Found problems: 85335
2014 Sharygin Geometry Olympiad, 4
Tanya has cut out a triangle from checkered paper as shown in the picture. The lines of the grid have faded. Can Tanya restore them without any instruments only folding the triangle (she remembers the triangle sidelengths)?
(T. Kazitsyna)
2003 Tournament Of Towns, 3
An ant crawls on the outer surface of the box in a shape of rectangular parallelepiped. From ant’s point of view, the distance between two points on a surface is defined by the length of the shortest path ant need to crawl to reach one point from the other. Is it true that if ant is at vertex then from ant’s point of view the opposite vertex be the most distant point on the surface?
1982 IMO Longlists, 43
[b](a)[/b] What is the maximal number of acute angles in a convex polygon?
[b](b)[/b] Consider $m$ points in the interior of a convex $n$-gon. The $n$-gon is partitioned into triangles whose vertices are among the $n + m$ given points (the vertices of the $n$-gon and the given points). Each of the $m$ points in the interior is a vertex of at least one triangle. Find the number of triangles obtained.
2007 Turkey Team Selection Test, 3
Let $a, b, c$ be positive reals such that their sum is $1$. Prove that \[\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ac+2b^{2}+2b}\geq \frac{1}{ab+bc+ac}.\]
2018 Pan-African Shortlist, G3
Given a triangle $ABC$, let $D$ be the intersection of the line through $A$ perpendicular to $AB$, and the line through $B$ perpendicular to $BC$. Let $P$ be a point inside the triangle. Show that $DAPB$ is cyclic if and only if $\angle BAP = \angle CBP$.
2011 BAMO, 3
Let $S$ be a finite, nonempty set of real numbers such that the distance between any two distinct points in $S$ is an element of $S$. In other words, $|x-y|$ is in $S$ whenever $x \ne y$ and $x$ and $y$ are both in $S$.
Prove that the elements of $S$ may be arranged in an arithmetic progression.
This means that there are numbers $a$ and $d$ such that $S = \{a, a+d, a+2d, a+3d, ..., a+kd, ...\}$.
2020 BMT Fall, 9
For any point $(x, y)$ with $0\le x < 1$ and $0 \le y < 1$, Jenny can perform a shuffle on that point, which takes the point to $(\{3x + y\} ,\{x + 2y\})$ where $\{a\}$ denotes the fractional or decimal part of $a$ (so for example,$\{\pi\} = \pi - 3 = 0.1415...$). How many points $p$ are there such that after $3$ shuffles on $p$, $p$ ends up in its original position?
MBMT Team Rounds, 2020.25
Let $\left \lfloor x \right \rfloor$ denote the greatest integer less than or equal to $x$. Find the sum of all positive integer solutions to $$\left \lfloor \frac{n^3}{27} \right \rfloor - \left \lfloor \frac{n}{3} \right \rfloor ^3=10.$$
[i]Proposed by Jason Hsu[/i]
2011 Argentina National Olympiad, 4
For each natural number $n$ we denote $a_n$ as the greatest perfect square less than or equal to $n$ and $b_n$ as the least perfect square greater than $n$. For example $a_9=3^2$, $b_9=4^2$ and $a_{20}=4^2$, $b_{20}=5^2$. Calculate: $$\frac{1}{a_1b_1}+\frac{1}{a_2b_2}+\frac{1}{a_3b_3}+\ldots +\frac{1}{a_{600}b_{600}}$$
2021 CCA Math Bonanza, L2.1
Let $ABC$ be a triangle with $AB=3$, $BC=4$, and $CA=5$. The line through $A$ perpendicular to $AC$ intersects line $BC$ at $D$, and the line through $C$ perpendicular to $AC$ intersects line $AB$ at $E$. Compute the area of triangle $BDE$.
[i]2021 CCA Math Bonanza Lightning Round #2.1[/i]
2023 Yasinsky Geometry Olympiad, 3
$ABC$ is a right triangle with $\angle C = 90^o$. Let $N$ be the middle of arc $BAC$ of the circumcircle and $K$ be the intersection point of $CN$ and $AB$. Assume $T$ is a point on a line $AK$ such that $TK=KA$. Prove that the circle with center $T$ and radius $TK$ is tangent to $BC$.
(Mykhailo Sydorenko)
1961 Putnam, B2
Let $a$ and $b$ be given positive real numbers, with $a<b.$ If two points are selected at random from a straight line segment of length $b,$ what is the probability that the distance between them is at least $a?$
MOAA Team Rounds, 2019.8
Suppose that $$\frac{(\sqrt2)^5 + 1}{\sqrt2 + 1} \times \frac{2^5 + 1}{2 + 1} \times \frac{4^5 + 1}{4 + 1} \times \frac{16^5 + 1}{16 + 1} =\frac{m}{7 + 3\sqrt2}$$ for some integer $m$. How many $0$’s are in the binary representation of $m$? (For example, the number $20 = 10100_2$ has three $0$’s in its binary representation.)
