Found problems: 85335
2003 District Olympiad, 4
a) Let $MNP$ be a triangle such that $\angle MNP> 60^o$. Show that the side $MP$ cannot be the smallest side of the triangle $MNP$.
b) In a plane the equilateral triangle $ABC$ is considered. The point $V$ that does not belong to the plane $(ABC)$ is chosen so that $\angle VAB = \angle VBC = \angle VCA$. Show that if $VA = AB$, the tetrahedron $VABC$ is regular.
Valentin Vornicu
2023 Yasinsky Geometry Olympiad, 5
Let $ABC$ be a triangle and $\ell$ be a line parallel to $BC$ that passes through vertex $A$. Draw two circles congruent to the circle inscribed in triangle $ABC$ and tangent to line $\ell$, $AB$ and $BC$ (see picture). Lines $DE$ and $FG$ intersect at point $P$. Prove that $P$ lies on $BC$ if and only if $P$ is the midpoint of $BC$.
(Mykhailo Plotnikov)
[img]https://cdn.artofproblemsolving.com/attachments/8/b/2dacf9a6d94a490511a2dc06fbd36f79f25eec.png[/img]
2003 Italy TST, 1
The incircle of a triangle $ABC$ touches the sides $AB,BC,CA$ at points $D,E,F$ respectively. The line through $A$ parallel to $DF$ meets the line through $C$ parallel to $EF$ at $G$.
$(a)$ Prove that the quadrilateral $AICG$ is cyclic.
$(b)$ Prove that the points $B,I,G$ are collinear.
1978 IMO Shortlist, 6
Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$
2015 JBMO Shortlist, A1
Let x; y; z be real numbers, satisfying the relations
$x \ge 20$
$y \ge 40$
$z \ge 1675$
x + y + z = 2015
Find the greatest value of the product P = $xy z$
1999 China Second Round Olympiad, 3
$n$ is a given positive integer, such that it’s possible to weigh out the mass of any product weighing $1,2,3,\cdots ,ng$ with a counter balance without sliding poise and $k$ counterweights, which weigh $x_ig(i=1,2,\cdots ,k),$ respectively, where $x_i\in \mathbb{N}^*$ for any $i \in \{ 1,2,\cdots ,k\}$ and $x_1\leq x_2\leq\cdots \leq x_k.$
$(1)$Let $f(n)$ be the least possible number of $k$. Find $f(n)$ in terms of $n.$
$(2)$Find all possible number of $n,$ such that sequence $x_1,x_2,\cdots ,x_{f(n)}$ is uniquely determined.
2002 India IMO Training Camp, 13
Let $ABC$ and $PQR$ be two triangles such that
[list]
[b](a)[/b] $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$.
[b](b)[/b] $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$
[/list]
Prove that $AB+AC=PQ+PR$.
2024 Indonesia TST, N
Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products
\[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\]
form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.
2017-2018 SDML (Middle School), 11
How many three-digit numbers leave remainder $2$ when divided by $5$ and leave remainder $7$ when divided by $9$?
$\mathrm{(A) \ } 20 \qquad \mathrm{(B) \ } 21 \qquad \mathrm {(C) \ } 22 \qquad \mathrm{(D) \ } 23 \qquad \mathrm{(E) \ } 24$
1983 AMC 12/AHSME, 9
In a certain population the ratio of the number of women to the number of men is 11 to 10. If the average (arithmetic mean) age of the women is 34 and the average age of the men is 32, then the average age of the population is
$ \textbf{(A)}\ 32\frac{9}{10}\qquad\textbf{(B)}\ 32\frac{20}{21}\qquad\textbf{(C)}\ 33\qquad\textbf{(D)}\ 33\frac{1}{21}\qquad\textbf{(E)}\ 33\frac{1}{10} $
1999 Harvard-MIT Mathematics Tournament, 5
Let $r$ be the inradius of triangle $ABC$. Take a point $D$ on side $BC$, and let $r_1$ and $r_2$ be the inradii of triangles $ABD$ and $ACD$. Prove that $r$, $r_1$, and $r_2$ can always be the side lengths of a triangle.
PEN E Problems, 2
Let $a, b, c, d$ be integers with $a>b>c>d>0$. Suppose that $ac+bd=(b+d+a-c)(b+d-a+c)$. Prove that $ab+cd$ is not prime.
2019 OMMock - Mexico National Olympiad Mock Exam, 5
There are $n\geq 2$ people at a party. Each person has at least one friend inside the party. Show that it is possible to choose a group of no more than $\frac{n}{2}$ people at the party, such that any other person outside the group has a friend inside it.
2012 239 Open Mathematical Olympiad, 5
Point $M$ is the midpoint of the base $AD$ of trapezoid $ABCD$ inscribed in circle $S$. Rays $AB$ and $DC$ intersect at point $P$, and ray $BM$ intersects $S$ at point $K$. The circumscribed circle of triangle $PBK$ intersects line $BC$ at point $L$. Prove that $\angle{LDP} = 90^{\circ}$.
