Found problems: 85335
2016 Croatia Team Selection Test, Problem 3
Let $P$ be a point inside a triangle $ABC$ such that
$$ \frac{AP + BP}{AB} = \frac{BP + CP}{BC} = \frac{CP + AP}{CA} .$$
Lines $AP$, $BP$, $CP$ intersect the circumcircle of triangle $ABC$ again in $A'$, $B'$, $C'$. Prove that the triangles $ABC$ and $A'B'C'$ have a common incircle.
2024 Israel Olympic Revenge, P4
Let $ABC$ be an acute triangle. Let $D$ be a point inside side $BC$. Let $E$ be the foot from $D$ to $AC$, and let $F$ be a point on $AB$ so that $FE\perp AB$. It is given that the lines $AD, BE, CF$ concur. $M_A, M_B, M_C$ are the midpoints of sides $BC, AC, AB$ respectively, and $O$ is the circumcenter of $ABC$. Moreover, we define $P=EF\cap M_AM_B, S=DE\cap M_AM_C$. Prove that $O, P, S$ are collinear.
1999 Slovenia National Olympiad, Problem 4
On a mountain, three shepherds cyclically alternate shearing the same herd of sheep. The shepherds agreed to obey the following rules:
(i) Every day a sheep can be shorn* on one side only;
(ii) Every day at least one sheep must be shorn;
(iii) No two days the same group of sheep can be shorn.
The shepherd who first breaks the agreement will have to accompany the herd in the valley next fall. Can anyone of the shepherds shear the sheep in such a way to make sure that he will avoid this punishment?
*shorn is the past tense of shear
2025 China National Olympiad, 5
Let $p$ be a prime number and $f$ be a bijection from $\left\{0,1,\ldots,p-1\right\}$ to itself. Suppose that for integers $a,b \in \left\{0,1,\ldots,p-1\right\}$, $|f(a) - f(b)|\leqslant 2024$ if $p \mid a^2 - b$. Prove that there exists infinite many $p$ such that there exists such an $f$ and there also exists infinite many $p$ such that there doesn't exist such an $f$.
1972 Canada National Olympiad, 1
Given three distinct unit circles, each of which is tangent to the other two, find the radii of the circles which are tangent to all three circles.
2011 Bosnia and Herzegovina Junior BMO TST, 3
In isosceles triangle $ABC$ ($AC=BC$), angle bisector $\angle BAC$ and altitude $CD$ from point $C$ intersect at point $O$, such that $CO=3 \cdot OD$. In which ratio does altitude from point $A$ on side $BC$ divide altitude $CD$ of triangle $ABC$
1990 IMO Longlists, 6
Let $S, T$ be the circumcenter and centroid of triangle $ABC$, respectively. $M$ is a point in the plane of triangle $ABC$ such that $90^\circ \leq \angle SMT < 180^\circ$. $A_1, B_1, C_1$ are the intersections of $AM, BM, CM$ with the circumcircle of triangle $ABC$ respectively. Prove that $MA_1 + MB_1 + MC_1 \geq MA + MB + MC.$
2015 Purple Comet Problems, 22
Let $x$ be a real number between 0 and $\tfrac{\pi}{2}$ for which the function $3\sin^2 x + 8\sin x \cos x + 9\cos^2 x$ obtains its maximum value, $M$. Find the value of $M + 100\cos^2x$.
2022 BMT, 9
What is the measure of the largest convex angle formed by the hour and minute hands of a clock between $1:45$ PM and $2:40$ PM, in degrees? Convex angles always have a measure of less than $180$ degrees.
2019 Purple Comet Problems, 6
A pentagon has four interior angles each equal to $110^o$. Find the degree measure of the fifth interior angle.
2010 Harvard-MIT Mathematics Tournament, 5
Suppose that $x$ and $y$ are complex numbers such that $x+y=1$ and $x^{20}+y^{20}=20$. Find the sum of all possible values of $x^2+y^2$.
2001 AMC 10, 6
Let $ P(n)$ and $ S(n)$ denote the product and the sum, respectively, of the digits of the integer $ n$. For example, $ P(23) \equal{} 6$ and $ S(23) \equal{} 5$. Suppose $ N$ is a two-digit number such that $ N \equal{} P(N) \plus{} S(N)$. What is the units digit of $ N$?
$ \textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }9$
1986 China Team Selection Test, 4
Given a triangle $ABC$ for which $C=90$ degrees, prove that given $n$ points inside it, we can name them $P_1, P_2 , \ldots , P_n$ in some way such that:
$\sum^{n-1}_{k=1} \left( P_K P_{k+1} \right)^2 \leq AB^2$ (the sum is over the consecutive square of the segments from $1$ up to $n-1$).
