Olympiad problems

2016 Sharygin Geometry Olympiad, P7

Tags: geometry , combinatorics

Let all distances between the vertices of a convex $n$-gon ($n > 3$) be different. a) A vertex is called uninteresting if the closest vertex is adjacent to it. What is the minimal possible number of uninteresting vertices (for a given $n$)? b) A vertex is called unusual if the farthest vertex is adjacent to it. What is the maximal possible number of unusual vertices (for a given $n$)? [i](Proposed by B.Frenkin)[/i]

2019 Switzerland Team Selection Test, 3

Tags: algebra , IMO Shortlist , rational number , Sets

Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.

2005 AMC 8, 20

Tags: number theory , least common multiple , modular arithmetic

Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise. The game ends when they stop on the same point. How many turns will this take? $ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 24 $

2001 National Olympiad First Round, 20

Tags:

If the sum of any $10$ of $21$ real numbers is less than the sum of remaining $11$ of them, at least how many of these $21$ numbers are positive? $ \textbf{(A)}\ 18 \qquad\textbf{(B)}\ 19 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ \text{None of the preceding} $

2011 Korea Junior Math Olympiad, 7

Tags: inequalities , n-variable inequality , convex function

For those real numbers $x_1 , x_2 , \ldots , x_{2011}$ where each of which satisfies $0 \le x_1 \le 1$ ($i = 1 , 2 , \ldots , 2011$), find the maximum of \[ x_1^3+x_2^3+ \cdots + x_{2011}^3 - \left( x_1x_2x_3 + x_2x_3x_4 + \cdots + x_{2011}x_1x_2 \right) \]

2010 Abels Math Contest (Norwegian MO) Final, 2b

Tags: algebra , inequalities

Show that $abc \le (ab + bc + ca)(a^2 + b^2 + c^2)^2$ for all positive real numbers $a, b$ and $c$ such that $a + b + c = 1$.

2014 Stars Of Mathematics, 2

Tags: number theory

Let $N$ be an arbitrary positive integer. Prove that if, from among any $n$ consecutive integers larger than $N$, one may select $7$ of them, pairwise co-prime, then $n\geq 22$. ([i]Dan Schwarz[/i])

1969 IMO, 4

Tags: geometry , circumcircle , reflection , tangent , IMO , IMO 1969

$C$ is a point on the semicircle diameter $AB$, between $A$ and $B$. $D$ is the foot of the perpendicular from $C$ to $AB$. The circle $K_1$ is the incircle of $ABC$, the circle $K_2$ touches $CD,DA$ and the semicircle, the circle $K_3$ touches $CD,DB$ and the semicircle. Prove that $K_1,K_2$ and $K_3$ have another common tangent apart from $AB$.

2013 Bosnia Herzegovina Team Selection Test, 5

Tags: inequalities proposed , inequalities

Let $x_1,x_2,\ldots,x_n$ be nonnegative real numbers of sum equal to $1$. Let $F_n=x_1^{2}+x_2^{2}+\cdots +x_n^{2}-2(x_1x_2+x_2x_3+\cdots +x_nx_1)$. Find: a) $\min F_3$; b) $\min F_4$; c) $\min F_5$.

2006 China Team Selection Test, 3

Tags: algebra , polynomial , pigeonhole principle , algebra unsolved

$k$ and $n$ are positive integers that are greater than $1$. $N$ is the set of positive integers. $A_1, A_2, \cdots A_k$ are pairwise not-intersecting subsets of $N$ and $A_1 \cup A_2 \cup \cdots \cup A_k = N$. Prove that for some $i \in \{ 1,2,\cdots,k \}$, there exsits infinity many non-factorable n-th degree polynomials so that coefficients of one polynomial are pairwise distinct and all the coeficients are in $A_i$.