2018 Mediterranean Mathematics OIympiad, 1
Let $a_1, a_2, ..., a_n$ be more than one real numbers, such that $0\leq a_i\leq \frac{\pi}{2}$. Prove that
$$\Bigg(\frac{1}{n}\sum_{i=1}^{n}\frac{1}{1+\sin a_i}\Bigg)\Bigg(1+\prod_{i=1}^{n}(\sin a_i)^{\frac{1}{n}}\Bigg)\leq1.$$
1955 AMC 12/AHSME, 20
The expression $ \sqrt{25\minus{}t^2}\plus{}5$ equals zero for:
$ \textbf{(A)}\ \text{no real or imaginary values of }t \qquad
\textbf{(B)}\ \text{no real values of }t\text{ only} \\
\textbf{(C)}\ \text{no imaginary values of }t\text{ only} \qquad
\textbf{(D)}\ t\equal{}0 \qquad
\textbf{(E)}\ t\equal{}\pm 5$
2014 China Team Selection Test, 1
Let the circumcenter of triangle $ABC$ be $O$. $H_A$ is the projection of $A$ onto $BC$. The extension of $AO$ intersects the circumcircle of $BOC$ at $A'$. The projections of $A'$ onto $AB, AC$ are $D,E$, and $O_A$ is the circumcentre of triangle $DH_AE$. Define $H_B, O_B, H_C, O_C$ similarly.
Prove: $H_AO_A, H_BO_B, H_CO_C$ are concurrent
2020 LMT Fall, A14
Two points $E$ and $F$ are randomly chosen in the interior of unit square $ABCD$. Let the line through $E$ parallel to $AB$ hit $AD$ at $E_1$, the line through $E$ parallel to $AD$ hit $CD$ at $E_2$, the line through $F$ parallel to $AB$ hit $BC$ at $F_1$, and the line through $F$ parallel to $BC$ hit $AB$ at $F_2$. The expected value of the overlap of the areas of rectangles $EE_1DE_2$ and $FF_1BF_2$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$.
[i]Proposed by Kevin Zhao[/i]
2021 South East Mathematical Olympiad, 2
In $\triangle ABC$,$AB=AC>BC$, point $O,H$ are the circumcenter and orthocenter of $\triangle ABC$ respectively,$G $ is the midpoint of segment $AH$ , $BE$ is the altitude on $AC$ . Prove that if $OE\parallel BC$, then $H$ is the incenter of $\triangle GBC$.
2014 Turkey Team Selection Test, 3
Let $r,R$ and $r_a$ be the radii of the incircle, circumcircle and A-excircle of the triangle $ABC$ with $AC>AB$, respectively. $I,O$ and $J_A$ are the centers of these circles, respectively. Let incircle touches the $BC$ at $D$, for a point $E \in (BD)$ the condition $A(IEJ_A)=2A(IEO)$ holds.
Prove that
\[ED=AC-AB \iff R=2r+r_a.\]
2021 CCA Math Bonanza, L1.2
A square is inscribed in a circle of radius $6$. A quarter circle is inscribed in the square, as shown in the diagram below. Given the area of the region inside the circle but outside the quarter circle is $n\pi$ for some positive integer $n$, what is $n$?
[asy]
size(5 cm);
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw(circle((1,1),1.41));
draw(arc((0,0),2,0,90));[/asy]
[i]2021 CCA Math Bonanza Lightning Round #1.2[/i]
2022 Harvard-MIT Mathematics Tournament, 9
Let $\Gamma_1$ and $\Gamma_2$ be two circles externally tangent to each other at $N$ that are both internally tangent to $\Gamma$ at points $U$ and $V$ , respectively. A common external tangent of $\Gamma_1$ and $\Gamma_2$ is tangent to $\Gamma_1$ and $\Gamma_2$ at $P$ and $Q$, respectively, and intersects $\Gamma$ at points $X$ and $Y$ . Let $M$ be the midpoint of the arc $XY$ that does not contain $U$ and $V$ . Let $Z$ be on $\Gamma$ such $MZ \perp NZ$, and suppose the circumcircles of $QVZ$ and $PUZ$ intersect at $T\ne Z$. Find, with proof, the value of $T U + T V$ , in terms of $R$, $r_1$, and $r_2$, the radii of $\Gamma$, $\Gamma_1$, and $\Gamma_2$, respectively.
2007 Brazil National Olympiad, 3
Consider $ n$ points in a plane which are vertices of a convex polygon. Prove that the set of the lengths of the sides and the diagonals of the polygon has at least $ \lfloor n/2\rfloor$ elements.
2021 Saudi Arabia IMO TST, 6
Find all functions $f : \mathbb{Z}\rightarrow \mathbb{Z}$ satisfying
\[f^{a^{2} + b^{2}}(a+b) = af(a) +bf(b)\]
for all integers $a$ and $b$
2016 EGMO, 6
Let $S$ be the set of all positive integers $n$ such that $n^4$ has a divisor in the range $n^2 +1, n^2 + 2,...,n^2 + 2n$. Prove that there are infinitely many elements of $S$ of each of the forms $7m, 7m+1, 7m+2, 7m+5, 7m+6$ and no elements of $S$ of the form $7m+3$ and $7m+4$, where $m$ is an integer.