Found problems: 85335
2014 Contests, 1
A sequence $a_0,a_1,a_2,\cdots$ satisfies the conditions $a_0 = 0$ , $a_{n-1}^2 - a_{n-1} = a_n^2 + a_n$
1) determine the two possible values of $a_1$ . then determine all possible values of $a_2$ .
2)for each $n$, prove that $a_{n+1}=a_n+1$ or $a_{n+1} = -a_n$
3)Describe the possible values of $a_{1435}$
4)Prove that the values that you got in (3) are correct
1992 Irish Math Olympiad, 1
Describe in geometric terms the set of points $(x,y)$ in the plane such that $x$ and $y$ satisfy the condition $t^2+yt+x\ge 0$ for all $t$ with $-1\le t\le 1$.
2013 Vietnam Team Selection Test, 6
A cube with size $10\times 10\times 10$ consists of $1000$ unit cubes, all colored white. $A$ and $B$ play a game on this cube. $A$ chooses some pillars with size $1\times 10\times 10$ such that no two pillars share a vertex or side, and turns all chosen unit cubes to black. $B$ is allowed to choose some unit cubes and ask $A$ their colors. How many unit cubes, at least, that $B$ need to choose so that for any answer from $A$, $B$ can always determine the black unit cubes?
2008 All-Russian Olympiad, 3
In a scalene triangle $ ABC, H$ and $ M$ are the orthocenter an centroid respectively. Consider the triangle formed by the lines through $ A,B$ and $ C$ perpendicular to $ AM,BM$ and $ CM$ respectively. Prove that the centroid of this triangle lies on the line $ MH$.
2021 China Second Round Olympiad, Problem 9
Let $\triangle ABC$ have its vertices at $A(0, 0), B(7, 0), C(3, 4)$ in the Cartesian plane. Construct a line through the point $(6-2\sqrt 2, 3-\sqrt 2)$ that intersects segments $AC, BC$ at $P, Q$ respectively. If $[PQC] = \frac{14}3$, what is $|CP|+|CQ|$?
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 9)[/i]
KoMaL A Problems 2024/2025, A. 894
In convex polyhedron $ABCDE$ line segment $DE$ intersects the plane of triangle $ABC$ inside the triangle. Rotate the point $D$ outward into the plane of triangle $ABC$ around the lines $AB$, $BC$, $CA$; let the resulting points be $D_1$, $D_2$, and $D_3$. Similarly, rotate the point $E$ outward into the plane of triangle $ABC$ around the lines $AB$, $BC$, $CA$; let the resulting points be $E_1$, $E_2$, and $E_3$. Show that if the polyhedron has an inscribed sphere, then the circumcircles of $D_1D_2D_3$ and $E_1E_2E_3$ are concentric.
[i]Proposed by: Géza Kós, Budapest[/i]
2010 IMO Shortlist, 7
Let $a_1, a_2, a_3, \ldots$ be a sequence of positive real numbers, and $s$ be a positive integer, such that
\[a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.\]
Prove there exist positive integers $\ell \leq s$ and $N$, such that
\[a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.\]
[i]Proposed by Morteza Saghafiyan, Iran[/i]
2005 AIME Problems, 11
A semicircle with diameter $d$ is contained in a square whose sides have length $8$. Given the maximum value of $d$ is $m- \sqrt{n}$, find $m+n$.
1983 Tournament Of Towns, (036) O5
A version of billiards is played on a right triangular table, with a pocket in each of the three corners, and one of the acute angles being $30^o$. A ball is played from just in front of the pocket at the $30^o$. vertex toward the midpoint of the opposite side. Prove that if the ball is played hard enough, it will land in the pocket of the $60^o$ vertex after $8$ reflections.
Russian TST 2015, P3
Let $0<\alpha<1$ be a fixed number. On a lake shaped like a convex polygon, at some point there is a duck and at another point a water lily grows. If the duck is at point $X{}$, then in one move it can swim towards one of the vertices $Y$ of the polygon a distance equal to a $\alpha\cdot XY$. Find all $\alpha{}$ for which, regardless of the shape of the lake and the initial positions of the duck and the lily, after a sequence of adequate moves, the distance between the duck and the lily will be at most one meter.
1991 AMC 8, 20
In the addition problem, each digit has been replaced by a letter. If different letters represent different digits then $C=$
[asy]
unitsize(18);
draw((-1,0)--(3,0));
draw((-3/4,1/2)--(-1/4,1/2)); draw((-1/2,1/4)--(-1/2,3/4));
label("$A$",(0.5,2.1),N); label("$B$",(1.5,2.1),N); label("$C$",(2.5,2.1),N);
label("$A$",(1.5,1.1),N); label("$B$",(2.5,1.1),N); label("$A$",(2.5,0.1),N);
label("$3$",(0.5,-.1),S); label("$0$",(1.5,-.1),S); label("$0$",(2.5,-.1),S);
[/asy]
$\text{(A)}\ 1 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 9$
2016 LMT, 25
Let $ABCD$ be a trapezoid with $AB\parallel DC$. Let $M$ be the midpoint of $CD$. If $AD\perp CD, AC\perp BM,$ and $BC\perp BD$, find $\frac{AB}{CD}$.
