Found problems: 85335
2011 IFYM, Sozopol, 2
On side $AB$ of $\Delta ABC$ is chosen point $M$. A circle is tangent internally to the circumcircle of $\Delta ABC$ and segments $MB$ and $MC$ in points $P$ and $Q$ respectively. Prove that the center of the inscribed circle of $\Delta ABC$ lies on line $PQ$.
2019 Azerbaijan Senior NMO, 4
Is it possible to construct a equilateral triangle such that:
$\text{a)}$ Coordinates of this triangle are integers in two dimensional plane?
$\text{b)}$ Coordinates of this triangle are integers in three dimensional plane?
2008 ITest, 46
Let $S$ be the sum of all $x$ in the interval $[0,2\pi)$ that satisfy \[\tan^2 x - 2\tan x\sin x=0.\] Compute $\lfloor10S\rfloor$.
2016 Putnam, A6
Find the smallest constant $C$ such that for every real polynomial $P(x)$ of degree $3$ that has a root in the interval $[0,1],$
\[\int_0^1|P(x)|\,dx\le C\max_{x\in[0,1]}|P(x)|.\]
2009 Princeton University Math Competition, 3
It is known that a certain mechanical balance can measure any object of integer mass anywhere between 1 and 2009 (both included). This balance has $k$ weights of integral values. What is the minimum $k$ for which there exist weights that satisfy this condition?
2021 Sharygin Geometry Olympiad, 19
A point $P$ lies inside a convex quadrilateral $ABCD$. Common internal tangents to the incircles of triangles $PAB$ and $PCD$ meet at point $Q$, and common internal tangents to the incircles of $PBC,PAD$ meet at point $R$. Prove that $P,Q,R$ are collinear.
1991 IberoAmerican, 2
A square is divided in four parts by two perpendicular lines, in such a way that three of these parts have areas equal to 1. Show that the square has area equal to 4.
2016 China Team Selection Test, 3
Let $n \geq 2$ be a natural. Define
$$X = \{ (a_1,a_2,\cdots,a_n) | a_k \in \{0,1,2,\cdots,k\}, k = 1,2,\cdots,n \}$$.
For any two elements $s = (s_1,s_2,\cdots,s_n) \in X, t = (t_1,t_2,\cdots,t_n) \in X$, define
$$s \vee t = (\max \{s_1,t_1\},\max \{s_2,t_2\}, \cdots , \max \{s_n,t_n\} )$$
$$s \wedge t = (\min \{s_1,t_1 \}, \min \{s_2,t_2,\}, \cdots, \min \{s_n,t_n\})$$
Find the largest possible size of a proper subset $A$ of $X$ such that for any $s,t \in A$, one has $s \vee t \in A, s \wedge t \in A$.
2023 Malaysian IMO Training Camp, 4
Find the largest constant $c>0$ such that for every positive integer $n\ge 2$, there always exist a positive divisor $d$ of $n$ such that $$d\le \sqrt{n}\hspace{0.5cm} \text{and} \hspace{0.5cm} \tau(d)\ge c\sqrt{\tau(n)}$$ where $\tau(n)$ is the number of divisors of $n$.
[i]Proposed by Mohd. Suhaimi Ramly[/i]
2018 EGMO, 4
A domino is a $ 1 \times 2 $ or $ 2 \times 1 $ tile.
Let $n \ge 3 $ be an integer. Dominoes are placed on an $n \times n$ board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. The value of a row or column is the number of dominoes that cover at least one cell of this row or column. The configuration is called balanced if there exists some $k \ge 1 $ such that each row and each column has a value of $k$. Prove that a balanced configuration exists for every $n \ge 3 $, and find the minimum number of dominoes needed in such a configuration.
2004 AMC 10, 4
A standard six-sided die is rolled, and $ P$ is the product of the fi ve numbers that are visible. What is the largest number that is certain to divide $ P$?
$ \textbf{(A)}\ 6\qquad
\textbf{(B)}\ 12\qquad
\textbf{(C)}\ 24\qquad
\textbf{(D)}\ 144\qquad
\textbf{(E)}\ 720$
1971 AMC 12/AHSME, 15
An aquarium on a level table has rectangular faces and is $10$ inches wide and $8$ inches high. When it was tilted, the water in it covered an $8"\times 10"$ end but only three-fourths of the rectangular room. The depth of the water when the bottom was again made level, was
$\textbf{(A) }2\textstyle{\frac{1}{2}}"\qquad\textbf{(B) }3"\qquad\textbf{(C) }3\textstyle{\frac{1}{4}}"\qquad\textbf{(D) }3\textstyle{\frac{1}{2}}"\qquad \textbf{(E) }4"$
2013 Singapore Junior Math Olympiad, 4
Let $a,b,$ be positive integers and $a>b>2$. Prove that $\frac{2^a+1}{2^b-1}$ is never an integer
2017 South Africa National Olympiad, 6
Determine all pairs $(P, d)$ of a polynomial $P$ with integer coefficients and an integer $d$ such that the equation $P(x) - P(y) = d$ has infinitely many solutions in integers $x$ and $y$ with $x \neq y$.