2019 CMIMC, 6
Let $a, b$ and $c$ be the distinct solutions to the equation $x^3-2x^2+3x-4=0$. Find the value of
$$\frac{1}{a(b^2+c^2-a^2)}+\frac{1}{b(c^2+a^2-b^2)}+\frac{1}{c(a^2+b^2-c^2)}.$$
2021 Latvia Baltic Way TST, P4
Determine the smallest positive constant $k$ such that no matter what $3$ lattice points we choose the following inequality holds:
$$ L_{\max} - L_{\min} \ge \frac{1}{\sqrt{k} \cdot L_{max}} $$
where $L_{\max}$, $L_{\min}$ is the maximal and minimal distance between chosen points.
2013 Purple Comet Problems, 11
After Jennifer walked $r$ percent of the way from her home to the store, she turned around and walked home, got on her bicycle, and bicycled to the store and back home. Jennifer bicycles two and a half times faster than she walks. Find the largest value of $r$ so that returning home for her bicycle was not slower than her walking all the way to and from the store without her bicycle.
1975 Dutch Mathematical Olympiad, 2
Let $T = \{n \in N|$n consists of $2$ digits $\}$ and $$P = \{x|x = n(n + 1)... (n + 7); n,n + 1,..., n + 7 \in T\}.$$
Determine the gcd of the elements of $P$.
2016 Iranian Geometry Olympiad, 1
Ali wants to move from point $A$ to point $B$. He cannot walk inside the black areas but he is free to move in any direction inside the white areas (not only the grid lines but the whole plane). Help Ali to find the shortest path between $A$ and $B$. Only draw the path and write its length.
[img]https://1.bp.blogspot.com/-nZrxJLfIAp8/W1RyCdnhl3I/AAAAAAAAIzQ/NM3t5EtJWMcWQS0ig0IghSo54DQUBH5hwCK4BGAYYCw/s1600/igo%2B2016.el1.png[/img]
by Morteza Saghafian
2011 Croatia Team Selection Test, 4
Find all pairs of integers $x,y$ for which
\[x^3+x^2+x=y^2+y.\]
2014 Contests, 1
Numbers $1$ through $2014$ are written on a board. A valid operation is to erase two numbers $a$ and $b$ on the board and replace them with the greatest common divisor and the least common multiple of $a$ and $b$.
Prove that, no matter how many operations are made, the sum of all the numbers that remain on the board is always larger than $2014$ $\times$ $\sqrt[2014]{2014!}$
2009 Canadian Mathematical Olympiad Qualification Repechage, 8
Determine an infinite family of quadruples $(a, b, c, d)$ of positive integers, each of which is a solution to $a^4+b^5+c^6=d^7$.
II Soros Olympiad 1995 - 96 (Russia), 9.6
There is a point inside a regular triangle located at distances $5$, $6$ and $7$ from its vertices. Find the area of this regular triangle.
2015 Iran Team Selection Test, 3
$a_1,a_2,\cdots ,a_n,b_1,b_2,\cdots ,b_n$ are $2n$ positive real numbers such that $a_1,a_2,\cdots ,a_n$ aren't all equal. And assume that we can divide $a_1,a_2,\cdots ,a_n$ into two subsets with equal sums.similarly $b_1,b_2,\cdots ,b_n$ have these two conditions. Prove that there exist a simple $2n$-gon with sides $a_1,a_2,\cdots ,a_n,b_1,b_2,\cdots ,b_n$ and parallel to coordinate axises Such that the lengths of horizontal sides are among $a_1,a_2,\cdots ,a_n$ and the lengths of vertical sides are among $b_1,b_2,\cdots ,b_n$.(simple polygon is a polygon such that it doesn't intersect itself)
2018 Baltic Way, 20
Find all the triples of positive integers $(a,b,c)$ for which the number
\[\frac{(a+b)^4}{c}+\frac{(b+c)^4}{a}+\frac{(c+a)^4}{b}\]
is an integer and $a+b+c$ is a prime.
2007 National Olympiad First Round, 23
A unit equilateral triangle is given. Divide each side into three equal parts. Remove the equilateral triangles whose bases are middle one-third segments. Now we have a new polygon. Remove the equilateral triangles whose bases are middle one-third segments of the sides of the polygon. After repeating these steps for infinite times, what is the area of the new shape?
$
\textbf{(A)}\ \dfrac {1}{2\sqrt 3}
\qquad\textbf{(B)}\ \dfrac {\sqrt 3}{8}
\qquad\textbf{(C)}\ \dfrac {\sqrt 3}{10}
\qquad\textbf{(D)}\ \dfrac {1}{4\sqrt 3}
\qquad\textbf{(E)}\ \text{None of the above}
$