Found problems: 85335
2021 Regional Olympiad of Mexico West, 3
The sequence of real numbers $a_1, a_2, a_3, ...$ is defined as follows: $a_1 = 2019$, $a_2 = 2020$, $a_3 = 2021$ and for all $n \ge 1$
$$a_{n+3} = 5a^6_{n+2} + 3a^3_{n+1} + a^2_n.$$
Show that this sequence does not contain numbers of the form $m^6$ where $m$ is a positive integer.
2019 Math Prize for Girls Problems, 13
Each side of a unit square (side length 1) is also one side of an equilateral triangle that lies in the square. Compute the area of the intersection of (the interiors of) all four triangles.
2003 Chile National Olympiad, 2
Find all primes $p, q$ such that $p + q = (p-q)^3$.
2012 Iran MO (3rd Round), 2
Prove that there exists infinitely many pairs of rational numbers $(\frac{p_1}{q},\frac{p_2}{q})$ with $p_1,p_2,q\in \mathbb N$ with the following condition:
\[|\sqrt{3}-\frac{p_1}{q}|<q^{-\frac{3}{2}}, |\sqrt{2}-\frac{p_2}{q}|< q^{-\frac{3}{2}}.\]
[i]Proposed by Mohammad Gharakhani[/i]
2012 National Olympiad First Round, 2
Find the sum of distinct residues of the number $2012^n+m^2$ on $\mod 11$ where $m$ and $n$ are positive integers.
$ \textbf{(A)}\ 55 \qquad \textbf{(B)}\ 46 \qquad \textbf{(C)}\ 43 \qquad \textbf{(D)}\ 39 \qquad \textbf{(E)}\ 37$
Kvant 2020, M2631
There is a convex quadrangle $ABCD$ such that no three of its sides can form a triangle. Prove that:
[list=a]
[*]one of its angles is not greater than $60^\circ{}$;
[*]one of its angles is at least $120^\circ$.
[/list]
[i]Maxim Didin[/i]
1996 Korea National Olympiad, 7
Let $A_n$ be the set of real numbers such that each element of $A_n$ can be expressed as $1+\frac{a_1}{\sqrt{2}}+\frac{a_2}{(\sqrt{2})^2}+\cdots +\frac{a_n}{(\sqrt{n})^n}$ for given $n.$ Find both $|A_n|$ and sum of the products of two distinct elements of $A_n$ where each $a_i$ is either $1$ or $-1.$
2021 Final Mathematical Cup, 2
The altitudes $BB_1$ and $CC_1$, are drawn in an acute triangle $ABC$. Let $X$ and $Y$ be the points, which are symmetrical to the points $B_1$ and $C_1$, with respect to the midpoints of the sides$ AB$ and $AC$ of the triangle $ABC$ respectively. Let's denote with $Z$ the point of intersection of the lines $BC$ and $XY$. Prove that the line $ZA$ is tangent to the circumscribed circle of the triangle $AXY$ .
2023 SG Originals, Q5
Determine all real numbers $x$ between $0$ and $180$ such that it is possible to partition an equilateral triangle into finitely many triangles, each of which has an angle of $x^{o}$.
2010 Sharygin Geometry Olympiad, 4
In triangle $ABC$, touching points $A', B'$ of the incircle with $BC, AC$ and common point $G$ of segments $AA'$ and $BB'$ were marked. After this the triangle was erased. Restore it by the ruler and the compass.
2013-2014 SDML (High School), 8
Twenty-four congruent squares are arranged as shown in the figure. In how many ways can we select $12$ of the squares so that no two are diagonally adjacent? Directly adjacent spaces are acceptable.
2004 AMC 10, 8
Minneapolis-St. Paul International Airport is $ 8$ miles southwest of downtown St. Paul and $ 10$ miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis?
$ \textbf{(A)}\ 13\qquad
\textbf{(B)}\ 14\qquad
\textbf{(C)}\ 15\qquad
\textbf{(D)}\ 16\qquad
\textbf{(E)}\ 17$
1988 Romania Team Selection Test, 12
The four vertices of a square are the centers of four circles such that the sum of theirs areas equals the square's area. Take an arbitrary point in the interior of each circle. Prove that the four arbitrary points are the vertices of a convex quadrilateral.
[i]Laurentiu Panaitopol[/i]
2002 Korea - Final Round, 1
For a prime $p$ of the form $12k+1$ and $\mathbb{Z}_p=\{0,1,2,\cdots,p-1\}$, let
\[\mathbb{E}_p=\{(a,b) \mid a,b \in \mathbb{Z}_p,\quad p\nmid 4a^3+27b^2\}\]
For $(a,b), (a',b') \in \mathbb{E}_p$ we say that $(a,b)$ and $(a',b')$ are equivalent if there is a non zero element $c\in \mathbb{Z}_p$ such that $p\mid (a' -ac^4)$ and $p\mid (b'-bc^6)$. Find the maximal number of inequivalent elements in $\mathbb{E}_p$.
