Olympiad problems

2004 USAMO, 3

Tags: geometry , ratio , function , calculus , quadratic

For what real values of $k>0$ is it possible to dissect a $1 \times k$ rectangle into two similar, but noncongruent, polygons?

2022 SG Originals, Q1

Tags: geometry , tangent , symmedian , circumcircle

For $\triangle ABC$ and its circumcircle $\omega$, draw the tangents at $B,C$ to $\omega$ meeting at $D$. Let the line $AD$ meet the circle with center $D$ and radius $DB$ at $E$ inside $\triangle ABC$. Let $F$ be the point on the extension of $EB$ and $G$ be the point on the segment $EC$ such that $\angle AFB=\angle AGE=\angle A$. Prove that the tangent at $A$ to the circumcircle of $\triangle AFG$ is parallel to $BC$. [i]Proposed by 61plus[/i]

Denmark (Mohr) - geometry, 2003.4

Tags: geometry , max , circles

Georg and his mother love pizza. They buy a pizza shaped as an equilateral triangle. Georg demands to be allowed to divide the pizza by a straight cut and then make the first choice. The mother accepts this reluctantly, but she wants to choose a point of the pizza through which the cut must pass. Determine the largest fraction of the pizza which the mother is certain to get by this procedure.

1894 Eotvos Mathematical Competition, 1

Tags: algebra

Let $x$ and $y$ be integers. Prove that one of the expressions $$2x+3y \text{ and } 9x+5y$$ is divisible by $17$ if and only if so is the other.

2016 BMT Spring, 4

Tags: graph theory , combinatorics

How many graphs are there on $6$ vertices with degrees $1,1,2,3,4,5$?

2023 Germany Team Selection Test, 1

Tags: algebra , polynomial

Let $P$ be a polynomial with integer coefficients. Assume that there exists a positive integer $n$ with $P(n^2)=2022$. Prove that there cannot be a positive rational number $r$ with $P(r^2)=2024$.

2007 Bulgarian Autumn Math Competition, Problem 11.2

Tags: inequalities , parameter , algebra

Find all values of the parameter $a$ for which the inequality \[\sqrt{x-x^2-a}+\sqrt{6a-2x-x^2}\leq \sqrt{10a-2x-4x^2}\] has a unique solution.

2022 MMATHS, 5

Tags: geometry

Equilateral triangle $\vartriangle ABC$ has side length $6$. Points $D$ and $E$ lie on $\overline{BC}$ such that $BD = CE$ and $B$, $D$, $E$, $C$ are collinear in that order. Points $F$ and $G$ lie on $\overline{AB}$ such that $\overline{FD} \perp \overline{BC}$, and $GF = GA$. If the minimum possible value of the sum of the areas of $\vartriangle BFD$ and $\vartriangle DGE$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b, c$ with $gcd (a, c) = 1$ and $b$ squarefree, find $a + b + c$.

Novosibirsk Oral Geo Oly IX, 2020.7

Tags: tangent , cyclic , geometry

The quadrilateral $ABCD$ is known to be inscribed in a circle, and that there is a circle with center on side $AD$ tangent to the other three sides. Prove that $AD = AB + CD$.

2014 Oral Moscow Geometry Olympiad, 3

Tags: geometry , convex , pentagon , diagonal

Is there a convex pentagon in which each diagonal is equal to a side?

2013 ELMO Shortlist, 4

Tags: number theory

Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$. [i]Proposed by Evan Chen[/i]

2001 Tuymaada Olympiad, 1

Tags: point , combinatorics , game

$16$ chess players held a tournament among themselves: every two chess players played exactly one game. For victory in the party was given $1$ point, for a draw $0.5$ points, for defeat $0$ points. It turned out that exactly 15 chess players shared the first place. How many points could the sixteenth chess player score?

