Olympiad problems

2020 Tuymaada Olympiad, 5

Tags: parabola , construction , geometry , algebra , analytic geometry , quadratic , conic

Coordinate axes (without any marks, with the same scale) and the graph of a quadratic trinomial $y = x^2 + ax + b$ are drawn in the plane. The numbers $a$ and $b$ are not known. How to draw a unit segment using only ruler and compass?

1978 IMO Longlists, 44

Tags: trigonometry , inequalities

In $ABC$ with $\angle C = 60^{\circ}$, prove that \[\frac{c}{a} + \frac{c}{b} \ge2.\]

2011 Spain Mathematical Olympiad, 1

Tags: combinatorics

Each pair of vertices of a regular $67$-gon is joined by a line segment. Suppose $n$ of these segments are selected, and each of them is painted one of ten available colors. Find the minimum possible value of $n$ for which, regardless of which $n$ segments were selected and how they were painted, there will always be a vertex of the polygon that belongs to seven segments of the same color.

2001 Slovenia National Olympiad, Problem 3

Tags: geometry

For an arbitrary point $P$ on a given segment $AB$, two isosceles right triangles $APQ$ and $PBR$ with the right angles at $Q$ and $R$ are constructed on the same side of the line $AB$. Prove that the distance from the midpoint $M$ of $QR$ to the line $AB$ does not depend on the choice of $P$.

VI Soros Olympiad 1999 - 2000 (Russia), 9.7

Tags: ratio , geometry

Points $A, B, C$ and $D$ are located on line $\ell$ so that $\frac{AB}{BC}=\frac{AC}{CD}=\lambda $. A certain circle is tangent to line $\ell$ at point $C$. A line is drawn through $A$ that intersects this circle at points $M$ and $N$ such that the bisector perpendiculars to segments $BM$ and $DN$ intersect at point $Q$ on line $\ell$ . In what ratio does point $Q$ divide segment $AD$?

2011 All-Russian Olympiad, 3

Tags: induction , floor function , combinatorics

A convex 2011-gon is drawn on the board. Peter keeps drawing its diagonals in such a way, that each newly drawn diagonal intersected no more than one of the already drawn diagonals. What is the greatest number of diagonals that Peter can draw?

2010 Contests, 2

Tags: linear algebra , matrix

How many ordered pairs of positive integers $(x,y)$ are there such that $y^2-x^2=2y+7x+4$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0 \qquad\textbf{(E)}\ \text{Infinitely many} $

Ukrainian From Tasks to Tasks - geometry, 2015.5

Tags: geometry , analytic geometry , coordinate geometry

A coordinate system was constructed on the board, points $A (1,2)$ and B $(3, 1)$ were marked, and then the coordinate system was erased. Restore the coordinate system at the two marked points.

2022 Malaysian IMO Team Selection Test, 6

Tags: geometry

Given a triangle $ABC$ with $AB=AC$ and circumcenter $O$. Let $D$ and $E$ be midpoints of $AC$ and $AB$ respectively, and let $DE$ intersect $AO$ at $F$. Denote $\omega$ to be the circle $(BOE)$. Let $BD$ intersect $\omega$ again at $X$ and let $AX$ intersect $\omega$ again at $Y$. Suppose the line parallel to $AB$ passing through $O$ meets $CY$ at $Z$. Prove that the lines $FX$ and $BZ$ meet at $\omega$. [i]Proposed by Ivan Chan Kai Chin[/i]

2021 Korea Junior Math Olympiad, 3

Tags: geometry , cyclic quadrilateral , circumcircle , perpendicular bisector , tangent

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\Omega$ and let diagonals $AC$ and $BD$ intersect at $X$. Suppose that $AEFB$ is inscribed in a circumcircle of triangle $ABX$ such that $EF$ and $AB$ are parallel. $FX$ meets the circumcircle of triangle $CDX$ again at $G$. Let $EX$ meets $AB$ at $P$, and $XG$ meets $CD$ at $Q$. Denote by $S$ the intersection of the perpendicular bisector of $\overline{EG}$ and $\Omega$ such that $S$ is closer to $A$ than $B$. Prove that line through $S$ parallel to $PQ$ is tangent to $\Omega$.

2017 CCA Math Bonanza, L4.2

Tags:

Find $\arctan\left(1\right)+\arctan\left(2\right)+\arctan\left(3\right)$ in radians. [i]2017 CCA Math Bonanza Lightning Round #4.2[/i]

2017 Romania Team Selection Test, P3

Tags: geometry

Let $ABCD$ be a convex quadrilateral with $\angle ABC = \angle ADC < 90^{\circ}$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $E$ and $F$ respectively, and meet each other at point $P$. Let $M$ be the midpoint of $AC$ and let $\omega$ be the circumcircle of triangle $BPD$. Segments $BM$ and $DM$ intersect $\omega$ again at $X$ and $Y$ respectively. Denote by $Q$ the intersection point of lines $XE$ and $YF$. Prove that $PQ \perp AC$.

