Found problems: 85335
2017 Sharygin Geometry Olympiad, 8
Let $ABCD$ be a square, and let $P$ be a point on the minor arc $CD$ of its circumcircle. The lines $PA, PB$ meet the diagonals $BD, AC$ at points $K, L$ respectively. The points $M, N$ are the projections of $K, L$ respectively to $CD$, and $Q$ is the common point of lines $KN$ and $ML$. Prove that $PQ$ bisects the segment $AB$.
2013 BMT Spring, 3
Suppose we have $2013$ piles of coins, with the $i$th pile containing exactly $i$ coins. We wish to remove the coins in a series of steps. In each step, we are allowed to take away coins from as many piles as we wish, but we have to take the same number of coins from each pile. We cannot take away more coins than a pile actually has. What is the minimum number of steps we have to take?
2019 Tournament Of Towns, 2
Given a convex pentagon $ABCDE$ such that $AE$ is parallel to $CD$ and $AB=BC$. Angle bisectors of angles $A$ and $C$ intersect at $K$. Prove that $BK$ and $AE$ are parallel.
2005 Abels Math Contest (Norwegian MO), 3b
In the parallelogram $ABCD$, all sides are equal, and $\angle A = 60^o$. Let $F$ be a point on line $AD, H$ a point on line $DC$, and $G$ a point on diagonal $AC$ such that $DFGH$ is a parallelogram. Show that then $\vartriangle BHF$ is equilateral.
2016 Saudi Arabia BMO TST, 2
A circle with center $O$ passes through points $A$ and $C$ and intersects the sides $AB$ and $BC$ of triangle $ABC$ at points $K$ and $N$, respectively. The circumcircles of triangles $ABC$ and $KBN$ meet at distinct points $B$ and $M$. Prove that $\angle OMB = 90^o$.
Kyiv City MO Seniors Round2 2010+ geometry, 2018.10.3.1
The point $O$ is the center of the circumcircle of the acute triangle $ABC$. The line $AC$ intersects the circumscribed circle $\Delta ABO$ for second time at the point $X$. Prove that $XO \bot BC$.
2006 IMO Shortlist, 6
Let $ a > b > 1$ be relatively prime positive integers. Define the weight of an integer $ c$, denoted by $ w(c)$ to be the minimal possible value of $ |x| \plus{} |y|$ taken over all pairs of integers $ x$ and $ y$ such that \[ax \plus{} by \equal{} c.\] An integer $ c$ is called a [i]local champion [/i]if $ w(c) \geq w(c \pm a)$ and $ w(c) \geq w(c \pm b)$.
Find all local champions and determine their number.
[i]Proposed by Zoran Sunic, USA[/i]
2008 Romanian Master of Mathematics, 1
Let $ ABC$ be an equilateral triangle and $ P$ in its interior. The distances from $ P$ to the triangle's sides are denoted by $ a^2, b^2,c^2$ respectively, where $ a,b,c>0$. Find the locus of the points $ P$ for which $ a,b,c$ can be the sides of a non-degenerate triangle.
2001 Baltic Way, 19
What is the smallest positive odd integer having the same number of positive divisors as $360$?
1999 Flanders Math Olympiad, 4
Let $a,b,m,n$ integers greater than 1. If $a^n-1$ and $b^m+1$ are both primes, give as much info as possible on $a,b,m,n$.
2010 Contests, 1
Consider a triangle $ABC$ such that $\angle A = 90^o, \angle C =60^o$ and $|AC|= 6$. Three circles with centers $A, B$ and $C$ are pairwise tangent in points on the three sides of the triangle.
Determine the area of the region enclosed by the three circles (the grey area in the figure).
[asy]
unitsize(0.2 cm);
pair A, B, C;
real[] r;
A = (6,0);
B = (6,6*sqrt(3));
C = (0,0);
r[1] = 3*sqrt(3) - 3;
r[2] = 3*sqrt(3) + 3;
r[3] = 9 - 3*sqrt(3);
fill(arc(A,r[1],180,90)--arc(B,r[2],270,240)--arc(C,r[3],60,0)--cycle, gray(0.7));
draw(A--B--C--cycle);
draw(Circle(A,r[1]));
draw(Circle(B,r[2]));
draw(Circle(C,r[3]));
dot("$A$", A, SE);
dot("$B$", B, NE);
dot("$C$", C, SW);
[/asy]
Estonia Open Junior - geometry, 2009.1.2
The feet of the altitudes drawn from vertices $A$ and $B$ of an acute triangle $ABC$ are $K$ and $L$, respectively. Prove that if $|BK| = |KL|$ then the triangle $ABC$ is isosceles.
