Found problems: 85335
2014 Regional Competition For Advanced Students, 2
You can determine all 4-ples $(a,b, c,d)$ of real numbers, which solve the following equation system $\begin{cases} ab + ac = 3b + 3c \\
bc + bd = 5c + 5d \\
ac + cd = 7a + 7d \\
ad + bd = 9a + 9b \end{cases} $
2022 Romania Team Selection Test, 3
Let $ABC$ be an acute triangle such that $AB < AC$. Let $\omega$ be the circumcircle of $ABC$
and assume that the tangent to $\omega$ at $A$ intersects the line $BC$ at $D$. Let $\Omega$ be the circle with
center $D$ and radius $AD$. Denote by $E$ the second intersection point of $\omega$ and $\Omega$. Let $M$ be the
midpoint of $BC$. If the line $BE$ meets $\Omega$ again at $X$, and the line $CX$ meets $\Omega$ for the second
time at $Y$, show that $A, Y$, and $M$ are collinear.
[i]Proposed by Nikola Velov, North Macedonia[/i]
TNO 2023 Senior, 6
The points inside a circle \( \Gamma \) are painted with \( n \geq 1 \) colors. A color is said to be dense in a circle \( \Omega \) if every circle contained within \( \Omega \) has points of that color in its interior. Prove that there exists at least one color that is dense in some circle contained within \( \Gamma \).
Indonesia Regional MO OSP SMA - geometry, 2004.5
The lattice point on the plane is a point that has coordinates in the form of a pair of integers.
Let $P_1, P_2, P_3, P_4, P_5$ be five different lattice points on the plane.
Prove that there is a pair of points $(P_i, P_j), i \ne j$, so that the line segment $P_iP_j$ contains a lattice point other than $P_i$ and $P_j$.
Kyiv City MO 1984-93 - geometry, 1991.8.5
The diagonals of the convex quadrilateral $ABCD$ are mutually perpendicular. Through the midpoint of the sides $AB$ and $AD$ draw lines, which are perpendicular to the opposite sides. Prove that they intersect on line $AC$.
2000 Tournament Of Towns, 4
Among a set of $2N$ coins, all identical in appearance, $2N - 2$ are real and $2$ are fake. Any two real coins have the same weight . The fake coins have the same weight, which is different from the weight of a real coin. How can one divide the coins into two groups of equal total weight by using a balance at most $4$ times, if
(a) $N = 16$,
( b ) $N = 11$ ?
(A Shapovalov)
1985 Tournament Of Towns, (090) T1
In quadrilateral ABCD it is given that $AB = BC = 1, \angle ABC = 100^o$ , and $\angle CDA = 130^o$ . Find the length of $BD$.
2004 Germany Team Selection Test, 2
Find all pairs of positive integers $\left(n;\;k\right)$ such that $n!=\left( n+1\right)^{k}-1$.
2018 AMC 8, 10
The [i]harmonic mean[/i] of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?
$\textbf{(A) }\frac{3}{7}\qquad\textbf{(B) }\frac{7}{12}\qquad\textbf{(C) }\frac{12}{7}\qquad\textbf{(D) }\frac{7}{4}\qquad \textbf{(E) }\frac{7}{3}$
2012 Online Math Open Problems, 18
There are 32 people at a conference. Initially nobody at the conference knows the name of anyone else. The conference holds several 16-person meetings in succession, in which each person at the meeting learns (or relearns) the name of the other fifteen people. What is the minimum number of meetings needed until every person knows everyone else's name?
[i]David Yang, Victor Wang.[/i]
[size=85][i]See the "odd version" [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=810&t=500914]here[/url].[/i][/size]
2001 National Olympiad First Round, 25
The circumradius of acute triangle $ABC$ is twice of the distance of its circumcenter to $AB$. If $|AC|=2$ and $|BC|=3$, what is the altitude passing through $C$?
$
\textbf{(A)}\ \sqrt {14}
\qquad\textbf{(B)}\ \dfrac{3}{7}\sqrt{21}
\qquad\textbf{(C)}\ \dfrac{4}{7}\sqrt{21}
\qquad\textbf{(D)}\ \dfrac{1}{2}\sqrt{21}
\qquad\textbf{(E)}\ \dfrac{2}{3}\sqrt{14}
$
2014-2015 SDML (High School), 14
What is the greatest integer $n$ such that $$n\leq1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{2014}}?$$
$\text{(A) }31\qquad\text{(B) }59\qquad\text{(C) }74\qquad\text{(D) }88\qquad\text{(E) }112$
2004 Romania Team Selection Test, 12
Let $n\geq 2$ be an integer and let $a_1,a_2,\ldots,a_n$ be real numbers. Prove that for any non-empty subset $S\subset \{1,2,3,\ldots, n\}$ we have
\[ \left( \sum_{i \in S} a_i \right)^2 \leq \sum_{1\leq i \leq j \leq n } (a_i + \cdots + a_j ) ^2 . \]
[i]Gabriel Dospinescu[/i]
1974 Miklós Schweitzer, 9
Let $ A$ be a closed and bounded set in the plane, and let $ C$ denote the set of points at a unit distance from $ A$. Let $ p \in
C$, and assume that the intersection of $ A$ with the unit circle $ K$ centered at $ p$ can be covered by an arc shorter that a semicircle of $ K$. Prove that the intersection of $ C$ with a suitable neighborhood of $ p$ is a simple arc which $ p$ is not an endpoint.