2018 Pan-African Shortlist, A4

Tags: inequalities , algebra

Let $a$, $b$, $c$ and $d$ be non-zero pairwise different real numbers such that $$ \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4 \text{ and } ac = bd. $$ Show that $$ \frac{a}{c} + \frac{b}{d} + \frac{c}{a} + \frac{d}{b} \leq -12 $$ and that $-12$ is the maximum.

1984 IMO Longlists, 67

Tags: geometry , circumcircle , inequalities

With the medians of an acute-angled triangle another triangle is constructed. If $R$ and $R_m$ are the radii of the circles circumscribed about the first and the second triangle, respectively, prove that \[R_m>\frac{5}{6}R\]

2017 HMNT, 7

Tags: combinatorics

Reimu has a wooden cube. In each step, she creates a new polyhedron from the previous one by cutting off a pyramid from each vertex of the polyhedron along a plane through the trisection point on each adjacent edge that is closer to the vertex. For example, the polyhedron after the first step has six octagonal faces and eight equilateral triangular faces. How many faces are on the polyhedron after the fifth step?

2001 Croatia National Olympiad, Problem 4

Tags: number theory

Let $S$ be a set of $100$ positive integers less than $200$. Prove that there exists a nonempty subset $T$ of $S$ the product of whose elements is a perfect square.

2018 Oral Moscow Geometry Olympiad, 3

Tags: orthocenter , fixed , geometry , circles

A circle is fixed, point $A$ is on it and point $K$ outside the circle. The secant passing through $K$ intersects circle at points $P$ and $Q$. Prove that the orthocenters of the triangle $APQ$ lie on a fixed circle.

1966 AMC 12/AHSME, 28

Tags:

Five points $O,A,B,C,D$ are taken in order on a straight line with distances $OA=a$, $OB=b$, $OC=c$, and $OD=d$. $P$ is a point on the line between $B$ and $C$ and such that $AP:PD=BP:PC$. Then $OP$ equals: $\text{(A)}\ \dfrac{b^2-bc}{a-b+c-d} \qquad \text{(B)}\ \dfrac{ac-b}{a-b+c-d} \qquad \text{(C)}\ -\dfrac{bd+c}{a-b+c-d}\qquad\\ \text{(D)}\ \dfrac{bc+ad}{a+b+c+d}\qquad \text{(E)}\ \dfrac{ac-bd}{a+b+c+d} \qquad$

2002 HKIMO Preliminary Selection Contest, 9

Tags: algebra

Let $x_1,x_2,y_1,y_2$ be real numbers satisfying the equations $x^2_1+5x^2_2=10$, $x_2y_1-x_1y_2=5$, and $x_1y_1+5x_2y_2=\sqrt{105}$. Find the value of $y_1^2+5y_2^2$

2015 Greece Team Selection Test, 2

Tags: combinatorics

Consider $111$ distinct points which lie on or in the internal of a circle with radius 1.Prove that there are at least $1998$ segments formed by these points with length $\leq \sqrt{3}$

2003 China Team Selection Test, 1

Tags: trigonometry , inequalities

Let $g(x)= \sum_{k=1}^{n} a_k \cos{kx}$, $a_1,a_2, \cdots, a_n, x \in R$. If $g(x) \geq -1$ holds for every $x \in R$, prove that $\sum_{k=1}^{n}a_k \leq n$.

2012 NIMO Problems, 10

Tags: geometry , incenter , trigonometry , circumcircle , perimeter , ratio , quadratic

In cyclic quadrilateral $ABXC$, $\measuredangle XAB = \measuredangle XAC$. Denote by $I$ the incenter of $\triangle ABC$ and by $D$ the projection of $I$ on $\overline{BC}$. If $AI = 25$, $ID = 7$, and $BC = 14$, then $XI$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$. [i]Proposed by Aaron Lin[/i]

2006 China Team Selection Test, 1

Tags: induction , ratio , inequalities , limit , algebra

Two positive valued sequences $\{ a_{n}\}$ and $\{ b_{n}\}$ satisfy: (a): $a_{0}=1 \geq a_{1}$, $a_{n}(b_{n+1}+b_{n-1})=a_{n-1}b_{n-1}+a_{n+1}b_{n+1}$, $n \geq 1$. (b): $\sum_{i=1}^{n}b_{i}\leq n^{\frac{3}{2}}$, $n \geq 1$. Find the general term of $\{ a_{n}\}$.

2012 Kazakhstan National Olympiad, 1

Tags: modular arithmetic , number theory

The number $\overline{13\ldots 3}$, with $k>1$ digits $3$, is a prime. Prove that $6\mid k^{2}-2k+3$.

2000 All-Russian Olympiad, 2

Tags: n-variable inequality , abel formula , inequalities

Let $-1 < x_1 < x_2 , \cdots < x_n < 1$ and $x_1^{13} + x_2^{13} + \cdots + x_n^{13} = x_1 + x_2 + \cdots + x_n$. Prove that if $y_1 < y_2 < \cdots < y_n$, then \[ x_1^{13}y_1 + \cdots + x_n^{13}y_n < x_1y_1 + x_2y_2 + \cdots + x_ny_n. \]