2014 Math Prize For Girls Problems, 9
Let $abc$ be a three-digit prime number whose digits satisfy $a < b < c$. The difference between every two of the digits is a prime number too. What is the sum of all the possible values of the three-digit number $abc$?
1993 USAMO, 1
For each integer $\, n \geq 2, \,$ determine, with proof, which of the two positive real numbers $\, a \,$ and $\, b \,$ satisfying \[ a^n = a + 1, \hspace{.3in} b^{2n} = b + 3a \] is larger.
2018 CMIMC Algebra, 8
Suppose $P$ is a cubic polynomial satisfying $P(0) = 3$ and \[(x^3 - 2x + 1 - P(x))(2x^3 - 5x^2 + 4 - P(x))\leq 0\] for all $x\in\mathbb R$. Determine all possible values of $P(-1)$.
Putnam 1939, B4
The axis of a parabola is its axis of symmetry and its vertex is its point of intersection with its axis. Find: the equation of the parabola which touches $y = 0$ at $(1,0)$ and $x = 0$ at $(0,2);$ the equation of its axis; and its vertex.
2021 OMMock - Mexico National Olympiad Mock Exam, 1
Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ that satisfy the following property for all real numbers $x$ and all polynomials $P$ with real coefficients:
If $P(f(x)) = 0$, then $f(P(x)) = 0$.
2002 AMC 12/AHSME, 2
Cindy was asked by her teacher to subtract $ 3$ from a certain number and then divide the result by $ 9$. Instead, she subtracted $ 9$ and then divided the result by $ 3$, giving an answer of $ 43$. What would her answer have been had she worked the problem correctly?
$ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 34 \qquad \textbf{(C)}\ 43 \qquad \textbf{(D)}\ 51 \qquad \textbf{(E)}\ 138$
2020 CIIM, 2
Find all triples of positive integers $(a,b,c)$ such that the following equations are both true:
I- $a^2+b^2=c^2$
II- $a^3+b^3+1=(c-1)^3$
2023 Oral Moscow Geometry Olympiad, 2
There is a square sheet of paper. How to get a rectangular sheet of paper with an aspect ratio equal to $\sqrt2$? (There are no tools, the sheet can only be bent.)
2016 Latvia Baltic Way TST, 11
Is it possible to cut a square with side $\sqrt{2015}$ into no more than five pieces so that these pieces can be rearranged into a rectangle with sides of integer length? (The cuts should be made using straight lines, and flipping of the pieces is disallowed.)
2013 Harvard-MIT Mathematics Tournament, 7
Let $ABC$ be an obtuse triangle with circumcenter $O$ such that $\angle ABC = 15^o$ and $\angle BAC > 90^o$. Suppose that $AO$ meets $BC$ at $D$, and that $OD^2 + OC \cdot DC = OC^2$. Find $\angle C$.
2016 Baltic Way, 11
Set $A$ consists of $2016$ positive integers. All prime divisors of these numbers are smaller than $30.$ Prove that there are four distinct numbers $a, b, c$ and $d$ in $A$ such that $abcd$ is a perfect square.
PEN J Problems, 17
Show that $\phi(n)+\sigma(n) \ge 2n$ for all positive integers $n$.
2025 AMC 8, 20
Sarika, Dev, and Rajiv are sharing a large block of cheese. They take turns cutting off half of what remains and eating it: first Sarika eats half of the cheese, then Dev eats half of the remaining half, then Rajiv eats half of what remains, then back to Sarika, and so on. They stop when the cheese is too small to see. About what fraction of the original block of cheese does Sarika eat in total?
$\hspace*{5mm}\text{(A) } \frac{4}{7} \quad \text{(B) } \frac{3}{5} \quad \text{(C) } \frac{2}{3} \quad \text{(D) } \frac{3}{4} \quad \text{(E) } \frac{7}{8}$