Found problems: 85335
2001 India Regional Mathematical Olympiad, 5
In a triangle $ABC$, $D$ is a point on $BC$ such that $AD$ is the internal bisector of $\angle A$. Suppose $\angle B = 2 \angle C$ and $CD =AB$. prove that $\angle A = 72^{\circ}$.
1980 Brazil National Olympiad, 4
Given $5$ points of a sphere radius $r$, show that two of the points are a distance $\le r \sqrt2$ apart.
1999 Harvard-MIT Mathematics Tournament, 3
How many non-empty subsets of $\{1, 2, 3, 4, 5, 6,7,8\}$ have exactly $k$ elements and do not contain the element $k$ for some $k = 1, 2,...,8$.
2011 AMC 10, 3
At a store, when a length is reported as $x$ inches that means the length is at least $x-0.5$ inches and at most $x+0.5$ inches. Suppose the dimensions of a rectangular tile are reported as $2$ inches by $3$ inches. In square inches, what is the minimum area for the rectangle?
$ \textbf{(A)}\ 3.75 \qquad
\textbf{(B)}\ 4.5 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 8.75 $
2006 Argentina National Olympiad, 5
The captain distributed $4000$ gold coins among $40$ pirates. A group of $5$ pirates is called poor if those $5$ pirates received, together, $500$ coins or less. The captain made the distribution so that there were the minimum possible number of poor groups of $5$ pirates. Determine how many poor $5$ pirate groups there are.
Clarification: Two groups of $5$ pirates are considered different if there is at least one pirate in one of them who is not in the other.
2019 Jozsef Wildt International Math Competition, W. 57
Let be $x_1=\frac{1}{\sqrt[n+1]{n!}}$ and $x_2=\frac{1}{\sqrt[n+1]{(n-1)!}}$ for all $n\in \mathbb{N}^*$ and $f:\left(\left .\frac{1}{\sqrt[n+1]{(n+1)!}},1\right.\right] \to \mathbb{R}$ where $$f(x)=\frac{n+1}{x\ln (n+1)!+(n+1)\ln \left(x^x\right)}$$Prove that the sequence $(a_n)_{n\geq1}$ when $a_n=\int \limits_{x_1}^{x_2}f(x)dx$ is convergent and compute $$\lim \limits_{n \to \infty}a_n$$
PEN E Problems, 9
Four integers are marked on a circle. On each step we simultaneously replace each number by the difference between this number and next number on the circle in a given direction (that is, the numbers $a$, $b$, $c$, $d$ are replaced by $a-b$, $b-c$, $c-d$, $d-a$). Is it possible after $1996$ such steps to have numbers $a$, $b$, $c$ and $d$ such that the numbers $|bc-ad|$, $|ac-bd|$ and $|ab-cd|$ are primes?
2014 ASDAN Math Tournament, 6
Compute the largest integer $N$ such that one can select $N$ different positive integers, none of which is larger than $17$, and no two of which share a common divisor greater than $1$.
2023 Assara - South Russian Girl's MO, 7
A parabola is drawn on the coordinate plane - the graph of a square trinomial. The vertices of triangle $ABC$ lie on this parabola so that the bisector of angle $\angle BAC$ is parallel to the axis $Ox$ . Prove that the midpoint of the median drawn from vertex $A$ lies on the axis of the parabola.
1972 IMO Longlists, 1
Find all integer solutions of the equation
\[1 + x + x^2 + x^3 + x^4 = y^4.\]
2007 IMO Shortlist, 6
Let $ a_1, a_2, \ldots, a_{100}$ be nonnegative real numbers such that $ a^2_1 \plus{} a^2_2 \plus{} \ldots \plus{} a^2_{100} \equal{} 1.$ Prove that
\[ a^2_1 \cdot a_2 \plus{} a^2_2 \cdot a_3 \plus{} \ldots \plus{} a^2_{100} \cdot a_1 < \frac {12}{25}.
\]
[i]Author: Marcin Kuzma, Poland[/i]
2021 Belarusian National Olympiad, 11.6
A convex quadrilateral $ABCD$ is given. $\omega_1$ is a circle with diameter $BC$, $\omega_2$ is a circle with diameter $AD$. $AC$ meets $\omega_1$ and $\omega_2$ for the second time at $B_1$ and $D_1$. $BD$ meets $\omega_1$ and $\omega_2$ for the second time at $C_1$ and $A_1$. $AA_1$ meets $DD_1$ at $X$, $BB_1$ meets $CC_1$ at $Y$. $\omega_1$ intersects $\omega_2$ at $P$ and $Q$. $XY$ meets $PQ$ at $N$.
Prove that $XN=NY$.
