Found problems: 235
1993 AMC 12/AHSME, 29
Which of the following sets could NOT be the lengths of the external diagonals of a right rectangular prism [a "box"]? (An [i]external diagonal[/i] is a diagonal of one of the rectangular faces of the box.)
$ \textbf{(A)}\ \{4, 5, 6\} \qquad\textbf{(B)}\ \{4, 5, 7\} \qquad\textbf{(C)}\ \{4, 6, 7\} \qquad\textbf{(D)}\ \{5, 6, 7\} \qquad\textbf{(E)}\ \{5, 7, 8\} $
1978 Vietnam National Olympiad, 6
Given a rectangular parallelepiped $ABCDA'B'C'D'$ with the bases $ABCD, A'B'C'D'$, the edges $AA',BB', CC',DD'$ and $AB = a,AD = b,AA' = c$. Show that there exists a triangle with the sides equal to the distances from $A,A',D$ to the diagonal $BD'$ of the parallelepiped. Denote those distances by $m_1,m_2,m_3$. Find the relationship between $a, b, c,m_1,m_2,m_3$.
2022 Novosibirsk Oral Olympiad in Geometry, 2
Faith has four different integer length segments. It turned out that any three of them can form a triangle. What is the smallest total length of this set of segments?
2010 Switzerland - Final Round, 4
Let $ x$, $ y$, $ z \in\mathbb{R}^+$ satisfying $ xyz = 1$. Prove that
\[ \frac {(x + y - 1)^2}{z} + \frac {(y + z - 1)^2}{x} + \frac {(z + x - 1)^2}{y}\geqslant x + y + z\mbox{.}\]
1967 IMO Longlists, 11
Let $n$ be a positive integer. Find the maximal number of non-congruent triangles whose sides lengths are integers $\leq n.$
1980 IMO Shortlist, 15
Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.
2012 AMC 12/AHSME, 23
Consider all polynomials of a complex variable, $P(z)=4z^4+az^3+bz^2+cz+d$, where $a, b, c$ and $d$ are integers, $0 \le d \le c \le b \le a \le 4$, and the polynomial has a zero $z_0$ with $|z_0|=1$. What is the sum of all values $P(1)$ over all the polynomials with these properties?
$ \textbf{(A)}\ 84\qquad\textbf{(B)}\ 92\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 108 \qquad\textbf{(E)}\ 120 $
2006 International Zhautykov Olympiad, 3
Let $ ABCDEF$ be a convex hexagon such that $ AD \equal{} BC \plus{} EF$, $ BE \equal{} AF \plus{} CD$, $ CF \equal{} DE \plus{} AB$. Prove that:
\[ \frac {AB}{DE} \equal{} \frac {CD}{AF} \equal{} \frac {EF}{BC}.
\]
2021 239 Open Mathematical Olympiad, 5
The median $AD$ is drawn in triangle $ABC$. Point $E$ is selected on segment $AC$, and on the ray $DE$ there is a point $F$, and $\angle ABC = \angle AED$ and $AF // BC$. Prove that from segments $BD, DF$ and $AF$, you can make a triangle, the area of which is not less half the area of triangle $ABC$.
2008 District Olympiad, 1
Let $ \{a_n\}_{n\geq 1}$ be a sequence of real numbers such that $ |a_{n\plus{}1}\minus{}a_n|\leq 1$, for all positive integers $ n$. Let $ \{b_n\}_{n\geq 1}$ be the sequence defined by \[ b_n \equal{} \frac { a_1\plus{} a_2 \plus{} \cdots \plus{}a_n} {n}.\] Prove that $ |b_{n\plus{}1}\minus{}b_n | \leq \frac 12$, for all positive integers $ n$.
2009 IMO Shortlist, 3
Determine all functions $ f$ from the set of positive integers to the set of positive integers such that, for all positive integers $ a$ and $ b$, there exists a non-degenerate triangle with sides of lengths
\[ a, f(b) \text{ and } f(b \plus{} f(a) \minus{} 1).\]
(A triangle is non-degenerate if its vertices are not collinear.)
[i]Proposed by Bruno Le Floch, France[/i]
2010 Tournament Of Towns, 5
For each side of a given pentagon, divide its length by the total length of all other sides. Prove that the sum of all the fractions obtained is less than 2.
