{k=1}^{n}{\frac{1}{[a_1,a_2,…,a_k]}} <4,$$ where $[a_1,a_2,…,a_k]$ is the least common multiple of the integers $a_1,a_2,…,a_k$.
1982 AMC 12/AHSME, 1
When the polynomial $x^3-2$ is divided by the polynomial $x^2-2$, the remainder is
$\textbf{(A)} \ 2 \qquad \textbf{(B)} \ -2 \qquad \textbf{(C)} \ -2x-2 \qquad \textbf{(D)} \ 2x+2 \qquad \textbf{(E)} \ 2x-2$
2023 Assara - South Russian Girl's MO, 5
In a $5 \times 5$ checkered square, the middle row and middle column are colored gray. You leave the corner cell and move to the cell adjacent to the side with each move. For each transition from a gray cell to a gray one you need to pay a ruble. What is the smallest number of rubles you need to pay to go around all the squares of the board exactly once (it is not necessary to return to the starting square)?
1983 IMO Shortlist, 25
Prove that every partition of $3$-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \in \mathbb R^+$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a.$
2019 Romania Team Selection Test, 2
Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.
1999 National Olympiad First Round, 1
Let $ ABC$ be a triangle with $ \left|AB\right| \equal{} 14$, $ \left|BC\right| \equal{} 12$, $ \left|AC\right| \equal{} 10$. Let $ D$ be a point on $ \left[AC\right]$ and $ E$ be a point on $ \left[BC\right]$ such that $ \left|AD\right| \equal{} 4$ and $ Area\left(ABC\right) \equal{} 2Area\left(CDE\right)$. Find $ Area\left(ABE\right)$.
$\textbf{(A)}\ 4\sqrt {6} \qquad\textbf{(B)}\ 6\sqrt {2} \qquad\textbf{(C)}\ 3\sqrt {6} \qquad\textbf{(D)}\ 4\sqrt {2} \qquad\textbf{(E)}\ 4\sqrt {5}$
Estonia Open Senior - geometry, 1997.2.3
The figure shows a square and three circles of equal radius tangent to each other and square passes. Find the radius of the circles if the square length is $1$.
[img]http://3.bp.blogspot.com/-iIjwupkz7DQ/XnrIRhKIJnI/AAAAAAAALhA/clERrIDqEtcujzvZk_qu975wsTjKaxCLQCK4BGAYYCw/s400/97%2Bestonia%2Bopen%2Bs2.3.png[/img]
2023 Korea National Olympiad, 2
Sets $A_0, A_1, \dots, A_{2023}$ satisfy the following conditions:
[list]
[*] $A_0 = \{ 3 \}$
[*] $A_n = \{ x + 2 \mid x \in A_{n - 1} \} \ \cup \{x(x+1) / 2 \mid x \in A_{n - 1} \}$ for each $n = 1, 2, \dots, 2023$.
[/list]
Find $|A_{2023}|$.
2019 IFYM, Sozopol, 5
Let $A$ be the number of 2019-digit numbers, that is made of 2 different digits (For example $10\underbrace{1...1}_{2016}0$ is such number). Determine the highest power of 3 that divides $A$.
Estonia Open Junior - geometry, 2014.2.5
In the plane there are six different points $A, B, C, D, E, F$ such that $ABCD$ and $CDEF$ are parallelograms. What is the maximum number of those points that can be located on one circle?
2015 Korea Junior Math Olympiad, 5
Let $I$ be the incenter of an acute triangle $\triangle ABC$, and let the incircle be $\Gamma$.
Let the circumcircle of $\triangle IBC$ hit $\Gamma$ at $D, E$, where $D$ is closer to $B$ and $E$ is closer to $C$.
Let $\Gamma \cap BE = K (\not= E)$, $CD \cap BI = T$, and $CD \cap \Gamma = L (\not= D)$.
Let the line passing $T$ and perpendicular to $BI$ meet $\Gamma$ at $P$, where $P$ is inside $\triangle IBC$.
Prove that the tangent to $\Gamma$ at $P$, $KL$, $BI$ are concurrent.
2005 Iran Team Selection Test, 3
Suppose $S= \{1,2,\dots,n\}$ and $n \geq 3$. There is $f:S^k \longmapsto S$ that if $a,b \in S^k$ and $a$ and $b$ differ in all of elements then $f(a) \neq f(b)$. Prove that $f$ is a function of one of its elements.
2022 BMT, 16
Let triangle $\vartriangle ABC$ be a triangle with $AB = 5$, $BC = 7$, and $CA = 8$, and let $I$ be the incenter of $\vartriangle ABC$. Let circle $C_A$ denote the circle with center $A$ and radius $\overline{AI}$, denote $C_B$ and circle $C_C$ similarly. Besides all intersecting at $I$, the circles $C_A$,$C_B$,$C_C$ also intersect pairwise at $F$, $G$, and $H$. Compute the area of triangle $\vartriangle FGH$.
1970 IMO Longlists, 13
Each side of an arbitrary $\triangle ABC$ is divided into equal parts, and lines parallel to $AB,BC,CA$ are drawn through each of these points, thus cutting $\triangle ABC$ into small triangles. Points are assigned a number in the following manner:
$(1)$ $A,B,C$ are assigned $1,2,3$ respectively
$(2)$ Points on $AB$ are assigned $1$ or $2$
$(3)$ Points on $BC$ are assigned $2$ or $3$
$(4)$ Points on $CA$ are assigned $3$ or $1$
Prove that there must exist a small triangle whose vertices are marked by $1,2,3$.
1986 Iran MO (2nd round), 1
Let $f$ be a function such that
\[f(x)=\frac{(x^2-2x+1) \sin \frac{1}{x-1}}{\sin \pi x}.\]
Find the limit of $f$ in the point $x_0=1.$
2024 USA TSTST, 2
Let $p$ be an odd prime number. Suppose $P$ and $Q$ are polynomials with integer coefficients such that $P(0)=Q(0)=1$, there is no nonconstant polynomial dividing both $P$ and $Q$, and
\[
1 + \cfrac{x}{1 + \cfrac{2x}{1 + \cfrac{\ddots}{1 +
(p-1)x}}}=\frac{P(x)}{Q(x)}.
\]
Show that all coefficients of $P$ except for the constant coefficient are divisible by $p$, and all coefficients of $Q$ are [i]not[/i] divisible by $p$.
[i]Andrew Gu[/i]