Found problems: 85335
2013 Sharygin Geometry Olympiad, 19
a) The incircle of a triangle $ABC$ touches $AC$ and $AB$ at points $B_0$ and $C_0$ respectively. The bisectors of angles $B$ and $C$ meet the perpendicular bisector to the bisector $AL$ in points $Q$ and $P$ respectively. Prove that the lines $PC_0, QB_0$ and $BC$ concur.
b) Let $AL$ be the bisector of a triangle $ABC$. Points $O_1$ and $O_2$ are the circumcenters of triangles $ABL$ and $ACL$ respectively. Points $B_1$ and $C_1$ are the projections of $C$ and $B$ to the bisectors of angles $B$ and $C$ respectively. Prove that the lines $O_1C_1, O_2B_1,$ and $BC$ concur.
c) Prove that the two points obtained in pp. a) and b) coincide.
1986 IMO Shortlist, 8
From a collection of $n$ persons $q$ distinct two-member teams are selected and ranked $1, \cdots, q$ (no ties). Let $m$ be the least integer larger than or equal to $2q/n$. Show that there are $m$ distinct teams that may be listed so that :
[b](i)[/b] each pair of consecutive teams on the list have one member in common and
[b](ii)[/b] the chain of teams on the list are in rank order.
[i]Alternative formulation.[/i]
Given a graph with $n$ vertices and $q$ edges numbered $1, \cdots , q$, show that there exists a chain of $m$ edges, $m \geq \frac{2q}{n}$ , each two consecutive edges having a common vertex, arranged monotonically with respect to the numbering.
2002 China Team Selection Test, 1
Given $ n \geq 3$, $ n$ is a integer. Prove that:
\[ (2^n \minus{} 2) \cdot \sqrt{2i\minus{}1} \geq \left( \sum_{j\equal{}0}^{i\minus{}1}C_n^j \plus{} C_{n\minus{}1}^{i\minus{}1} \right) \cdot \sqrt{n}\]
where if $ n$ is even, then $ \displaystyle 1 \leq i \leq \frac{n}{2}$; if $ n$ is odd, then $ \displaystyle 1 \leq i \leq \frac{n\minus{}1}{2}$.
2002 Moldova National Olympiad, 3
Consider a circle $ \Gamma(O,R)$ and a point $ P$ found in the interior of this circle. Consider a chord $ AB$ of $ \Gamma$ that passes through $ P$. Suppose that the tangents to $ \Gamma$ at the points $ A$ and $ B$ intersect at $ Q$. Let $ M\in QA$ and $ N\in QB$ s.t. $ PM\perp QA$ and $ PN\perp QB$. Prove that the value of $ \frac {1}{PN} \plus{} \frac {1}{PM}$ doesn't depend of choosing the chord $ AB$.
2011 ELMO Shortlist, 4
Consider the infinite grid of lattice points in $\mathbb{Z}^3$. Little D and Big Z play a game, where Little D first loses a shoe on an unmunched point in the grid. Then, Big Z munches a shoe-free plane perpendicular to one of the coordinate axes. They continue to alternate turns in this fashion, with Little D's goal to lose a shoe on each of $n$ consecutive lattice points on a line parallel to one of the coordinate axes. Determine all $n$ for which Little D can accomplish his goal.
[i]David Yang.[/i]
2025 India STEMS Category B, 5
Let $ABC$ be an acute scalene triangle. Let $D, E$ be points on segments $AB, AC$ respectively, such that $BD=CE$. Prove that the nine-point centers of $ADE$, $ACD$, $ABC$, $AEB$ form a rhombus.
[i]Proposed by Malay Mahajan and Siddharth Choppara[/i]
2014 Contests, 1
Let $ABCD$ be a convex quadrilateral. Diagonals $AC$ and $BD$ meet at point $P$. The inradii of triangles $ABP$, $BCP$, $CDP$ and $DAP$ are equal. Prove that $ABCD$ is a rhombus.
