Found problems: 85335
2022 Irish Math Olympiad, 1
1. For [i]n[/i] a positive integer, [i]n[/i]! = 1 $\cdot$ 2 $\cdot$ 3 $\dots$ ([i]n[/i] - 1) $\cdot$ [i]n[/i] is the product of the positive integers from 1 to [i]n[/i].
Determine, with proof, all positive integers [i]n[/i] for which [i]n[/i]! + 3 is a power of 3.
2021 European Mathematical Cup, 4
Find all positive integers $d$ for which there exist polynomials $P(x)$ and $Q(x)$ with real coefficients such that degree of $P$ equals $d$ and
$$P(x)^2+1=(x^2+1)Q(x)^2.$$
2012 AMC 12/AHSME, 18
Triangle $ABC$ has $AB=27$, $AC=26$, and $BC=25$. Let $I$ denote the intersection of the internal angle bisectors of $\triangle ABC$. What is $BI$?
$ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 5+\sqrt{26}+3\sqrt{3}\qquad\textbf{(C)}\ 3\sqrt{26}\qquad\textbf{(D)}\ \frac{2}{3}\sqrt{546}\qquad\textbf{(E)}\ 9\sqrt{3} $
VI Soros Olympiad 1999 - 2000 (Russia), 8.3
$72$ was added to the natural number $n$ and in the sum we got a number written in the same digits as the number $n$, but in the reverse order. Find all numbers $n$ that satisfy the given condition.
1987 AMC 12/AHSME, 2
A triangular corner with side lengths $DB=EB=1$ is cut from equilateral triangle $ABC$ of side length $3$. The perimeter of the remaining quadrilateral is
[asy]
draw((0,0)--(2,0)--(2.5,.87)--(1.5,2.6)--cycle, linewidth(1));
draw((2,0)--(3,0)--(2.5,.87));
label("3", (0.75,1.3), NW);
label("1", (2.5, 0), S);
label("1", (2.75,.44), NE);
label("A", (1.5,2.6), N);
label("B", (3,0), S);
label("C", (0,0), W);
label("D", (2.5,.87), NE);
label("E", (2,0), S);[/asy]
$\text{(A)} \ 6 \qquad \text{(B)} \ 6\frac12 \qquad \text{(C)} \ 7 \qquad \text{(D)} \ 7\frac12 \qquad \text{(E)} \ 8$
1976 Czech and Slovak Olympiad III A, 5
Let $\mathbf{P}_1,\mathbf{P}_2$ be convex polygons with perimeters $o_1,o_2,$ respectively. Show that if $\mathbf P_1\subseteq\mathbf P_2,$ then $o_1\le o_2.$
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 $1994$?
2011 Indonesia MO, 6
Let a sequence of integers $a_0, a_1, a_2, \cdots, a_{2010}$ such that $a_0 = 1$ and $2011$ divides $a_{k-1}a_k - k$ for all $k = 1, 2, \cdots, 2010$. Prove that $2011$ divides $a_{2010} + 1$.
2017 ELMO Shortlist, 2
Let $ABC$ be a scalene triangle with $\angle A = 60^{\circ}$. Let $E$ and $F$ be the feet of the angle bisectors of $\angle ABC$ and $\angle ACB$, respectively, and let $I$ be the incenter of $\triangle ABC$. Let $P,Q$ be distinct points such that $\triangle PEF$ and $\triangle QEF$ are equilateral. If $O$ is the circumcenter of of $\triangle APQ$, show that $\overline{OI}\perp \overline{BC}$.
[i]Proposed by Vincent Huang
2004 AMC 10, 10
A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains $ 100$ cans, how many rows does it contain?
$ \textbf{(A)}\ 5\qquad
\textbf{(B)}\ 8\qquad
\textbf{(C)}\ 9\qquad
\textbf{(D)}\ 10\qquad
\textbf{(E)}\ 11$
1987 Tournament Of Towns, (141) 1
Is it possible to represent the number $1986$ as the sum of squares of $6$ odd integers?
2021 Ukraine National Mathematical Olympiad, 2
Find all natural numbers $n \ge 3$ for which in an arbitrary $n$-gon one can choose $3$ vertices dividing its boundary into three parts, the lengths of which can be the lengths of the sides of some triangle.
(Fedir Yudin)
2009 QEDMO 6th, 1
Solve $y^5 - x^2 = 4$ in integers numbers $x,y$.