2020 Yasinsky Geometry Olympiad, 6

Tags: geometry , concurrency , concurrent , incircle , isosceles , circumcircle

In an isosceles triangle $ABC, I$ is the center of the inscribed circle, $M_1$ is the midpoint of the side $BC, K_2, K_3$ are the points of contact of the inscribed circle of the triangle with segments $AC$ and $AB$, respectively. The point $P$ lies on the circumcircle of the triangle $BCI$, and the angle $M_1PI$ is right. Prove that the lines $BC, PI, K_2K_3$ intersect at one point. (Mikhail Plotnikov)

2009 Serbia Team Selection Test, 1

Tags: inequalities , trigonometry , calculus , derivative , geometry , angle bisector , geometry unsolved

Let $ \alpha$ and $ \beta$ be the angles of a non-isosceles triangle $ ABC$ at points $ A$ and $ B$, respectively. Let the bisectors of these angles intersect opposing sides of the triangle in $ D$ and $ E$, respectively. Prove that the acute angle between the lines $ DE$ and $ AB$ isn't greater than $ \frac{|\alpha\minus{}\beta|}3$.

2013 IFYM, Sozopol, 1

Tags: geometry , collinear , Pascal s theorem

Let point $T$ be on side $AB$ of $\Delta ABC$ be such that $AT-BT=AC-BC$. The perpendicular from point $T$ to $AB$ intersects $AC$ in point $E$ and the angle bisectors of $\angle B$ and $\angle C$ intersect the circumscribed circle $k$ of $ABC$ in points $M$ and $L$. If $P$ is the second intersection point of the line $ME$ with $k$, then prove that $P,T,L$ are collinear.

2016 CCA Math Bonanza, L2.2

Tags: CCA Math Bonanza

In triangle $ABC$, $AB=7$, $AC=9$, and $BC=8$. The angle bisector of $\angle{BAC}$ intersects side $BC$ at $D$, and the angle bisector of $\angle{ABC}$ intersects $AD$ at $E$. Compute $AE^2$. [i]2016 CCA Math Bonanza Lightning #2.2[/i]

1976 IMO Longlists, 21

Tags: inequalities , inequalities unsolved

Find the largest positive real number $p$ (if it exists) such that the inequality \[x^2_1+ x_2^2+ \cdots + x^2_n\ge p(x_1x_2 + x_2x_3 + \cdots + x_{n-1}x_n)\] is satisfied for all real numbers $x_i$, and $(a) n = 2; (b) n = 5.$ Find the largest positive real number $p$ (if it exists) such that the inequality holds for all real numbers $x_i$ and all natural numbers $n, n \ge 2.$

2018 Pan-African Shortlist, G3

Tags: geometry , cyclic quadrilateral , Angle Chasing

Given a triangle $ABC$, let $D$ be the intersection of the line through $A$ perpendicular to $AB$, and the line through $B$ perpendicular to $BC$. Let $P$ be a point inside the triangle. Show that $DAPB$ is cyclic if and only if $\angle BAP = \angle CBP$.

1968 AMC 12/AHSME, 15

Tags: AMC

Let $P$ be the product of any three consecutive positive odd integers. The largest integer dividing all such $P$ is: $\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 1 $

2009 Balkan MO Shortlist, G3

Tags: diagonals , projections , geometry , midpoints , convex quadrilateral , Circumcenter , Isogonal conjugate

Let $ABCD$ be a convex quadrilateral, and $P$ be a point in its interior. The projections of $P$ on the sides of the quadrilateral lie on a circle with center $O$. Show that $O$ lies on the line through the midpoints of $AC$ and $BD$.

2016 Ecuador Juniors, 2

Tags: diophantine , Diophantine equation

Prove that there are no positive integers $x, y$ such that: $(x + 1)^2 + (x + 2)^2 +...+ (x + 9)^2 = y^2$

1995 All-Russian Olympiad Regional Round, 10.6

Tags: geometry , trapezoid , vector , geometry proposed

Let a quardilateral $ABCD$ with $AB=AD$ and $\widehat B=\widehat D=90$. At $CD$ we take point $E$ and at $BC$ we take point $Z$ such that $AE\bot DZ$. Prove that $AZ\bot BE$

2007 Moldova National Olympiad, 11.4

Tags: function , algebra unsolved , algebra

The function $f: \mathbb{R}\rightarrow\mathbb{R}$ satisfies $f(\textrm{cot}x)=\sin2x+\cos2x$, for any $x\in(0,\pi)$. Find the minimum and maximum value of $g: [-1;1]\rightarrow\mathbb{R}$, $g(x)=f(x)\cdot f(1-x)$.

MBMT Team Rounds, 2020.1