Found problems: 85335
2010 Tournament Of Towns, 1
There are $100$ points on the plane. All $4950$ pairwise distances between two points have been recorded.
$(a)$ A single record has been erased. Is it always possible to restore it using the remaining records?
$(b)$ Suppose no three points are on a line, and $k$ records were erased. What is the maximum value of $k$ such that restoration of all the erased records is always possible?
2024 Canadian Mathematical Olympiad Qualification, 7a
In triangle $ABC$, let $I$ be the incentre. Let $H$ be the orthocentre of triangle $BIC$. Show that $AH$ is parallel to $BC$ if and only if $H$ lies on the circle with diameter $AI$.
2012 Dutch BxMO/EGMO TST, 4
Let $ABCD$ a convex quadrilateral (this means that all interior angles are smaller than $180^o$), such that there exist a point $M$ on line segment $AB$ and a point $N$ on line segment $BC$ having the property that $AN$ cuts the quadrilateral in two parts of equal area, and such that the same property holds for $CM$.
Prove that $MN$ cuts the diagonal $BD$ in two segments of equal length.
1993 Cono Sur Olympiad, 2
Prove that there exists a succession $a_1, a_2, ... , a_k, ...$, where each $a_i$ is a digit ($a_i \in (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)$ ) and $a_0=6$, such that, for each positive integrer $n$, the number $x_n=a_0+10a_1+100a_2+...+10^{n-1}a_{n-1}$ verify that $x_n^2-x_n$ is divisible by $10^n$.
2007 Junior Balkan Team Selection Tests - Romania, 4
Find all integer positive numbers $n$ such that:
$n=[a,b]+[b,c]+[c,a]$, where $a,b,c$ are integer positive numbers and $[p,q]$ represents the least common multiple of numbers $p,q$.
2014 Flanders Math Olympiad, 1
(a) Prove the parallelogram law that says that in a parallelogram the sum of the squares of the lengths of the four sides equals the sum of the squares of the lengths of the two diagonals.
(b) The edges of a tetrahedron have lengths $a, b, c, d, e$ and $f$. The three line segments connecting the centers of intersecting edges have lengths $x, y$ and $z$. Prove that
$$4 (x^2 + y^2 + z^2) = a^2 + b^2 + c^2 + d^2 + e^2 + f^2$$
2008 ITest, 43
Alexis notices Joshua working with Dr. Lisi and decides to join in on the fun. Dr. Lisi challenges her to compute the sum of all $2008$ terms in the sequence. Alexis thinks about the problem and remembers a story one of her teahcers at school taught her about how a young Karl Gauss quickly computed the sum \[1+2+3+\cdots+98+99+100\] in elementary school. Using Gauss's method, Alexis correctly finds the sum of the $2008$ terms in Dr. Lisi's sequence. What is this sum?
2024 Bulgaria MO Regional Round, 12.2
Let $N$ be a positive integer. The sequence $x_1, x_2, \ldots$ of non-negative reals is defined by $$x_n^2=\sum_{i=1}^{n-1} \sqrt{x_ix_{n-i}}$$ for all positive integers $n>N$. Show that there exists a constant $c>0$, such that $x_n \leq \frac{n} {2}+c$ for all positive integers $n$.
2006 ISI B.Math Entrance Exam, 2
Prove that there is no non-constant polynomial $P(x)$ with integer coefficients such that $P(n)$ is a prime number for all positive integers $n$.
2011 Saudi Arabia IMO TST, 3
In acute triangle $ABC$, $\angle A = 20^o$. Prove that the triangle is isosceles if and only if $$\sqrt[3]{a^3 + b^3 + c^3 -3abc} = \min\{b, c\}$$, where $a,b, c$ are the side lengths of triangle $ABC$.
2017 Bulgaria EGMO TST, 3
Let $ ABC$ be a fixed triangle, and let $ A_1$, $ B_1$, $ C_1$ be the midpoints of sides $ BC$, $ CA$, $ AB$, respectively. Let $ P$ be a variable point on the circumcircle. Let lines $ PA_1$, $ PB_1$, $ PC_1$ meet the circumcircle again at $ A'$, $ B'$, $ C'$, respectively. Assume that the points $ A$, $ B$, $ C$, $ A'$, $ B'$, $ C'$ are distinct, and lines $ AA'$, $ BB'$, $ CC'$ form a triangle. Prove that the area of this triangle does not depend on $ P$.
[i]Author: Christopher Bradley, United Kingdom [/i]
1985 National High School Mathematics League, 1
In rectangular coordinate system $xOy, A(x_1,y_1), B(x_2,y_2)$, where $x_1,y_1,x_2,y_2$ are 1-digit-numbers. Intersection angle between $OA$ and $x$-axis positive direction is larger than $\frac{\pi}{4}$, intersection angle between $OB$ and $x$-axis positive direction is smaller than $\frac{\pi}{4}$. Projection of $A$ on $y$-axis is $A'$, projection of $B$ on $x$-axis is $B'$. Area of $\triangle OBB'$ is $33.5$ larger than $\triangle OAA'$. Find all 4-digit-number $\overline{x_1x_2y_1y_2}$.
