Found problems: 85335
2024 Mexico National Olympiad, 6
Ana and Beto play on a blackboard where all integers from 1 to 2024 (inclusive) are written. On each turn, Ana chooses three numbers $a,b,c$ written on the board and then Beto erases them and writes one of the following numbers:
$$a+b-c, a-b+c, ~\text{or}~ -a+b+c.$$
The game ends when only two numbers are left on the board and Ana cannot play. If the sum of the final numbers is a multiple of 3, Beto wins. Otherwise, Ana wins. ¿Who has a winning strategy?
1992 Miklós Schweitzer, 10
We place n points in the unit square independently, according to a uniform distribution. These points are the vertices of a graph $G_n$. Two points are connected by an edge if the slope of the segment connecting them is nonnegative. Denote by $M_n$ the event that the graph $G_n$ has a 1-factor. Prove that $\lim_{n \to \infty} P(M_ {2n}) = 1$.
Mathley 2014-15, 8
Two circles $(U)$ and $(V)$ intersect at $A,B$. A line d meets $(U), (V)$ at $P, Q$ and $R,S$ respectively. Let $t_P, t_Q, t_R,t_S$ be the tangents at $P,Q,R, S$ of the two circles. Another circle $(W)$ passes through through $A, B$. Prove that if the circumcircle of triangle that is formed by the intersections of $t_P,t_R, AB$ is tangent to $(W)$ then the circumcircle of triangle formed by $t_Q, t_S, AB$ is also tangent to $(W)$.
Tran Minh Ngoc, a student of Ho Chi Minh City College, Ho Chi Minh
2020 Bundeswettbewerb Mathematik, 2
Konstantin moves a knight on a $n \times n$- chess board from the lower left corner to the lower right corner with the minimal number of moves.
Then Isabelle takes the knight and moves it from the lower left corner to the upper right corner with the minimal number of moves.
For which values of $n$ do they need the same number of moves?
1977 Putnam, A4
For $0<x<1,$ express $$\sum_{n=0}^{\infty} \frac{x^{2^n}}{1-x^{2^{n+1}}}$$ as a rational function of $x.$
2023 India National Olympiad, 2
Suppose $a_0,\ldots, a_{100}$ are positive reals. Consider the following polynomial for each $k$ in $\{0,1,\ldots, 100\}$:
$$a_{100+k}x^{100}+100a_{99+k}x^{99}+a_{98+k}x^{98}+a_{97+k}x^{97}+\dots+a_{2+k}x^2+a_{1+k}x+a_k,$$where indices are taken modulo $101$, [i]i.e.[/i], $a_{100+i}=a_{i-1}$ for any $i$ in $\{1,2,\dots, 100\}$. Show that it is impossible that each of these $101$ polynomials has all its roots real.
[i]Proposed by Prithwijit De[/i]
2012 All-Russian Olympiad, 3
Consider the parallelogram $ABCD$ with obtuse angle $A$. Let $H$ be the feet of perpendicular from $A$ to the side $BC$. The median from $C$ in triangle $ABC$ meets the circumcircle of triangle $ABC$ at the point $K$. Prove that points $K,H,C,D$ lie on the same circle.
1993 Tournament Of Towns, (393) 1
Two tangents $CA$ and $CB$ are drawn to a circle ($A$ and $B$ being the tangent points). Consider a “triangle” bounded by an arc $AB$ (the smaller one) and segments $CA$ and $CB$. Prove that the length of any segment inside the triangle is not greater than the length of $CA = CB$.
(Folklore)
2024 Bulgarian Winter Tournament, 10.2
Find all positive integers $k$ for which there exist positive integers $x, y$, such that $\frac{x^ky}{x^2+y^2}$ is a prime.
1993 All-Russian Olympiad, 4
On a board, there are $n$ equations in the form $*x^2+*x+*$. Two people play a game where they take turns. During a turn, you are aloud to change a star into a number not equal to zero. After $3n$ moves, there will be $n$ quadratic equations. The first player is trying to make more of the equations not have real roots, while the second player is trying to do the opposite. What is the maximum number of equations that the first player can create without real roots no matter how the second player acts?
1996 China Team Selection Test, 1
Let side $BC$ of $\bigtriangleup ABC$ be the diameter of a semicircle which cuts $AB$ and $AC$ at $D$ and $E$ respectively. $F$ and $G$ are the feet of the perpendiculars from $D$ and $E$ to $BC$ respectively. $DG$ and $EF$ intersect at $M$. Prove that $AM \perp BC$.
1998 Harvard-MIT Mathematics Tournament, 8
Find the slopes of all lines passing through the origin and tangent to the curve $y^2=x^3+39x-35$.
1999 Italy TST, 4
Let $X$ be an $n$-element set and let $A_1,\ldots ,A_m$ be subsets of $X$ such that
i) $|A_i|=3$ for each $i=1,\ldots ,m$.
ii) $|A_i\cap A_j|\le 1$ for any two distinct indices $i,j$.
Show that there exists a subset of $X$ with at least $\lfloor\sqrt{2n}\rfloor$ elements which does not contain any of the $A_i$’s.
2008 Princeton University Math Competition, A10/B10
What is the smallest number $n$ such that you can choose $n$ distinct odd integers $a_1, a_2,..., a_n$, none of them $1$, with $\frac{1}{a_1}+ \frac{1}{a_2}+ ...+ \frac{1}{a_n}= 1$?