1984 Tournament Of Towns, (061) O2
Six altitudes are constructed from the three vertices of the base of a tetrahedron to the opposite sides of the three lateral faces. Prove that all three straight lines joining two base points of the altitudes in each lateral face are parallel to a certain plane.
(IF Sharygin, Moscow)
Brazil L2 Finals (OBM) - geometry, 2022.3
Let $ABC$ be a triangle with incenter $I$ and let $\Gamma$ be its circumcircle. Let $M$ be the midpoint of $BC$, $K$ the midpoint of the arc $BC$ which does not contain $A$, $L$ the midpoint of the arc $BC$ which contains $A$ and $J$ the reflection of $I$ by the line $KL$. The line $LJ$ intersects $\Gamma$ again at the point $T\neq L$. The line $TM$ intersects $\Gamma$ again at the point $S\neq T$. Prove that $S, I, M, K$ lie on the same circle.
1992 National High School Mathematics League, 2
Define set $S_n=\{1,2,\cdots,n\}$. $X$ is a subset of $S_n$. We call sum of all numbers in $X$ [i]capacity[/i] ([i]capacity[/i] of empty set is $0$). If [i]capacity[/i] of $X$ is odd/even, then we call it [i]odd/even subset[/i].
[b](a)[/b] Prove that the number of [i]odd subsets[/i] and [i]even subsets[/i] of $S_n$ are the same.
[b](b)[/b] Prove that the sum of [i]capacity[/i] of all [i]odd subsets[/i] and [i]even subsets[/i] are the same when $n\geq3$.
[b](c)[/b] Calculate the sum of [i]capacity[/i] of all [i]odd subsets[/i] when $n\geq3$.
1990 IMO Longlists, 35
Prove that if $|x| < 1$, then
\[ \frac{x}{(1-x)^2}+\frac{x^2}{(1+x^2)^2} + \frac{x^3}{(1-x^3)^2}+\cdots=\frac{x}{1-x}+\frac{2x^2}{1+x^2}+\frac{3x^3}{1-x^3}+\cdots\]
2023 Israel TST, P3
Let $ABC$ be a fixed triangle. Three similar (by point order) isosceles trapezoids are built on its sides: $ABXY, BCZW, CAUV$, such that the sides of the triangle are bases of the respective trapezoids. The circumcircles of triangles $XZU, YWV$ meet at two points $P, Q$. Prove that the line $PQ$ passes through a fixed point independent of the choice of trapezoids.
2004 AMC 10, 5
In the expression $ c\cdot a^b\minus{}d$, the values of $ a$, $ b$, $ c$, and $ d$ are $ 0$, $ 1$, $ 2$, and $ 3$, although not necessarily in that order. What is the maximum possible value of the result?
$ \textbf{(A)}\ 5\qquad
\textbf{(B)}\ 6\qquad
\textbf{(C)}\ 8\qquad
\textbf{(D)}\ 9\qquad
\textbf{(E)}\ 10$
JOM 2023, 3
Given an acute triangle $ABC$ with $AB<AC$, let $D$ be the foot of altitude from $A$ to $BC$ and let $M\neq D$ be a point on segment $BC$.$\,J$ and $K$ lie on $AC$ and $AB$ respectively such that $K,J,M$ lies on a common line perpendicular to $BC$. Let the circumcircles of $\triangle ABJ$ and $\triangle ACK$ intersect at $O$. Prove that $J,O,M$ are collinear if and only if $M$ is the midpoint of $BC$.
[i]Proposed by Wong Jer Ren[/i]
2009 Putnam, B3
Call a subset $ S$ of $ \{1,2,\dots,n\}$ [i]mediocre[/i] if it has the following property: Whenever $ a$ and $ b$ are elements of $ S$ whose average is an integer, that average is also an element of $ S.$ Let $ A(n)$ be the number of mediocre subsets of $ \{1,2,\dots,n\}.$ [For instance, every subset of $ \{1,2,3\}$ except $ \{1,3\}$ is mediocre, so $ A(3)\equal{}7.$] Find all positive integers $ n$ such that $ A(n\plus{}2)\minus{}2A(n\plus{}1)\plus{}A(n)\equal{}1.$
2018 ASDAN Math Tournament, 6
Given that $x > 1$, compute $x$ such that
$$\log_{16}(x) + \log_x(2)$$
is minimal.
2012 District Olympiad, 1
Let $a_1, a_2, ... , a_{2012}$ be odd positive integers. Prove that the number
$$A=\sqrt{a^2_1+ a^2_2+ ...+ a^2_{2012}-1}$$ is irrational.
2017 Irish Math Olympiad, 3
Four circles are drawn with the sides of quadrilateral $ABCD$ as diameters. The two circles passing through $A$ meet again at $A'$, two circles through $B$ at $B'$ , two circles at $C$ at $C'$ and the two circles at $D$ at $D'$. Suppose the points $A',B',C'$ and $D'$ are distinct. Prove quadrilateral $A'B'C'D'$ is similar to $ABCD$.