2021 Oral Moscow Geometry Olympiad, 4
On the diagonal $AC$ of cyclic quadrilateral $ABCD$ a point $E$ is chosen such that $\angle ABE = \angle CBD$. Points $O,O_1,O_2$ are the circumcircles of triangles $ABC, ABE$ and $CBE$ respectively. Prove that lines $DO,AO_{1}$ and $CO_{2}$ are concurrent.
1994 National High School Mathematics League, 4
Give a point set on a plane $P=\{P_1,P_2,\cdots,P_{1994}\}$. Any three points in $P$ are not colinear. Divide points in $P$ into $83$ groups, in each group there are at least three points, and any point exactly belongs to one group. Connect any two points in the same group with a line segment, but we do not connect points not in the same group. Now we get a figure $G$. Note the number of triangles in $G$ : $m(G)$.
[b](a)[/b] Find the minumum value of $m(G)$ : $m_0$.
[b](b)[/b] $G*$ is one of the figures that $m(G)=m_0$, color the line segments in $G*$ with four colors. Prove that there exists a proper way, satisfying that no triangle is made up of three sides in the same color.
1988 Nordic, 1
The positive integer $ n$ has the following property:
if the three last digits of $n$ are removed, the number $\sqrt[3]{n}$ remains.
Find $n$.
1998 Korea Junior Math Olympiad, 1
Show that there exist no integer solutions $(x, y, z)$ to the equation
$$x^3+2y^3+4z^3=9$$
2014 Online Math Open Problems, 4
The integers $1, 2, \dots, n$ are written in order on a long slip of paper. The slip is then cut into five pieces, so that each piece consists of some (nonempty) consecutive set of integers. The averages of the numbers on the five slips are $1234$, $345$, $128$, $19$, and $9.5$ in some order. Compute $n$.
[i]Proposed by Evan Chen[/i]
1987 AMC 12/AHSME, 5
A student recorded the exact percentage frequency distribution for a set of measurements, as shown below. However, the student neglected to indicate $N$, the total number of measurements. What is the smallest possible value of $N$?
\[ \begin{tabular}{c c}
\text{measured value} & \text{percent frequency} \\
\hline
0 & 12.5 \\
1 & 0\\
2 & 50\\
3 & 25 \\
4 & 12.5 \\ \hline
\ & 100 \\
\end{tabular}
\]
$ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 25 \qquad\textbf{(E)}\ 50 $
2005 Purple Comet Problems, 23
Let $a = \sqrt[401]{4} - 1$ and for each $n \ge 2$, let $b_n = \binom{n}{1} + \binom{n}{2} a + \ldots + \binom{n}{n} a^{n-1}$. Find $b_{2006} - b_{2005}$.
2001 Irish Math Olympiad, 2
Three hoops are arranged concentrically as in the diagram. Each hoop is threaded with $ 20$ beads, $ 10$ of which are black and $ 10$ are white. On each hoop the positions of the beads are labelled $ 1$ through $ 20$ as shown. We say there is a match at position $ i$ if all three beads at position $ i$ have the same color. We are free to slide beads around a hoop, not breaking the hoop. Show that it is always possible to move them into a configuration involving no less than $ 5$ matches.
2019 Stanford Mathematics Tournament, 3
In triangle $ABC$ with $AB = 10$, let$ D$ be a point on side BC such that $AD$ bisects $\angle BAC$. If $\frac{CD}{BD} = 2$ and the area of $ABC$ is $50$, compute the value of $\angle BAD$ in degrees.
1978 Germany Team Selection Test, 4
Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$ is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.
2018 Sharygin Geometry Olympiad, 8
Restore a triangle $ABC$ by the Nagel point, the vertex $B$ and the foot of the altitude from this vertex.
2015 NIMO Summer Contest, 13
Let $\triangle ABC$ be a triangle with $AB=85$, $BC=125$, $CA=140$, and incircle $\omega$. Let $D$, $E$, $F$ be the points of tangency of $\omega$ with $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ respectively, and furthermore denote by $X$, $Y$, and $Z$ the incenters of $\triangle AEF$, $\triangle BFD$, and $\triangle CDE$, also respectively. Find the circumradius of $\triangle XYZ$.
[i] Proposed by David Altizio [/i]
2007 National Olympiad First Round, 9
Let $|AB|=3$ and the length of the altitude from $C$ be $2$ in $\triangle ABC$. What is the maximum value of the product of the lengths of the other two altitudes?
$
\textbf{(A)}\ \frac{144}{25}
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 3\sqrt 2
\qquad\textbf{(D)}\ 6
\qquad\textbf{(E)}\ \text{None of the above}
$