May Olympiad L1 - geometry, 2021.1
In a forest there are $5$ trees $A, B, C, D, E$ that are in that order on a straight line. At the midpoint of $AB$ there is a daisy, at the midpoint of $BC$ there is a rose bush, at the midpoint of $CD$ there is a jasmine, and at the midpoint of $DE$ there is a carnation. The distance between $A$ and $E$ is $28$ m; the distance between the daisy and the carnation is $20$ m. Calculate the distance between the rose bush and the jasmine.
2019 Brazil EGMO TST, 3
Let $ABC$ be a triangle and $E$ and $F$ two arbitrary points on sides $AB$ and $AC$, respectively. The circumcircle of triangle $AEF$ meets the circumcircle of triangle $ABC$ again at point $M$. The point $D$ is such that $EF$ bisects the segment $MD$ . Finally, $O$ is the circumcenter of triangle $ABC$. Prove that $D$ lies on line $BC$ if and only if $O$ lies on the circumcircle of triangle $AEF$.
1991 Arnold's Trivium, 86
Through the centre of a cube (tetrahedron, icosahedron) draw a straight line in such a way that the sum of the squares of its distances from the vertices is a) minimal, b) maximal.
2013 HMNT, 5
In triangle $ABC$, $\angle BAC=60^o$/ Let $\omega$ be a circle tangent to segment $AB$ at point $D$ and segment $AC$ at point $E$. Suppose $\omega$ intersects segment $BC$ at points $F$ and $G$ such that$ F$ lies in between $B$ and $G$. Given that $AD = FG = 4$ and $BF = \frac12$ , find the length of $CG$.
1980 IMO, 4
Prove that $\sum \frac{1}{i_1i_2 \ldots i_k} = n$ is taken over all non-empty subsets $\left\{i_1,i_2, \ldots, i_k\right\}$ of $\left\{1,2,\ldots,n\right\}$. (The $k$ is not fixed, so we are summing over all the $2^n-1$ possible nonempty subsets.)
2004 Purple Comet Problems, 5
Write the number $2004_{(5)}$ [ $2004$ base $5$ ] as a number in base $6$.
2012 Sharygin Geometry Olympiad, 15
Given triangle $ABC$. Consider lines $l$ with the next property: the reflections of $l$ in the sidelines of the triangle concur. Prove that all these lines have a common point.
2007 Croatia Team Selection Test, 6
$\displaystyle 2n$ students $\displaystyle (n \geq 5)$ participated at table tennis contest, which took $\displaystyle 4$ days. In every day, every student played a match. (It is possible that the same pair meets twice or more times, in different days) Prove that it is possible that the contest ends like this:
- there is only one winner;
- there are $\displaystyle 3$ students on the second place;
- no student lost all $\displaystyle 4$ matches.
How many students won only a single match and how many won exactly $\displaystyle 2$ matches? (In the above conditions)
2011 VJIMC, Problem 4
Let $a,b,c$ be elements of finite order in some group. Prove that if $a^{-1}ba=b^2$, $b^{-2}cb^2=c^2$, and $c^{-3}ac^3=a^2$ then $a=b=c=e$, where $e$ is the unit element.
2009 Moldova Team Selection Test, 2
[color=darkblue]Let $ M$ be a set of aritmetic progressions with integer terms and ratio bigger than $ 1$.
[b]a)[/b] Prove that the set of the integers $ \mathbb{Z}$ can be written as union of the finite number of the progessions from $ M$ with different ratios.
[b]b)[/b] Prove that the set of the integers $ \mathbb{Z}$ can not be written as union of the finite number of the progessions from $ M$ with ratios integer numbers, any two of them coprime.[/color]
2010 Peru Iberoamerican Team Selection Test, P2
For each positive integer $k$, let $S(k)$ be the sum of the digits of $k$ in the decimal system.
Find all positive integers N for which there exist positive integers $a$,$b$,$c$, coprime two by two, such that:
$S(ab) = S(bc) = S(ca) = N$.
Indonesia Regional MO OSP SMA - geometry, 2005.1
The length of the largest side of the cyclic quadrilateral $ABCD$ is $a$, while the radius of the circumcircle of $\vartriangle ACD$ is $1$. Find the smallest possible value for $a$. Which cyclic quadrilateral $ABCD$ gives the value $a$ equal to the smallest value?