2014 Mediterranean Mathematics Olympiad, 1
Let $a_1,\ldots,a_n$ and $b_1\ldots,b_n$ be $2n$ real numbers. Prove that there exists an integer $k$ with $1\le k\le n$ such that
$ \sum_{i=1}^n|a_i-a_k| ~~\le~~ \sum_{i=1}^n|b_i-a_k|.$
(Proposed by Gerhard Woeginger, Austria)
1991 Spain Mathematical Olympiad, 3
What condition must be satisfied by the coefficients $u,v,w$ if the roots of the polynomial $x^3 -ux^2+vx-w$ are the sides of a triangle
1986 AMC 12/AHSME, 29
Two of the altitudes of the scalene triangle $ABC$ have length $4$ and $12$. If the length of the third altitude is also an integer, what is the biggest it can be?
$ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ \text{none of these} $
2015 Purple Comet Problems, 7
How many non-congruent isosceles triangles (including equilateral triangles) have positive integer side
lengths and perimeter less than 20?
1969 IMO Shortlist, 55
For each of $k=1,2,3,4,5$ find necessary and sufficient conditions on $a>0$ such that there exists a tetrahedron with $k$ edges length $a$ and the remainder length $1$.
1990 IMO Longlists, 6
Let function $f : \mathbb Z_{\geq 0}^0 \to \mathbb N$ satisfy the following conditions:
(i) $ f(0, 0, 0) = 1;$
(ii) $f(x, y, z) = f(x - 1, y, z) + f(x, y - 1, z) + f(x, y, z - 1);$
(iii) when applying above relation iteratively, if any of $x', y', z$' is negative, then $f(x', y', z') = 0.$
Prove that if $x, y, z$ are the side lengths of a triangle, then $\frac{\left(f(x,y,z) \right) ^k}{ f(mx ,my, mz)}$ is not an integer for any integers $k, m > 1.$
2020 Novosibirsk Oral Olympiad in Geometry, 2
It is known that four of these sticks can be assembled into a quadrilateral. Is it always true that you can make a triangle out of three of them?
May Olympiad L2 - geometry, 2012.4
Six points are given so that there are not three on the same line and that the lengths of the segments determined by these points are all different. We consider all the triangles that they have their vertices at these points. Show that there is a segment that is both the shortest side of one of those triangles and the longest side of another.
2012 China Team Selection Test, 1
Complex numbers ${x_i},{y_i}$ satisfy $\left| {{x_i}} \right| = \left| {{y_i}} \right| = 1$ for $i=1,2,\ldots ,n$. Let $x=\frac{1}{n}\sum\limits_{i=1}^n{{x_i}}$, $y=\frac{1}{n}\sum\limits_{i=1}^n{{y_i}}$ and $z_i=x{y_i}+y{x_i}-{x_i}{y_i}$. Prove that $\sum\limits_{i=1}^n{\left| {{z_i}}\right|}\leqslant n$.
2023 Peru MO (ONEM), 2
For each positive real number $x$, let $f(x)=\frac{x}{1+x}$ . Prove that if $a$, $b,$ $c$ are the sidelengths of a triangle, then $f(a)$, $f(b),$ $f(c)$ are sidelengths of a triangle.
1999 Brazil Team Selection Test, Problem 4
Let Q+ and Z denote the set of positive rationals and the set of inte-
gers, respectively. Find all functions f : Q+ → Z satisfying the following
conditions:
(i) f(1999) = 1;
(ii) f(ab) = f(a) + f(b) for all a, b ∈ Q+;
(iii) f(a + b) ≥ min{f(a), f(b)} for all a, b ∈ Q+.
2014 Saudi Arabia BMO TST, 4
Let $n$ be an integer greater than $2$. Consider a set of $n$ different points, with no three collinear, in the plane. Prove that we can label the points $P_1,~ P_2, \dots , P_n$ such that $P_1P_2 \dots P_n$ is not a self-intersecting polygon. ([i]A polygon is self-intersecting if one of its side intersects the interior of another side. The polygon is not necessarily convex[/i] )
2025 Macedonian TST, Problem 5
Let $\triangle ABC$ be a triangle with side‐lengths $a,b,c$, incenter $I$, and circumradius $R$. Denote by $P$ the area of $\triangle ABC$, and let $P_1,\;P_2,\;P_3$ be the areas of triangles $\triangle ABI$, $\triangle BCI$, and $\triangle CAI$, respectively. Prove that
\[
\frac{abc}{12R}
\;\le\;
\frac{P_1^2 + P_2^2 + P_3^2}{P}
\;\le\;
\frac{3R^3}{4\sqrt[3]{abc}}.
\]