[i]Proposed by Nathan Ramesh
2012 Thailand Mathematical Olympiad, 9
Let $n$ be a positive integer and let $P(x) = x^n + a_{n-1}x^{n-1} +... + a_1x + 1$ be a polynomial with positive real coefficients. Under the assumption that the roots of $P$ are all real, show that $P(x) \ge (x + 1)^n$ for all $x > 0$.
1994 IberoAmerican, 1
Let $A,\ B$ and $C$ be given points on a circumference $K$ such that the triangle $\triangle{ABC}$ is acute. Let $P$ be a point in the interior of $K$. $X,\ Y$ and $Z$ be the other intersection of $AP, BP$ and $CP$ with the circumference. Determine the position of $P$ such that $\triangle{XYZ}$ is equilateral.
2022 Auckland Mathematical Olympiad, 2
The number $12$ is written on the whiteboard. Each minute, the number on the board is either multiplied or divided by one of the numbers $2$ or $3$ (a division is possible only if the result is an integer) . Prove that the number that will be written on the board in exactly one hour will not be equal to $54$.
2020 Putnam, A6
For a positive integer $N$, let $f_N$ be the function defined by
\[ f_N (x)=\sum_{n=0}^N \frac{N+1/2-n}{(N+1)(2n+1)} \sin\left((2n+1)x \right). \]
Determine the smallest constant $M$ such that $f_N (x)\le M$ for all $N$ and all real $x$.
2004 Mexico National Olympiad, 6
What is the maximum number of possible change of directions in a path traveling on the edges of a rectangular array of $2004 \times 2004$, if the path does not cross the same place twice?.
2019 May Olympiad, 2
There is a board with $2020$ squares in the bottom row and $2019$ in the top row, located as shown shown in the figure.
[img]https://cdn.artofproblemsolving.com/attachments/f/3/516ad5485c399427638c3d1783593d79d83002.png[/img]
In the bottom row the integers numbers from $ 1$ to $2020$ are placed in some order. Then in each box in the top row records the multiplication of the two numbers below it. How can they place the numbers in the bottom row so that the sum of the numbers in the top row be the smallest possible?
2006 Switzerland Team Selection Test, 2
Let $D$ be inside $\triangle ABC$ and $E$ on $AD$ different to $D$. Let $\omega_1$ and $\omega_2$ be the circumscribed circles of $\triangle BDE$ and $\triangle CDE$ respectively. $\omega_1$ and $\omega_2$ intersect $BC$ in the interior points $F$ and $G$ respectively. Let $X$ be the intersection between $DG$ and $AB$ and $Y$ the intersection between $DF$ and $AC$. Show that $XY$ is $\|$ to $BC$.
2003 AIME Problems, 13
A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$
2010 Postal Coaching, 3
In a quadrilateral $ABCD$, we have $\angle DAB = 110^{\circ} , \angle ABC = 50^{\circ}$ and $\angle BCD = 70^{\circ}$ . Let $ M, N$ be the mid-points of $AB$ and $CD$ respectively. Suppose $P$ is a point on the segment $M N$ such that $\frac{AM}{CN} = \frac{MP}{PN}$ and $AP = CP$ . Find $\angle AP C$.
2017 Sharygin Geometry Olympiad, P2
A circle cuts off four right-angled triangles from rectangle $ABCD$.Let $A_0, B_0, C_0$ and $D_0$ be the midpoints of the correspondent hypotenuses. Prove that $A_0C_0 = B_0D_0$
[i]Proosed by L.Shteingarts[/i]
2012 Saint Petersburg Mathematical Olympiad, 2
Points $C,D$ are on side $BE$ of triangle $ABE$, such that $BC=CD=DE$. Points $X,Y,Z,T$ are circumcenters of $ABE,ABC,ADE,ACD$. Prove, that $T$ - centroid of $XYZ$
2015 Junior Regional Olympiad - FBH, 2
One day students in school organised a exchange between them such that : $11$ strawberries change for $14$ raspberries, $22$ cherries change for $21$ raspberries, $10$ cherries change for $3$ bananas and $5$ pears for $2$ bananas. How many pears has Amila to give to get $7$ strawberries
2008 IMS, 5
Prove that there does not exist a ring with exactly 5 regular elements.
($ a$ is called a regular element if $ ax \equal{} 0$ or $ xa \equal{} 0$ implies $ x \equal{} 0$.)
A ring is not necessarily commutative, does not necessarily contain unity element, or is not necessarily finite.