2018 Costa Rica - Final Round, F2
Consider $f (n, m)$ the number of finite sequences of $ 1$'s and $0$'s such that each sequence that starts at $0$, has exactly n $0$'s and $m$ $ 1$'s, and there are not three consecutive $0$'s or three $ 1$'s. Show that if $m, n> 1$, then
$$f (n, m) = f (n-1, m-1) + f (n-1, m-2) + f (n-2, m-1) + f (n-2, m-2)$$
2005 All-Russian Olympiad, 2
Find the number of subsets $A\subset M=\{2^0,\,2^1,\,2^2,\dots,2^{2005}\}$ such that equation $x^2-S(A)x+S(B)=0$ has integral roots, where $S(M)$ is the sum of all elements of $M$, and $B=M\setminus A$ ($A$ and $B$ are not empty).
1992 IMO Longlists, 24
[i](a)[/i] Show that there exists exactly one function $ f : \mathbb Q^+ \to \mathbb Q^+$ satisfying the following conditions:
[b](i)[/b] if $0 < q < \frac 12$, then $f(q)=1+f \left( \frac{q}{1-2q} \right);$
[b](ii)[/b] if $1 < q \leq 2$, then $f(q) = 1+f(q + 1);$
[b](iii)[/b] $f(q)f(1/q) = 1$ for all $q \in \mathbb Q^+.$
[i](b)[/i] Find the smallest rational number $q \in \mathbb Q^+$ such that $f(q) = \frac{19}{92}.$
2006 South africa National Olympiad, 3
Determine all positive integers whose squares end in $196$.
2017 Dutch Mathematical Olympiad, 3
Six teams participate in a hockey tournament. Each team plays exactly once against each other team. A team is awarded $3$ points for each game they win, $1$ point for each draw, and $0$ points for each game they lose. After the tournament, a ranking is made. There are no ties in the list. Moreover, it turns out that each team (except the very last team) has exactly $2$ points more than the team ranking one place lower.
Prove that the team that fi nished fourth won exactly two games.
2008 Miklós Schweitzer, 6
Is it possible to draw circles on the plane so that every line intersects at least one of them but no more than $100$ of them?
2022 Saint Petersburg Mathematical Olympiad, 3
Ivan and Kolya play a game, Ivan starts. Initially, the polynomial $x-1$ is written of the blackboard. On one move, the player deletes the current polynomial $f(x)$ and replaces it with $ax^{n+1}-f(-x)-2$, where $\deg(f)=n$ and $a$ is a real root of $f$. The player who writes a polynomial which does not have real roots loses. Can Ivan beat Kolya?
2025 USAMO, 6
Let $m$ and $n$ be positive integers with $m\geq n$. There are $m$ cupcakes of different flavors arranged around a circle and $n$ people who like cupcakes. Each person assigns a nonnegative real number score to each cupcake, depending on how much they like the cupcake. Suppose that for each person $P$, it is possible to partition the circle of $m$ cupcakes into $n$ groups of consecutive cupcakes so that the sum of $P$'s scores of the cupcakes in each group is at least $1$. Prove that it is possible to distribute the $m$ cupcakes to the $n$ people so that each person $P$ receives cupcakes of total score at least $1$ with respect to $P$.
1989 AIME Problems, 6
Two skaters, Allie and Billie, are at points $A$ and $B$, respectively, on a flat, frozen lake. The distance between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a speed of $8$ meters per second on a straight line that makes a $60^\circ$ angle with $AB$. At the same time Allie leaves $A$, Billie leaves $B$ at a speed of $7$ meters per second and follows the straight path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie?
[asy]
defaultpen(linewidth(0.8));
draw((100,0)--origin--60*dir(60), EndArrow(5));
label("$A$", origin, SW);
label("$B$", (100,0), SE);
label("$100$", (50,0), S);
label("$60^\circ$", (15,0), N);[/asy]
2006 Junior Balkan Team Selection Tests - Romania, 2
Consider the integers $a_1, a_2, a_3, a_4, b_1, b_2, b_3, b_4$ with $a_k \ne b_k$ for all $k = 1, 2, 3, 4$. If
$\{a_1, b_1\} + \{a_2, b_2\} = \{a_3, b_3\} + \{a_4, b_4\}$, show that the number $|(a_1 - b_1)(a_2 - b_2)(a_3 - b_3)(a_4 - b_4)|$ is a square.
