Found problems: 85335
1979 Romania Team Selection Tests, 3.
Let $a,b,c\in \mathbb{R}$ with $a^2+b^2+c^2=1$ and $\lambda\in \mathbb{R}_{>0}\setminus\{1\}$. Then for each solution $(x,y,z)$ of the system of equations:
\[
\begin{cases}
x-\lambda y=a,\\
y-\lambda z=b,\\
z-\lambda x=c.
\end{cases}
\]
we have $\displaystyle x^2+y^2+z^2\leqslant \frac1{(\lambda-1)^2}$.
[i]Radu Gologan[/i]
2024 MMATHS, 8
Let circle $A$ have radius $9,$ and let circle $B$ have radius $5$ and be internally tangent to circle $A.$ The largest radius $r$ such that there are two circles with radius $r$ that lie inside circle $A,$ are externally tangent to each other, and externally tangent with circle $B$ can be expressed as a fraction $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2025 Bangladesh Mathematical Olympiad, P6
Let the incircle of triangle $ABC$ touch sides $BC, CA$ and $AB$ at the points $D, E$ and $F$ respectively and let $I$ be the center of that circle. Furthermore, let $P$ be the foot of the perpendicular from point $I$ to line $AD$ and let $M$ be the midpoint of $DE$. If $N$ is the intersection point of $PM$ and $AC$, prove that $DN \parallel EF$.
2008 China Team Selection Test, 1
Let $ ABC$ be a triangle, line $ l$ cuts its sides $ BC,CA,AB$ at $ D,E,F$, respectively. Denote by $ O_{1},O_{2},O_{3}$ the circumcenters of triangle $ AEF,BFD,CDE$, respectively. Prove that the orthocenter of triangle $ O_{1}O_{2}O_{3}$ lies on line $ l$.
1992 Dutch Mathematical Olympiad, 1
Four dice are thrown. What is the probability that the product of the number equals $ 36?$
2024 Harvard-MIT Mathematics Tournament, 30
Let $ABC$ be an equilateral triangle with side length $1.$ Points $D, E,$ and $F$ lie inside triangle $ABC$ such that $A, E, F$ are collinear, $B, F, D$ are collinear, $C, D, E$ are collinear, and triangle $DEF$ is equilateral. Suppose that there exists a unique equilateral triangle $XYZ$ with $X$ on side $\overline{BC},$ $Y$ is on side $\overline{AB},$ and $Z$ is on side $\overline{AC}$ such that $D$ lies on side $\overline{XZ},$ $E$ lies on side $\overline{YZ},$ and $F$ lies on side $\overline{XY}.$ Compute $AZ.$
2002 Chile National Olympiad, 7
A convex polygon of sides $\ell_1, \ell_2, ..., \ell_n$ is called [i]ordered [/i] if for all reordering $( \sigma (1), \sigma (2), ..., \sigma (n))$ of the set $(1, 2,..., n)$ there exists a point $P$ inside the polygon such that $d_{\sigma (1)} < _{\sigma (2)} <...< d_{\sigma (n)}$ , where $d_i$ represents the distance between $P$ and side $\ell_i$. Find all the convex ordered polygons.
2021 USMCA, 14
Derek the Dolphin and Kevin the Frog are playing a game where they take turns taking coins from a stack of $N$ coins, except with one rule: The number of coins someone takes each turn must be a power of $6$. The person who cannot take any more coins loses. If Derek goes first, how many integers $N$ from $1$ to $6^{2021}$ inclusive will guarantee him a win? (Example: If $N = 37$, then a possible sequence of turns is: Derek takes one coin, Kevin takes $36$ coins, and Kevin wins.)
2024 Assara - South Russian Girl's MO, 1
There is a set of $50$ cards. Each card on both sides is colored in one of three colors — red, blue or white, and for each card its two sides are colored in different colors. The cards were laid out on the table. The card [i]lies beautifully[/i] if at least one of two conditions is met: its upper side — red; its underside is blue. It turned out that exactly $25$ cards are lying beautifully. Then all the cards were turned over. Now some of the cards are lying beautifully on the table. How many of them can there be?
