Olympiad problems

1994 Bundeswettbewerb Mathematik, 3

Tags: geometry , 3d geometry , sphere , cone , touching

Let $A$ and $B$ be two spheres of different radii, both inscribed in a cone $K$. There are $m$ other, congruent spheres arranged in a ring such that each of them touches $A, B, K$ and two of the other spheres. Prove that this is possible for at most three values of $m.$

Kharkiv City MO Seniors - geometry, 2016.11.5

Tags: geometry , circumcenter , circles , equal segments

The circle $\omega$ passes through the vertices $B$ and $C$ of triangle $ABC$ and intersects its sides $AC,AB$ at points $A,E$, respectively. On the ray $BD$, a point $K$ such that $BK = AC$ is chosen , and on the ray $CE$, a point $L$ such that $CL = AB$ is chosen. Prove that the center $O$ of the circumscribed circle of the triangle $AKL$ lies on the circle $\omega$.

2023 Puerto Rico Team Selection Test, 2

Tags: geometry , perpendicular , perpendicularity

Let $I$ be the incenter of a triangle $ABC$ and let $D$ and $E$ be the touchpoints of the incircle with sides $BC$ and $AC$, respectively. The lines $DE$ and $BI$ intersect at point $P$. Prove that $AP$ is perpendicular to $BP$.

2009 Baltic Way, 5

Tags: function , algebra

Let $f_0=f_1=1$ and $f_{i+2}=f_{i+1}+f_i$ for all $n\ge 0$. Find all real solutions to the equation \[x^{2010}=f_{2009}\cdot x+f_{2008}\]

1996 Flanders Math Olympiad, 2

Tags: greatest common divisor , number theory

Determine the gcd of all numbers of the form $p^8-1$, with p a prime above 5.

1985 Polish MO Finals, 2

Tags: combinatorics , combinatorial geometry , tiling

Given a square side $1$ and $2n$ positive reals $a_1, b_1, ... , a_n, b_n$ each $\le 1$ and satisfying $\sum a_ib_i \ge 100$. Show that the square can be covered with rectangles $R_i$ with sides length $(a_i, b_i)$ parallel to the square sides.

1988 National High School Mathematics League, 2

Tags: ellipse , conic

If the coordinate origin is inside the ellipse $k^2x^2+y^2-4kx+2ky+k^2-1=0$, then the range value of $k$ is $\text{(A)}|k|>1\qquad\text{(B)}|k|\neq1\qquad\text{(C)}-1<k<1\qquad\text{(D)}0<|k|<1$

1951 Moscow Mathematical Olympiad, 190

Tags: algebra , compare

Which number is greater: $\frac{2.00 000 000 004}{(1.00 000 000 004)^2 + 2.00 000 000 004}$ or $\frac{2.00 000 000 002}{(1.00 000 000 002)^2 + 2.00 000 000 002}$ ?

2009 Tuymaada Olympiad, 4

Tags: floor function , inequalities , function , algebra

The sum of several non-negative numbers is not greater than 200, while the sum of their squares is not less than 2500. Prove that among them there are four numbers whose sum is not less than 50. [i]Proposed by A. Khabrov[/i]

2019 Bulgaria National Olympiad, 5

Tags: combinatorics , combinatorial geometry , geometry

Let $P$ be a $2019-$gon, such that no three of its diagonals concur at an internal point. We will call each internal intersection point of diagonals of $P$ a knot. What is the greatest number of knots one can choose, such that there doesn't exist a cycle of chosen knots? ( Every two adjacent knots in a cycle must be on the same diagonal and on every diagonal there are at most two knots from a cycle.)

2008 IMO Shortlist, 4

Tags: binomial coefficient , number theory , modular arithmetic

Let $ n$ be a positive integer. Show that the numbers \[ \binom{2^n \minus{} 1}{0},\; \binom{2^n \minus{} 1}{1},\; \binom{2^n \minus{} 1}{2},\; \ldots,\; \binom{2^n \minus{} 1}{2^{n \minus{} 1} \minus{} 1}\] are congruent modulo $ 2^n$ to $ 1$, $ 3$, $ 5$, $ \ldots$, $ 2^n \minus{} 1$ in some order. [i]Proposed by Duskan Dukic, Serbia[/i]

2014 Contests, 3

Tags: circumcircle , trapezoid , geometry

Let $ABC$ be an acute,non-isosceles triangle with $AB<AC<BC$.Let $D,E,Z$ be the midpoints of $BC,AC,AB$ respectively and segments $BK,CL$ are altitudes.In the extension of $DZ$ we take a point $M$ such that the parallel from $M$ to $KL$ crosses the extensions of $CA,BA,DE$ at $S,T,N$ respectively (we extend $CA$ to $A$-side and $BA$ to $A$-side and $DE$ to $E$-side).If the circumcirle $(c_{1})$ of $\triangle{MBD}$ crosses the line $DN$ at $R$ and the circumcirle $(c_{2})$ of $\triangle{NCD}$ crosses the line $DM$ at $P$ prove that $ST\parallel PR$.