Note. For any sets $A$ and $B$, we denote $A + B = \{x + y | x \in A, y \in B\}$.
2022 Thailand TSTST, 1
Let $n\geq 3$ be an integer. Each vertex of a regular $n$-gon is labelled with a real number not exceeding $1$. For real numbers $a,b,c$ on any three consecutive vertices which are arranged clockwise in such an order, we have $c=|a-b|$. Determine the maximum value of the sum of all numbers in terms of $n$.
2023 Canadian Mathematical Olympiad Qualification, 8
A point starts at the origin of the coordinate plane. Every minute, it either moves one unit in the $x$-direction or is rotated $\theta$ degrees counterclockwise about the origin.
(a) If $\theta = 90^o$, determine all locations where the point could end up.
(b) If $\theta = 45^o$, prove that for every location $ L$ in the coordinate plane and every positive number $\varepsilon$, there is a sequence of moves after which the point has distance less than $\varepsilon$ from $L$.
(c) Determine all rational numbers $\theta$ such that for every location $L$ in the coordinate plane and every positive number $\varepsilon$, there is a sequence of moves after which the point has distance less than $\varepsilon$ from $L$.
(d) Prove that when $\theta$ is irrational, for every location $L$ in the coordinate plane and every positive number $\varepsilon$, there is a sequence of moves after which the point has distance less than $\varepsilon$ from $L.$
1984 AMC 12/AHSME, 12
If the sequence $\{a_n\}$ is defined by \begin{align*}a_1 &= 2,\\ a_{n+1} &= a_n + 2n\qquad (n\geq 1),\end{align*} then $a_{100}$ equals
$\textbf{(A) }9900\qquad
\textbf{(B) }9902\qquad
\textbf{(C) }9904\qquad
\textbf{(D) }10100\qquad
\textbf{(E) }10102$
1988 IMO Longlists, 73
A two-person game is played with nine boxes arranged in a $3 \times 3$ square and with white and black stones. At each move a player puts three stones, not necessarily of the same colour, in three boxes in either a horizontal or a vertical line. No box can contain stones of different colours: if, for instance, a player puts a white stone in a box containing black stones the white stone and one of the black stones are removed from the box. The game is over when the centrebox and the cornerboxes contain one black stone and the other boxes are empty. At one stage of a game $x$ boxes contained one black stone each and the other boxes were empty. Determine all possible values for $x.$
May Olympiad L2 - geometry, 2008.4
In the plane we have $16$ lines(not parallel and not concurrents), we have $120$ point(s) of intersections of this lines.
Sebastian has to paint this $120$ points such that in each line all the painted points are with colour differents, find the minimum(quantity) of colour(s) that Sebastian needs to paint this points.
If we have have $15$ lines(in this situation we have $105$ points), what's the minimum(quantity) of colour(s)?
2017 Purple Comet Problems, 9
In $\triangle{ADE}$ points $B$ and $C$ are on side $AD$ and points $F$ and $G$ are on side $AE$ so that $BG \parallel CF \parallel DE$, as shown. The area of $\triangle{ABG}$ is $36$, the area of trapezoid $CFED$ is $144$, and $AB = CD$. Find the area of trapezoid $BGFC$.
[center][img]https://snag.gy/SIuOLB.jpg[/img][/center]
2000 Italy TST, 4
On a mathematical competition $ n$ problems were given. The final results showed that:
(i) on each problem, exactly three contestants scored $ 7$ points;
(ii) for each pair of problems, exactly one contestant scored $ 7$ points on both problems.
Prove that if $ n \geq 8$, then there is a contestant who got $ 7$ points on each problem. Is this statement necessarily true if $ n \equal{} 7$?
2005 AMC 10, 1
While eating out, Mike and Joe each tipped their server $ 2$ dollars. Mike tipped $ 10\%$ of his bill and Joe tipped $ 20\%$ of his bill. What was the difference, in dollars between their bills?
$ \textbf{(A)}\ 2\qquad
\textbf{(B)}\ 4\qquad
\textbf{(C)}\ 5\qquad
\textbf{(D)}\ 10\qquad
\textbf{(E)}\ 20$