Kyiv City MO Juniors 2003+ geometry, 2016.9.5
On the sides $BC$ and $AB$ of the triangle $ABC$ the points ${{A} _ {1}}$ and ${{C} _ {1}} $ are selected accordingly so that the segments $A {{A} _ {1}}$ and $C {{C} _ {1}}$ are equal and perpendicular. Prove that if $\angle ABC = 45 {} ^ \circ$, then $AC = A {{A} _ {1}} $.
(Gogolev Andrew)
2016 Online Math Open Problems, 23
Let $\mathbb N$ denote the set of positive integers. Let $f: \mathbb N \to \mathbb N$ be a function such that the following conditions hold:
(a) For any $n\in \mathbb N$, we have $f(n) | n^{2016}$.
(b) For any $a,b,c\in \mathbb N$ satisfying $a^2+b^2=c^2$, we have $f(a)f(b)=f(c)$.
Over all possible functions $f$, determine the number of distinct values that can be achieved by $f(2014)+f(2)-f(2016)$.
[i]Proposed by Vincent Huang[/i]
2021 Iranian Geometry Olympiad, 1
Let $ABC$ be a triangle with $AB = AC$. Let $H$ be the orthocenter of $ABC$. Point
$E$ is the midpoint of $AC$ and point $D$ lies on the side $BC$ such that $3CD = BC$. Prove that
$BE \perp HD$.
[i]Proposed by Tran Quang Hung - Vietnam[/i]
2024 Belarusian National Olympiad, 8.6
For each number $x$ we denote by $S(x)$ the sum of digits from its decimal representation. Find all positive integers $m$ for each of which there exists a positive integer $n$, such that $$S(n^2-2n+10)=m$$
[i]Chernov S.[/i]
2004 Italy TST, 2
Let $\mathcal{P}_0=A_0A_1\ldots A_{n-1}$ be a convex polygon such that $A_iA_{i+1}=2^{[i/2]}$ for $i=0, 1,\ldots ,n-1$ (where $A_n=A_0$). Define the sequence of polygons $\mathcal{P}_k=A_0^kA_1^k\ldots A_{n-1}^k$ as follows: $A_i^1$ is symmetric to $A_i$ with respect to $A_0$, $A_i^2$ is symmetric to $A_i^1$ with respect to $A_1^1$, $A_i^3$ is symmetric to $A_i^2$ with respect to $A_2^2$ and so on. Find the values of $n$ for which infinitely many polygons $\mathcal{P}_k$ coincide with $\mathcal{P}_0$.
2003 USAMO, 1
Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.
1992 Tournament Of Towns, (356) 5
The bisector of the angle $A$ of triangle $ABC$ intersects its circumscribed circle at the point $D$. Suppose $P$ is the point symmetric to the incentre of the triangle with respect to the midpoint of the side $BC$, and $M$ is the second intersection point of the line $PD$ with the circumscribed circle. Prove that one of the distances $AM$, $BM$, $CM$ is equal to the sum of two other distances.
(VO Gordon)
2009 Germany Team Selection Test, 1
In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a [i]box[/i]. Two boxes [i]intersect[/i] if they have a common point in their interior or on their boundary. Find the largest $ n$ for which there exist $ n$ boxes $ B_1$, $ \ldots$, $ B_n$ such that $ B_i$ and $ B_j$ intersect if and only if $ i\not\equiv j\pm 1\pmod n$.
[i]Proposed by Gerhard Woeginger, Netherlands[/i]
1997 Canada National Olympiad, 3
Prove that $\frac{1}{1999}< \prod_{i=1}^{999}{\frac{2i-1}{2i}}<\frac{1}{44}$.
2011 Junior Balkan Team Selection Tests - Romania, 2
We consider an $n \times n$ ($n \in N, n \ge 2$) square divided into $n^2$ unit squares. Determine all the values of $k \in N$ for which we can write a real number in each of the unit squares such that the sum of the $n^2$ numbers is a positive number, while the sum of the numbers from the unit squares of any $k \times k$ square is a negative number.
2012 Thailand Mathematical Olympiad, 12
Let $a, b, c$ be positive integers. Show that if $\frac{a}{b} +\frac{b}{c} +\frac{c}{a}$ is an integer then $abc$ is a perfect cube.
1992 National High School Mathematics League, 2
The equation of unit circle in Quadrant I, III, IV ($(-1,0),(1,0),(0,-1),(0,1)$ included) is
$\text{(A)}(x+\sqrt{1-y^2})(y+\sqrt{1-x^2})=0$
$\text{(B)}(x-\sqrt{1-y^2})(y-\sqrt{1-x^2})=0$
$\text{(C)}(x+\sqrt{1-y^2})(y-\sqrt{1-x^2})=0$
$\text{(D)}(x-\sqrt{1-y^2})(y+\sqrt{1-x^2})=0$
1966 IMO Shortlist, 21
Prove that the volume $V$ and the lateral area $S$ of a right circular cone satisfy the inequality
\[\left( \frac{6V}{\pi}\right)^2 \leq \left( \frac{2S}{\pi \sqrt 3}\right)^3\]
When does equality occur?