2006 Sharygin Geometry Olympiad, 16

Tags: geometry , equilateral , equilateral triangle

Regular triangles are built on the sides of the triangle $ABC$. It turned out that their vertices form a regular triangle. Is the original triangle regular also?

1999 Kurschak Competition, 1

Tags: number theory

For any positive integer $m$, denote by $d_i(m)$ the number of positive divisors of $m$ that are congruent to $i$ modulo $2$. Prove that if $n$ is a positive integer, then \[\left|\sum_{k=1}^n \left(d_0(k)-d_1(k)\right)\right|\le n.\]

2000 JBMO ShortLists, 22

Tags: geometry

Consider a quadrilateral with $\angle DAB=60^{\circ}$, $\angle ABC=90^{\circ}$ and $\angle BCD=120^{\circ}$. The diagonals $AC$ and $BD$ intersect at $M$. If $MB=1$ and $MD=2$, find the area of the quadrilateral $ABCD$.

2013 Denmark MO - Mohr Contest, 4

Tags: number theory , perfect square

The positive integer $a$ is greater than $10$, and all its digits are equal. Prove that $a$ is not a perfect square. (A perfect square is a number which can be expressed as $n^2$ , where $n$ is an integer.)

2001 Romania National Olympiad, 1

Tags: algebra

Let $A$ be a set of real numbers which verifies: \[ a)\ 1 \in A \\ b) \ x\in A\implies x^2\in A\\ c)\ x^2-4x+4\in A\implies x\in A \] Show that $2000+\sqrt{2001}\in A$.

2017 Online Math Open Problems, 12

Tags:

Bill draws two circles which intersect at $X,Y$. Let $P$ be the intersection of the common tangents to the two circles and let $Q$ be a point on the line segment connecting the centers of the two circles such that lines $PX$ and $QX$ are perpendicular. Given that the radii of the two circles are $3,4$ and the distance between the centers of these two circles is $5$, then the largest distance from $Q$ to any point on either of the circles can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$. [i]Proposed by Tristan Shin[/i]

1998 Bosnia and Herzegovina Team Selection Test, 4

Tags: tangent , perpendicular , circles , geometry

Circle $k$ with radius $r$ touches the line $p$ in point $A$. Let $AB$ be a dimeter of circle and $C$ an arbitrary point of circle distinct from points $A$ and $B$. Let $D$ be a foot of perpendicular from point $C$ to line $AB$. Let $E$ be a point on extension of line $CD$, over point $D$, such that $ED=BC$. Let tangents on circle from point $E$ intersect line $p$ in points $K$ and $N$. Prove that length of $KN$ does not depend from $C$

1983 National High School Mathematics League, 11

Tags: geometry , 3d geometry , octahedron , inradius

For a regular hexahedron and a regular octahedron, all their faces are regular triangles, whose lengths of each side are $a$. Their inradius are $r_1,r_2$. $\frac{r_1}{r_2}=\frac{m}{n}, \gcd(m,n)=1$. Then $mn=$________.

2019 Jozsef Wildt International Math Competition, W. 51

Tags: inequalities

Let $a$, $b$, $c$, $d$, $e$ be real strictly positive real numbers such that $abcde = 1$. Then is true the following inequality:$$\frac{de}{a(b+1)}+\frac{ea}{b(c+1)}+\frac{ab}{c(d+1)}+\frac{bc}{d(e+1)}+\frac{cd}{e(a+1)}\geq \frac{5}{2}$$

2011 Bosnia Herzegovina Team Selection Test, 2

Tags: inequalities

Let $a, b, c$ be positive reals such that $a+b+c=1$. Prove that the inequality \[a \sqrt[3]{1+b-c} + b\sqrt[3]{1+c-a} + c\sqrt[3]{1+a-b} \leq 1\] holds.

2019 JBMO Shortlist, C4

Tags: combinatorics

We have a group of $n$ kids. For each pair of kids, at least one has sent a message to the other one. For each kid $A$, among the kids to whom $A$ has sent a message, exactly $25 \% $ have sent a message to $A$. How many possible two-digit values of $n$ are there? [i]Proposed by Bulgaria[/i]

2022 Sharygin Geometry Olympiad, 2

Tags: geometry , circumcircle

Let $ABCD$ be a curcumscribed quadrilateral with incenter $I$, and let $O_{1}, O_{2}$ be the circumcenters of triangles $AID$ and $CID$. Prove that the circumcenter of triangle $O_{1}IO_{2}$ lies on the bisector of angle $ABC$

2021 Brazil Team Selection Test, 3

Tags: algebra , polynomial , algorithm , lagrange s interpolation

A magician intends to perform the following trick. She announces a positive integer $n$, along with $2n$ real numbers $x_1 < \dots < x_{2n}$, to the audience. A member of the audience then secretly chooses a polynomial $P(x)$ of degree $n$ with real coefficients, computes the $2n$ values $P(x_1), \dots , P(x_{2n})$, and writes down these $2n$ values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick?