Found problems: 85335
2019 Rioplatense Mathematical Olympiad, Level 3, 5
Let $ABC$ be a triangle with $AB<AC$ and circuncircle $\omega$. Let $M$ and $N$ be the midpoints of $AC$ and $AB$ respectively and $G$ is the centroid of $ABC$. Let $P$ be the foot of perpendicular of $A$ to the line $BC$, and the point $Q$ is the intersection of $GP$ and $\omega$($Q,P,G$ are collinears in this order). The line $QM$ cuts $\omega$ in $M_1$ and the line $QN$ cuts $\omega$ in $N_1$. If $K$ is the intersection of $BM_1$ and $CN_1$ prove that $P$, $G$ and $K$ are collinears.
2023 Harvard-MIT Mathematics Tournament, 9
Point $Y$ lies on line segment $XZ$ such that $XY = 5$ and $Y Z = 3$. Point $G$ lies on line $XZ$ such that there exists a triangle $ABC$ with centroid $G$ such that $X$ lies on line $BC$, $Y$ lies on line $AC$, and $Z$ lies on line $AB$. Compute the largest possible value of $XG$.
2022 Assara - South Russian Girl's MO, 8
In parallelogram $ABCD$, angle $A$ is acute. Let $X$ be a point, symmetrical to point $C$ wrt to straight line $AD$, $Y$ is a point symmetrical to the point $C$ wrt point $D$, and $M$ is the intersection point of $AC$ and $BD$. It turned out, that the circumcircles of triangles $BMC$ and $AXY$ are tangent internally. Prove that $AM = AB$.
2011 Kosovo National Mathematical Olympiad, 2
Find all solutions to the equation:
\[ \left(\left\lfloor x+\frac{7}{3} \right\rfloor \right)^2-\left\lfloor x-\frac{9}{4} \right\rfloor = 16 \]
2024 Taiwan TST Round 3, 6
Find all positive integers $n$ and sequence of integers $a_0,a_1,\ldots, a_n$ such that the following hold:
1. $a_n\neq 0$;
2. $f(a_{i-1})=a_i$ for all $i=1,\ldots, n$, where $f(x) = a_nx^n+a_{n-1}x^{n-1}+\cdots +a_0$.
[i]
Proposed by usjl[/i]
2014 Belarus Team Selection Test, 3
Do there exist functions $f$ and $g$, $f : R \to R$, $g : R \to R$ such that $f(x + f(y)) = y^2 + g(x)$ for all real $x$ and $y$ ?
(I. Gorodnin)
1986 Tournament Of Towns, (129) 4
We define $N !!$ to be $N(N - 2)(N -4)...5 \cdot 3 \cdot 1$ if $N$ is odd and $N(N -2)(N -4)... 6\cdot 4\cdot 2$ if $N$ is even .
For example, $8 !! = 8 \cdot 6\cdot 4\cdot 2$ , and $9 !! = 9v 7 \cdot 5\cdot 3 \cdot 1$ .
Prove that $1986 !! + 1985 !!$ i s divisible by $1987$.
(V.V . Proizvolov , Moscow)
1983 IMO, 3
Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that
\[ a^{2}b(a \minus{} b) \plus{} b^{2}c(b \minus{} c) \plus{} c^{2}a(c \minus{} a)\ge 0.
\]
Determine when equality occurs.
1981 Dutch Mathematical Olympiad, 3
We want to split the set of natural numbers from $1$ to $3n$, where $n$ is a natural number, into $n$ mutually disjoint sets $\{x,y,z\}$ of three elements such that always holds: $x + y = 3z$. Is this possible for :
a) $n = 5$?
b) $n=10$?
In both cases, provide either such a split or proof that such a split is not possible.
2017 Azerbaijan EGMO TST, 1
Given an equilateral triangle $ABC$ and a point $P$ so that the distances $P$ to $A$ and to $C$ are not farther than the distances $P$ to $B$. Prove that $PB = PA + PC$ if and only if $P$ lies on the circumcircle of $\vartriangle ABC$.
1991 APMO, 5
Given are two tangent circles and a point $P$ on their common tangent perpendicular to the lines joining their centres. Construct with ruler and compass all the circles that are tangent to these two circles and pass through the point $P$.
2024-IMOC, N5
Find all positive integers $n$ such that
$$2^n+15|3^n+200$$
2024-25 IOQM India, 11
The positive real numbers $a,b,c,$ satisfy: $$\frac{a}{2b+1} + \frac{2b}{3c+1} + \frac{3c}{a+1} = 1$$ $$\frac{1}{a+1} + \frac{1}{2b+1} + \frac{1}{3c+1} = 2$$ What is the value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$
2020 Vietnam National Olympiad, 4
Let a non-isosceles acute triangle ABC with the circumscribed cycle (O) and the orthocenter H. D, E, F are the reflection of O in the lines BC, CA and AB.
a) $H_a$ is the reflection of H in BC, A' is the reflection of A at O and $O_a$ is the center of (BOC). Prove that $H_aD$ and OA' intersect on (O).
b) Let X is a point satisfy AXDA' is a parallelogram. Prove that (AHX), (ABF), (ACE) have a comom point different than A
2014 NIMO Problems, 3
In triangle $ABC$, we have $AB=AC=20$ and $BC=14$. Consider points $M$ on $\overline{AB}$ and $N$ on $\overline{AC}$. If the minimum value of the sum $BN + MN + MC$ is $x$, compute $100x$.
[i]Proposed by Lewis Chen[/i]
2009 F = Ma, 13
Lucy (mass $\text{33.1 kg}$), Henry (mass $\text{63.7 kg}$), and Mary (mass $\text{24.3 kg}$) sit on a lightweight seesaw at evenly spaced $\text{2.74 m}$ intervals (in the order in which they are listed; Henry is between Lucy and Mary) so that the seesaw balances. Who exerts the most torque (in terms of magnitude) on the seesaw? Ignore the mass of the seesaw.
(A) Henry
(B) Lucy
(C) Mary
(D) They all exert the same torque.
(E) There is not enough information to answer the question.
2015 Iran Geometry Olympiad, 1
We have four wooden triangles with sides $3, 4, 5$ centimeters. How many convex polygons can we make by all of these triangles? (Just draw the polygons without any proof)
A convex polygon is a polygon which all of it's angles are less than $180^o$ and there isn't any hole in it. For example:
[img]https://1.bp.blogspot.com/-JgvF_B-uRag/W1R4f4AXxTI/AAAAAAAAIzc/Fo3qu3pxXcoElk01RTYJYZNwj0plJaKPQCK4BGAYYCw/s640/igo%2B2015.el1.png[/img]
2005 Estonia National Olympiad, 4
A sequence of natural numbers $a_1, a_2, a_3,..$ is called [i]periodic modulo[/i] $n$ if there exists a positive integer $k$ such that, for any positive integer $i$, the terms $a_i$ and $a_{i+k}$ are equal modulo $n$. Does there exist a strictly increasing sequence of natural numbers that
a) is not periodic modulo finitely many positive integers and is periodic modulo all the other positive integers?
b) is not periodic modulo infinitely many positive integers and is periodic modulo infinitely many positive integers?
1994 Swedish Mathematical Competition, 1
$x\sqrt8 + \frac{1}{x\sqrt8} = \sqrt8$ has two real solutions $x_1, x_2$. The decimal expansion of $x_1$ has the digit $6$ in place $1994$. What digit does $x_2$ have in place $1994$?
2010 Oral Moscow Geometry Olympiad, 4
From the vertex $A$ of the parallelogram $ABCD$, the perpendiculars $AM,AN$ on sides $BC,CD$ respectively. $P$ is the intersection point of $BN$ and $DM$. Prove that the lines $AP$ and $MN$ are perpendicular.
2016 ASDAN Math Tournament, 2
The largest factor of $n$ not equal to $n$ is $35$. Compute the largest possible value of $n$.
2000 BAMO, 2
Let $ABC$ be a triangle with $D$ the midpoint of side $AB, E$ the midpoint of side $BC$, and $F$ the midpoint of side $AC$. Let $k_1$ be the circle passing through points $A, D$, and $F$, let $k_2$ be the circle passing through points $B, E$, and $D$, and let $k_3$ be the circle passing through $C, F$, and $E$. Prove that circles $k_1, k_2$, and $k_3$ intersect in a point.
2006 ITAMO, 5
Consider the inequality
\[(a_1+a_2+\dots+a_n)^2\ge 4(a_1a_2+a_2a_3+\cdots+a_na_1).\]
a) Find all $n\ge 3$ such that the inequality is true for positive reals.
b) Find all $n\ge 3$ such that the inequality is true for reals.
2012 Tournament of Towns, 5
A car rides along a circular track in the clockwise direction. At noon Peter and Paul took their positions at two different points of the track. Some moment later they simultaneously ended their duties and compared their notes. The car passed each of them at least $30$ times. Peter noticed that each circle was passed by the car $1$ second faster than the preceding one while Paul’s observation was opposite: each circle was passed $1$ second slower than the preceding one.
Prove that their duty was at least an hour and a half long.
2015 Paraguayan Mathematical Olympiad, Problem 3
A cube is divided into $8$ smaller cubes of the same size, as shown in the figure. Then, each of these small cubes is divided again into $8$ smaller cubes of the same size. This process is done $4$ more times to each resulting cube. What is the ratio between the sum of the total areas of all the small cubes resulting from the last division and the total area of the initial cube?