1986 IMO Shortlist, 9
Given a finite set of points in the plane, each with integer coordinates, is it always possible to color the points red or white so that for any straight line $L$ parallel to one of the coordinate axes the difference (in absolute value) between the numbers of white and red points on $L$ is not greater than $1$?
2020 CCA Math Bonanza, I13
Let $n$ be a positive integer. Compute, in terms of $n$, the number of sequences $(x_1,\ldots,x_{2n})$ with each $x_i\in\{0,1,2,3,4\}$ such that $x_1^2+\dots+x_{2n}^2$ is divisible by $5$.
[i]2020 CCA Math Bonanza Individual Round #13[/i]
2015 Saudi Arabia GMO TST, 3
Let $BD$ and $CE$ be altitudes of an arbitrary scalene triangle $ABC$ with orthocenter $H$ and circumcenter $O$. Let $M$ and $N$ be the midpoints of sides $AB$, respectively $AC$, and $P$ the intersection point of lines $MN$ and $DE$. Prove that lines $AP$ and $OH$ are perpendicular.
Liana Topan
MOAA Gunga Bowls, 2023.17
Call a polynomial with real roots [i]n-local[/i] if the greatest difference between any pair of its roots is $n$. Let $f(x)=x^2+ax+b$ be a 1-[i]local[/i] polynomial with distinct roots such that $a$ and $b$ are non-zero integers. If $f(f(x))$ is a 23-[i]local[/i] polynomial, find the sum of the roots of $f(x)$.
[i]Proposed by Anthony Yang[/i]
2018 Hanoi Open Mathematics Competitions, 12
Let ABCD be a rectangle with $45^o < \angle ADB < 60^o$. The diagonals $AC$ and$ BD$ intersect at $O$. A line passing through $O$ and perpendicular to $BD$ meets $AD$ and $CD$ at $M$ and $N$ respectively. Let $K$ be a point on side $BC$ such that $MK \parallel AC$. Show that $\angle MKN = 90^o$.
[img]https://cdn.artofproblemsolving.com/attachments/4/1/1d37b96cebaea3409ade7ce6711ac2d3fc2ef9.png[/img]
2004 Romania Team Selection Test, 16
Three circles $\mathcal{K}_1$, $\mathcal{K}_2$, $\mathcal{K}_3$ of radii $R_1,R_2,R_3$ respectively, pass through the point $O$ and intersect two by two in $A,B,C$. The point $O$ lies inside the triangle $ABC$.
Let $A_1,B_1,C_1$ be the intersection points of the lines $AO,BO,CO$ with the sides $BC,CA,AB$ of the triangle $ABC$. Let $ \alpha = \frac {OA_1}{AA_1} $, $ \beta= \frac {OB_1}{BB_1} $ and $ \gamma = \frac {OC_1}{CC_1} $ and let $R$ be the circumradius of the triangle $ABC$. Prove that
\[ \alpha R_1 + \beta R_2 + \gamma R_3 \geq R. \]
2020 AMC 8 -, 15
Suppose $15\%$ of $x$ equals $20\%$ of $y$. What percentage of $x$ is $y$?
$\textbf{(A)}\ 5~~\qquad\textbf{(B)}\ 35~~\qquad~~\textbf{(C)}\ 75\qquad~~\textbf{(D)}\ 133\frac13\qquad~~ \textbf{(E)}\ 300$
Durer Math Competition CD Finals - geometry, 2017.D4
The convex quadrilateral $ABCD$ is has angle $A$ equal to $60^o$ , angle bisector of $A$ the diagonal $AC$ and $\angle ACD= 40^o$ and $\angle ACB = 120^o$. Inside the quadrilateral the point $P$ lies such that $\angle PDA = 40^o$ and $\angle PBA = 10^o$;
a) Find the angle $\angle DPB$?
b) Prove that $P$ lies on the diagonal $AC$.
2019 ELMO Shortlist, N5
Given an even positive integer $m$, find all positive integers $n$ for which there exists a bijection $f:[n]\to [n]$ so that, for all $x,y\in [n]$ for which $n\mid mx-y$, $$(n+1)\mid f(x)^m-f(y).$$
Note: For a positive integer $n$, we let $[n] = \{1,2,\dots, n\}$.
[i]Proposed by Milan Haiman and Carl Schildkraut[/i]
2016 AIME Problems, 7
Squares $ABCD$ and $EFGH$ have a common center and $\overline{AB}\parallel \overline{EF}$. The area of $ABCD$ is $2016$, and the area of $EFGH$ is a smaller positive integer. Square $IJKL$ is constructed so that each of its vertices lies on a side of $ABCD$ and each vertex of $EFGH$ lies on a side of $IJKL$. Find the difference between the largest and smallest possible integer values of the area of $IJKL$.
2009 Moldova Team Selection Test, 3
[color=darkred]Quadrilateral $ ABCD$ is inscribed in the circle of diameter $ BD$. Point $ A_1$ is reflection of point $ A$ wrt $ BD$ and $ B_1$ is reflection of $ B$ wrt $ AC$. Denote $ \{P\}\equal{}CA_1 \cap BD$ and $ \{Q\}\equal{}DB_1\cap AC$. Prove that $ AC\perp PQ$.[/color]