2022 Saint Petersburg Mathematical Olympiad, 1
The positive integers $a$ and $b$ are such that $a+k$ is divisible by $b+k$ for all positive integers numbers $k<b$. Prove that $a-k$ is divisible by $b-k$ for all positive integers $k<b$.
2017 Irish Math Olympiad, 1
Does there exist an even positive integer $n$ for which $n+1$ is divisible by $5$ and the two numbers $2^n + n$ and $2^n -1$ are co-prime?
1995 Tournament Of Towns, (469) 3
Let $AK$, $BL$ and $CM$ be the angle bisectors of a triangle $ABC$, with $K$ on $BC$. Let $P$ and $Q$ be the points on the lines $BL$ and $CM$ respectively such that $AP = PK$ and $AQ = QK$. Prove that $\angle PAQ = 90^o -\frac12 \angle B AC.$
(I Sharygin)
Novosibirsk Oral Geo Oly VIII, 2020.1
Three squares of area $4, 9$ and $36$ are inscribed in the triangle as shown in the figure. Find the area of the big triangle
[img]https://cdn.artofproblemsolving.com/attachments/9/7/3e904a9c78307e1be169ec0b95b1d3d24c1aa2.png[/img]
2018 Romania National Olympiad, 2
In the square $ABCD$ the point $E$ is located on the side $[AB]$, and $F$ is the foot of the perpendicular from $B$ on the line $DE$. The point $L$ belongs to the line $DE$, such that $F$ is between $E$ and $L$, and $FL = BF$. $N$ and $P$ are symmetric of the points $A , F$ with respect to the lines $DE, BL$, respectively. Prove that:
a) The quadrilateral $BFLP$ is square and the quadrilateral $ALND$ is rhombus.
b) The area of the rhombus $ALND$ is equal to the difference between the areas of the squares $ABCD$ and $BFLP$.
2015 CHMMC (Fall), 1
$3$ players take turns drawing lines that connect vertices of a regular $n$-gon. No player may draw a line that intersects another line at a point other than a vertex of the $n-$gon. The last player able to draw a line wins. For how many $n$ in the range $4\le n \le 100$ does the first player have a winning strategy?
2005 Postal Coaching, 9
In how many ways can $n$ identical balls be distributed to nine persons $A,B,C,D,E,F,G,H,I$ so that the number of balls recieved by $A$ is the same as the total number of balls recieved by $B,C,D,E$ together,.
2022 Kosovo & Albania Mathematical Olympiad, 3
Let $ABCD$ be a square and let $M$ be the midpoint of $BC$. Let $X$ and $Y$ be points on the segments $AB$ and $CD$, respectively. Prove that $\angle XMY = 90^\circ$ if and only if $BX + CY = XY$.
[i]Note: In the competition, students were only asked to prove the 'only if' direction.[/i]
2006 ISI B.Stat Entrance Exam, 5
Let $A,B$ and $C$ be three points on a circle of radius $1$.
(a) Show that the area of the triangle $ABC$ equals
\[\frac12(\sin(2\angle ABC)+\sin(2\angle BCA)+\sin(2\angle CAB))\]
(b) Suppose that the magnitude of $\angle ABC$ is fixed. Then show that the area of the triangle $ABC$ is maximized when $\angle BCA=\angle CAB$
(c) Hence or otherwise, show that the area of the triangle $ABC$ is maximum when the triangle is equilateral.
2005 National Olympiad First Round, 19
What is the greatest real root of the equation $x^3-x^2-x-\frac 13 = 0$?
$
\textbf{(A)}\ \dfrac{\sqrt {3} - \sqrt{2}}{2}
\qquad\textbf{(B)}\ \dfrac{\sqrt [3]{3} - \sqrt[3]{2}}{2}
\qquad\textbf{(C)}\ \dfrac 1{\sqrt[3] {3} - 1}
\qquad\textbf{(D)}\ \dfrac 1{\sqrt[3] {4} - 1}
\qquad\textbf{(E)}\ \text{None of above}
$
Ukrainian TYM Qualifying - geometry, 2020.10
In triangle $ABC$, point $I$ is the center, point $I_a$ is the center of the excircle tangent to the side $BC$. From the vertex $A$ inside the angle $BAC$ draw rays $AX$ and $AY$. The ray $AX$ intersects the lines $BI$, $CI$, $BI_a$, $CI_a$ at points $X_1$, $...$, $X_4$, respectively, and the ray $AY$ intersects the same lines at points $Y_1$, $...$, $Y_4$ respectively. It turned out that the points $X_1,X_2,Y_1,Y_2$ lie on the same circle. Prove the equality $$\frac{X_1X_2}{X_3X_4}= \frac{Y_1Y_2}{Y_3Y_4}.$$