[i]K.A.Sukhov[/i]
2011 Tournament of Towns, 5
Given that $0 < a, b, c, d < 1$ and $abcd = (1 - a)(1 - b)(1 - c)(1 - d)$, prove that $(a + b + c + d) -(a + c)(b + d) \ge 1$
2008 IberoAmerican Olympiad For University Students, 5
Find all positive integers $n$ such that there are positive integers $a_1,\cdots,a_n, b_1,\cdots,b_n$ that satisfy
\[(a_1^2+\cdots+a_n^2)(b_1^2+\cdots+b_n^2)-(a_1b_1+\cdots+a_nb_n)^2=n\]
1991 IMO Shortlist, 3
Let $ S$ be any point on the circumscribed circle of $ PQR.$ Then the feet of the perpendiculars from S to the three sides of the triangle lie on the same straight line. Denote this line by $ l(S, PQR).$ Suppose that the hexagon $ ABCDEF$ is inscribed in a circle. Show that the four lines $ l(A,BDF),$ $ l(B,ACE),$ $ l(D,ABF),$ and $ l(E,ABC)$ intersect at one point if and only if $ CDEF$ is a rectangle.
1969 IMO Shortlist, 57
Given triangle $ ABC $ with points $ M $ and $ N $ are in the sides $ AB $ and $ AC $ respectively.
If $ \dfrac{BM}{MA} +\dfrac{CN}{NA} = 1 $ , then prove that the centroid of $ ABC $ lies on $ MN $ .
2014 Turkey Team Selection Test, 2
A circle $\omega$ cuts the sides $BC,CA,AB$ of the triangle $ABC$ at $A_1$ and $A_2$; $B_1$ and $B_2$; $C_1$ and $C_2$, respectively. Let $P$ be the center of $\omega$. $A'$ is the circumcenter of the triangle $A_1A_2P$, $B'$ is the circumcenter of the triangle $B_1B_2P$, $C'$ is the circumcenter of the triangle $C_1C_2P$. Prove that $AA', BB'$ and $CC'$ concur.
2015 Estonia Team Selection Test, 1
Let $n$ be a natural number, $n \ge 5$, and $a_1, a_2, . . . , a_n$ real numbers such that all possible sums $a_i + a_j$, where $1 \le i < j \le n$, form $\frac{n(n-1)}{2}$ consecutive members of an arithmetic progression when taken in some order. Prove that $a_1 = a_2 = . . . = a_n$.
2019 JBMO Shortlist, A4
Let $a$, $b$ be two distinct real numbers and let $c$ be a positive real numbers such that
$a^4 - 2019a = b^4 - 2019b = c$.
Prove that $- \sqrt{c} < ab < 0$.
2012 Princeton University Math Competition, B2
Define a sequence $a_n$ such that $a_n = a_{n-1} - a_{n-2}$. Let $a_1 = 6$ and $a_2 = 5$. Find $\Sigma_{n=1}^{1000}a_n$.
1960 Polish MO Finals, 1
Prove that if $ n $ is an integer greater than $ 4 $, then $ 2^n $ is greater than $ n^2 $.
2021 JHMT HS, 5
For real numbers $x,$ let $T_x$ be the triangle with vertices $(5, 5^3),$ $(8, 8^3),$ and $(x, x^3)$ in $\mathbb{R}^2.$ Over all $x$ in the interval $[5, 8],$ the area of the triangle $T_x$ is maximized at $x = \sqrt{n},$ for some positive integer $n.$ Compute $n.$
1992 China Team Selection Test, 2
A $(3n + 1) \times (3n + 1)$ table $(n \in \mathbb{N})$ is given. Prove that deleting any one of its squares yields a shape cuttable into pieces of the following form and its rotations: ''L" shape formed by cutting one square from a $2 \times 2$ squares.
2015 Peru IMO TST, 2
Ana chose some unit squares of a $50 \times 50$ board and placed a chip on each of them. Prove that Beto can always choose at most $99$ empty unit squares and place a chip on each so that each row and each column of the board contains an even number of chips.
1998 North Macedonia National Olympiad, 2
Prove that the numbers $1,2,...,1998$ cannot be separated into three classes whose sums of elements are divisible by $2000,3999$, and $5998$, respectively.
2018 Flanders Math Olympiad, 2
Prove that for every acute angle $\alpha$, $\sin (\cos \alpha) < \cos(\sin \alpha)$.
2016 IMO Shortlist, C3
Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.
2017 ASDAN Math Tournament, 9
Let $f(x)=x^3+ax^2+bx$ for some $a,b$. For some $c$, $f(c)$ achieves a local maximum of $539$ (in other words, $f(c)$ is the maximum value of $f$ for some open interval around $c$). In addition, at some $d$, $f(d)$ achieves a local minimum of $-325$. Given that $c$ and $d$ are integers, compute $a+b$.