[i]Edited by orl.[/i]
2020 Greece JBMO TST, 2
Let $a,b,c$ be positive real numbers such that $\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}=3$. Prove that
$$\frac{a+b}{a^2+ab+b^2}+ \frac{b+c}{b^2+bc+c^2}+ \frac{c+a}{c^2+ca+a^2}\le 2$$
When is the equality valid?
1999 Spain Mathematical Olympiad, 3
A one player game is played on the triangular board shown on the picture. A token is placed on each circle. Each token is white on one side and black on the other. Initially, the token at one vertex of the triangle has the black side up, while the others have the white sides up. A move consists of removing a token with the black side up and turning over the adjacent tokens (two tokens are adjacent if they are joined by a segment). Is it possible to remove all the tokens by a sequence of moves?
[img]https://cdn.artofproblemsolving.com/attachments/d/2/aabf82a0ddd6907482f27e6e0f1e1b56cd931d.png[/img]
2001 Baltic Way, 15
Let $a_0,a_1,a_2,\ldots $ be a sequence of positive real numbers satisfying $i\cdot a_2\ge (i + 1)\cdot a_{i_1}a_{i+1}$ for $i=1, 2, \ldots $ Furthermore, let $x$ and $y$ be positive reals, and let $b_i=xa_i+ya_{i-1}$ for $i=1, 2, \ldots $
Prove that the inequality $i\cdot b_2\ge (i + 1)\cdot b_{i-1}b_{i+1}$ holds for all integers $i\ge 2$.
2015 Cuba MO, 5
Let $a, b$ and $c$ be real numbers such that $0 < a, b, c < 1$. Prove that:
$$\min \ \ \{ab(1 -c)^2, bc(1 - a)^2, ca(1 - b)^2 \} \le \frac{1}{16}.$$
2024 Nepal Mathematics Olympiad (Pre-TST), Problem 1
Nirajan is trapped in a magical dungeon. He has infinitely many magical cards with arbitrary MPs(Mana Points) which is always an integer $\mathbb{Z}$. To escape, he must give the dungeon keeper some magical cards whose MPs add up to an integer with at least $2024$ divisors. Can Nirajan always escape?
[i]( Proposed by Vlad Spǎtaru, Romania)[/i]
2022 Yasinsky Geometry Olympiad, 2
In the triangle $ABC$, angle $C$ is four times smaller than each of the other two angle The altitude $AK$ and the angle bisector $AL$ are drawn from the vertex of the angle $A$. It is known that the length of $AL$ is equal to $\ell$. Find the length of the segment $LK$.
(Gryhoriy Filippovskyi)
2018 Mexico National Olympiad, 1
Let $A$ and $B$ be two points on a line $\ell$, $M$ the midpoint of $AB$, and $X$ a point on segment $AB$ other than $M$. Let $\Omega$ be a semicircle with diameter $AB$. Consider a point $P$ on $\Omega$ and let $\Gamma$ be the circle through $P$ and $X$ that is tangent to $AB$. Let $Q$ be the second intersection point of $\Omega$ and $\Gamma$. The internal angle bisector of $\angle PXQ$ intersects $\Gamma$ at a point $R$. Let $Y$ be a point on $\ell$ such that $RY$ is perpendicular to $\ell$. Show that $MX > XY$
2024 Saint Petersburg Mathematical Olympiad, 2
$32$ real and $32$ fake coins are given the same appearance. All fake coins weigh equally and less than the real ones, which also all weigh the same. How to determine the type of at least seven coins in six weighings on a scale with two bowls?
2011 China Second Round Olympiad, 3
Given $n\ge 4$ real numbers $a_{n}>...>a_{1} > 0$. For $r > 0$, let $f_{n}(r)$ be the number of triples $(i,j,k)$ with $1\leq i<j<k\leq n$ such that $\frac{a_{j}-a_{i}}{a_{k}-a_{j}}=r$. Prove that ${f_{n}(r)}<\frac{n^{2}}{4}$.
2013 Saint Petersburg Mathematical Olympiad, 4
There are $100$ numbers from $(0,1)$ on the board. On every move we replace two numbers $a,b$ with roots of $x^2-ax+b=0$(if it has two roots). Prove that process is not endless.
2015 HMNT, 7
Let $ABCD$ be a convex quadrilateral whose diagonals $AC$ and $BD$ meet at $P$. Let the area of triangle $APB$ be $24$ and let the area of triangle $CPD$ be $25$. What is the minimum possible area of quadrilateral $ABCD$?
2016 Japan MO Preliminary, 11
How many pairs $(a, b)$ for integers $a, b \ge 2$ which exist the sequence $x_1, x_2, . . . , x_{1000}$ which satisfy conditions as below?
1.Terms $x_1, x_2, . . . , x_{1000}$ are sorting of $1, 2, . . . , 1000$.
2.For each integers $1 \le i < 1000$, the sequence forms $x_{i+1} = x_i + a$ or $x_{i+1} = x_i - b$.