Found problems: 85335
2021-IMOC, G10
Let $O$, $I$ be the circumcenter and the incenter of triangle $ABC$, respectively, and let the incircle tangents $BC$ at $D$. Furthermore, suppose that $H$ is the orthocenter of triangle $BIC$, $N$ is the midpoint of the arc $BAC$, and $X$ is the intersection of $OI$ and $NH$. If $P$ is the reflection of $A$ with respect to $OI$, show that $\odot(IDP)$ and $\odot(IHX)$ are tangent to each other.
1992 Tournament Of Towns, (343) 1
Numbers in an $n$ by $n$ table may be changed by adding $1$ to each number on an arbitrary closed non-selfintersecting “rook path” (a broken line with segments parallel to the borders of the table). Originally $1$’s stand on one of the diagonals, and $0S’s in the other cells of the table. Can one get (after several transformations) a table in which all numbers are equal to each other? (A “rook path” contains all cells through which it passes.)
(AA Egorov)
2017 Saudi Arabia JBMO TST, 1
Given a polynomial $f(x) = x^4+ax^3+bx^2+cx$.
It is known that each of the equations $f(x) = 1$ and $f(x) = 2$ has four real roots (not necessarily distinct).
Prove that if the roots of the first equation satisfy the equality $x_1 + x_2 = x_3 + x_4$, then the same equation holds for the roots of the second equation
2007 Estonia National Olympiad, 4
The figure shows a figure of $5$ unit squares, a Greek cross. What is the largest number of Greek crosses that can be placed on a grid of dimensions $8 \times 8$ without any overlaps, with each unit square covering just one square in a grid?
1982 AMC 12/AHSME, 3
Evaluate $(x^x)^{(x^x)}$ at $x = 2$.
$\textbf{(A)} \ 16 \qquad \textbf{(B)} \ 64 \qquad \textbf{(C)} \ 256 \qquad \textbf{(D)} \ 1024 \qquad \textbf{(E)} \ 65,536$
2017 NZMOC Camp Selection Problems, 7
Let $a, b, c, d, e$ be distinct positive integers such that $$a^4 + b^4 = c^4 + d^4 = e^5.$$
Show that $ac + bd$ is composite.
Croatia MO (HMO) - geometry, 2018.7
Given an acute-angled triangle $ABC$ in which $|AB| <|AC|$. Point $D$ is the midpoint of the shorter arc $BC$ of its circumcircle. The point $I$ is the center of its incircle, and the point $J$ is symmetric point of $I$ wrt line $BC$. The line $DJ$ intersects the circumcircle of the triangle $ABC$ at the point $E$ belonging to the arc $AB$. Prove that $|AI |= |IE|$.
1992 Spain Mathematical Olympiad, 1
Determine the smallest number N, multiple of 83, such that N^2 has 63 positive divisors.
2015 Belarus Team Selection Test, 3
Let the incircle of the triangle $ABC$ touch the side $AB$ at point $Q$. The incircles of the triangles $QAC$ and $QBC$ touch $AQ,AC$ and $BQ,BC$ at points $P,T$ and $D,F$ respectively. Prove that $PDFT$ is a cyclic quadrilateral.
I.Gorodnin
1958 Miklós Schweitzer, 6
[b]6.[/b] Prove that if $a_n \geq 0$ and
$\frac{1}{n}\sum_{k=1}^{n} a_k \geq \sum_{k=n+1}^{2n}a_k$ $(n=1, 2, \dots)$ ,
then $\sum_{k=1}^{\infty} a_k $ is convergent and its sum is less than $2ea_1$. [b](S. 9)[/b]
2024 HMNT, 10
Let $S = \{1, 2, 3, . . . , 64\}.$ Compute the number of ways to partition $S$ into $16$ arithmetic sequences such that each arithmetic sequence has length $4$ and common difference $1, 4,$ or $16.$
1987 Traian Lălescu, 2.3
Prove that $ C_G\left( N_G(H) \right)\subset N_G\left( C_G(H) \right) , $ for any subgroup $ H $ of $ G, $ and characterize the groups $ G $ for which equality in this relation holds for all $ H\le G. $
[i]Here,[/i] $ C_G,N_G $ [i]are the centralizer, respectively, the normalizer of[/i] $ G. $
1950 AMC 12/AHSME, 34
When the circumference of a toy balloon is increased from $20$ inches to $25$ inches, the radius is increased by:
$\textbf{(A)}\ 5\text{ in} \qquad
\textbf{(B)}\ 2\dfrac{1}{2}\text{ in} \qquad
\textbf{(C)}\ \dfrac{5}{\pi}\text{ in} \qquad
\textbf{(D)}\ \dfrac{5}{2\pi}\text{ in} \qquad
\textbf{(E)}\ \dfrac{\pi}{5}\text{ in}$
2004 CHKMO, 1
Find the greatest real number $K$ such that for all positive real number $u,v,w$ with $u^{2}>4vw$ we have $(u^{2}-4vw)^{2}>K(2v^{2}-uw)(2w^{2}-uv)$
2012 USAMTS Problems, 5
A unit square $ABCD$ is given in the plane, with $O$ being the intersection of its diagonals. A ray $l$ is drawn from $O$. Let $X$ be the unique point on $l$ such that $AX + CX = 2$, and let $Y$ be the point on $l$ such that $BY + DY = 2$. Let $Z$ be the midpoint of $\overline{XY}$, with $Z = X$ if $X$ and $Y$ coincide. Find, with proof, the minimum value of the length of $OZ$.