[i]M. Bognar[/i]
2011 Putnam, A5
Let $F:\mathbb{R}^2\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be twice continuously differentiable functions with the following properties:
• $F(u,u)=0$ for every $u\in\mathbb{R};$
• for every $x\in\mathbb{R},g(x)>0$ and $x^2g(x)\le 1;$
• for every $(u,v)\in\mathbb{R}^2,$ the vector $\nabla F(u,v)$ is either $\mathbf{0}$ or parallel to the vector $\langle g(u),-g(v)\rangle.$
Prove that there exists a constant $C$ such that for every $n\ge 2$ and any $x_1,\dots,x_{n+1}\in\mathbb{R},$ we have
\[\min_{i\ne j}|F(x_i,x_j)|\le\frac{C}{n}.\]
2011 IFYM, Sozopol, 8
Let $S$ be the set of all 9-digit natural numbers, which are written only with the digits 1, 2, and 3. Find all functions $f:S\rightarrow \{1,2,3\}$ which satisfy the following conditions:
(1) $f(111111111)=1$, $f(222222222)=2$, $f(333333333)=3$, $f(122222222)=1$;
(2) If $x,y\in S$ differ in each digit position, then $f(x)\neq f(y)$.
2015 JBMO TST - Turkey, 3
In a country consisting of $2015$ cities, between any two cities there is exactly one direct round flight operated by some air company. Find the minimal possible number of air companies if direct flights between any three cities are operated by three different air companies.
2019 PUMaC Team Round, 8
The curves $y = x + 5$ and $y = x^2 - 3x$ intersect at points $A$ and $B$. $C$ is a point on the lower curve between $A$ and $B$. The maximum possible area of the quadrilateral $ABCO$ can be written as $A/B$ for coprime $A, B$. Find $A + B$.
2020 Bulgaria Team Selection Test, 6
In triangle $\triangle ABC$, $BC>AC$, $I_B$ is the $B$-excenter, the line through $C$ parallel to $AB$ meets $BI_B$ at $F$. $M$ is the midpoint of $AI_B$ and the $A$-excircle touches side $AB$ at $D$. Point $E$ satisfies $\angle BAC=\angle BDE, DE=BC$, and lies on the same side as $C$ of $AB$. Let $EC$ intersect $AB,FM$ at $P,Q$ respectively. Prove that $P,A,M,Q$ are concyclic.
2018 Israel Olympic Revenge, 1
Let $n$ be a positive integer.
Prove that every prime $p > 2$ that divides $(2-\sqrt{3})^n + (2+\sqrt{3})^n$ satisfy $p=1 (mod3)$
2012-2013 SDML (Middle School), 10
Palmer correctly computes the product of the first $1,001$ prime numbers. Which of the following is NOT a factor of Palmer's product?
$\text{(A) }2,002\qquad\text{(B) }3,003\qquad\text{(C) }5,005\qquad\text{(D) }6,006\qquad\text{(E) }7,007$
2000 CentroAmerican, 3
Let $ ABCDE$ be a convex pentagon. If $ P$, $ Q$, $ R$ and $ S$ are the respective centroids of the triangles $ ABE$, $ BCE$, $ CDE$ and $ DAE$, show that $ PQRS$ is a parallelogram and its area is $ 2/9$ of that of $ ABCD$.
1970 Regional Competition For Advanced Students, 2
In the plane seven different points $P_1, P_2, P_3, P_4, Q_1, Q_2, Q_3$ are given. The points $P_1, P_2, P_3, P_4$ are on the straight line $p$, the points $Q_1, Q_2, Q_3$ are not on $p$. By each of the three points $Q_1, Q_2, Q_3$ the perpendiculars are drawn on the straight lines connecting points different of them. Prove that the maximum's number of the possibles intersections of all perpendiculars is to 286, if the points $Q_1, Q_2, Q_3$ are taken in account as intersections.
2005 National Olympiad First Round, 24
There are $20$ people in a certain community. $10$ of them speak English, $10$ of them speak German, and $10$ of them speak French. We call a [i]committee[/i] to a $3$-subset of this community if there is at least one who speaks English, at least one who speaks German, and at least one who speaks French in this subset. At most how many commitees are there in this community?
$
\textbf{(A)}\ 120
\qquad\textbf{(B)}\ 380
\qquad\textbf{(C)}\ 570
\qquad\textbf{(D)}\ 1020
\qquad\textbf{(E)}\ 1140
$
1999 AIME Problems, 8
Let $\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.$ Let us say that $(x,y,z)$ supports $(a,b,c)$ when exactly two of the following are true: $x\ge a, y\ge b, z\ge c.$ Let $\mathcal{S}$ consist of those triples in $\mathcal{T}$ that support $\left(\frac 12,\frac 13,\frac 16\right).$ The area of $\mathcal{S}$ divided by the area of $\mathcal{T}$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$