2019 Sharygin Geometry Olympiad, 15
The incircle $\omega$ of triangle $ABC$ touches the sides $BC$, $CA$ and $AB$ at points $D$, $E$ and $F$ respectively. The perpendicular from $E$ to $DF$ meets $BC$ at point $X$, and the perpendicular from $F$ to $DE$ meets $BC$ at point $Y$. The segment $AD$ meets $\omega$ for the second time at point $Z$. Prove that the circumcircle of the triangle $XYZ$ touches $\omega$.
2009 Jozsef Wildt International Math Competition, W. 15
Let a triangle $\triangle ABC$ and the real numbers $x$, $y$, $z>0$. Prove that $$x^n\cos\frac{A}{2}+y^n\cos\frac{B}{2}+z^n\cos\frac{C}{2}\geq (yz)^{\frac{n}{2}}\sin A +(zx)^{\frac{n}{2}}\sin B +(xy)^{\frac{n}{2}}\sin C$$
1985 IMO Longlists, 57
[i]a)[/i] The solid $S$ is defined as the intersection of the six spheres with the six edges of a regular tetrahedron $T$, with edge length $1$, as diameters. Prove that $S$ contains two points at a distance $\frac{1}{\sqrt 6}.$
[i]b)[/i] Using the same assumptions in [i]a)[/i], prove that no pair of points in $S$ has a distance larger than $\frac{1}{\sqrt 6}.$
2017 ASDAN Math Tournament, 2
Two distinct positive factors of $144$ are selected at random. What is the probability that their product is greater than $144$?
2014 India IMO Training Camp, 1
Let $p$ be an odd prime and $r$ an odd natural number.Show that $pr+1$ does not divide $p^p-1$
2013 India Regional Mathematical Olympiad, 5
Let $n \ge 3$ be a natural number and let $P$ be a polygon with $n$ sides. Let $a_1,a_2,\cdots, a_n$ be the lengths of sides of $P$ and let $p$ be its perimeter. Prove that \[\frac{a_1}{p-a_1}+\frac{a_2}{p-a_2}+\cdots + \frac{a_n}{p-a_n} < 2 \]
2013 BMT Spring, 7
Given real numbers $a, b, c$ such that $a + b - c = ab- bc - ca = abc = 8$. Find all possible values of $a$.
2021 Malaysia IMONST 2, 2
The five numbers $a, b, c, d,$ and $e$ satisfy the inequalities
$$a+b < c+d < e+a < b+c < d+e.$$
Among the five numbers, which one is the smallest, and which one is the largest?
1991 AMC 12/AHSME, 11
Jack and Jill run $10$ kilometers. They start at the same point, run $5$ kilometers up a hill, and return to the starting point by the same route. Jack has a $10$ minute head start and runs at the rate of $15$ km/hr uphill and $20$ km/hr downhill. Jill runs $16$ km/hr uphill and $22$ km/hr downhill. How far from the top of the hill are they when they pass going in opposite directions?
$ \textbf{(A)}\ \frac{5}{4}\ km\qquad\textbf{(B)}\ \frac{35}{27}\ km\qquad\textbf{(C)}\ \frac{27}{20}\ km\qquad\textbf{(D)}\ \frac{7}{3}\ km\qquad\textbf{(E)}\ \frac{28}{9}\ km $
2020 Balkan MO Shortlist, C2
Let $k$ be a positive integer. Determine the least positive integer $n$, with $n\geq k+1$, for which the game below can be played indefinitely:
Consider $n$ boxes, labelled $b_1,b_2,...,b_n$. For each index $i$, box $b_i$ contains exactly $i$ coins. At each step, the following three substeps are performed in order:
[b](1)[/b] Choose $k+1$ boxes;
[b](2)[/b] Of these $k+1$ boxes, choose $k$ and remove at least half of the coins from each, and add to the remaining box, if labelled $b_i$, a number of $i$ coins.
[b](3)[/b] If one of the boxes is left empty, the game ends; otherwise, go to the next step.
[i]Proposed by Demetres Christofides, Cyprus[/i]