1979 IMO Longlists, 35

Tags: geometry , induction , area of a triangle , heron's formula , inequalities

Given a sequence $(a_n)$, with $a_1 = 4$ and $a_{n+1} = a_n^2-2 (\forall n \in\mathbb{N})$, prove that there is a triangle with side lengths $a_{n-1}, a_n, a_{n+1},$ and that its area is equal to an integer.

2022 Junior Balkan Team Selection Tests - Moldova, 5

Tags: number theory

Determine all nonzero natural numbers $n$, for which the number $\sqrt{n! + 5}$ is a natural number.

2022 Kyiv City MO Round 2, Problem 4

Tags: geometry

Points $D, E, F$ are selected on sides $BC, CA, AB$ correspondingly of triangle $ABC$ with $\angle C = 90^\circ$ such that $\angle DAB = \angle CBE$ and $\angle BEC = \angle AEF$. Show that $DB = DF$. [i](Proposed by Mykhailo Shtandenko)[/i]

2022 Irish Math Olympiad, 6

Tags: algebraic manipulations , algebra , inequalities

6. Suppose [i]a[/i], [i]b[/i], [i]c[/i] are real numbers such that [i]a[/i] + [i]b[/i] + [i]c[/i] = 1. Prove that \[a^3 + b^3 + c^3 + 3(1-a)(1-b)(1-c) = 1.\]

2010 National Olympiad First Round, 22

Tags:

How many pairs of integers $(x,y)$ are there such that $\frac{x}{y+7}+\frac{y}{x+7}=1$? $ \textbf{(A)}\ 18 \qquad\textbf{(B)}\ 17 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 11 $

1969 Miklós Schweitzer, 6

Tags: function , integration , complex analysis

Let $ x_0$ be a fixed real number, and let $ f$ be a regular complex function in the half-plane $ \Re z>x_0$ for which there exists a nonnegative function $ F \in L_1(- \infty, +\infty)$ satisfying $ |f(\alpha+i\beta)| \leq F(\beta)$ whenever $ \alpha > x_0$ , $ -\infty <\beta < +\infty$. Prove that \[ \int_{\alpha-i \infty} ^{\alpha+i \infty} f(z)dz=0.\] [i]L. Czach[/i]

1953 AMC 12/AHSME, 5

Tags: logarithm

If $ \log_6 x\equal{}2.5$, the value of $ x$ is: $ \textbf{(A)}\ 90 \qquad\textbf{(B)}\ 36 \qquad\textbf{(C)}\ 36\sqrt{6} \qquad\textbf{(D)}\ 0.5 \qquad\textbf{(E)}\ \text{none of these}$

2007 Tournament Of Towns, 4

Tags: number theory , combinatorics , invariant

There three piles of pebbles, containing 5, 49, and 51 pebbles respectively. It is allowed to combine any two piles into a new one or to split any pile consisting of even number of pebbles into two equal piles. Is it possible to have 105 piles with one pebble in each in the end? [i](3 points)[/i]

2016 IFYM, Sozopol, 4

Tags: algebra

Prove that for each $n\geq 3$ the equation: $x^n+y^n+z^n+u^n=v^{n-1}$ has infinitely many solutions in natural numbers.

2015 PAMO, Problem 3

Tags: combinatorial number theory , number theory

Let $a_1,a_2,...,a_{11}$ be integers. Prove that there are numbers $b_1,b_2,...,b_{11}$, each $b_i$ equal $-1,0$ or $1$, but not all being $0$, such that the number $$N=a_1b_1+a_2b_2+...+a_{11}b_{11}$$ is divisible by $2015$.

2012 IMO Shortlist, N4

Tags: modular arithmetic , number theory , equation

An integer $a$ is called friendly if the equation $(m^2+n)(n^2+m)=a(m-n)^3$ has a solution over the positive integers. [b]a)[/b] Prove that there are at least $500$ friendly integers in the set $\{ 1,2,\ldots ,2012\}$. [b]b)[/b] Decide whether $a=2$ is friendly.

1998 Mediterranean Mathematics Olympiad, 3

Tags: incenter , geometry

In a triangle $ABC$, $I$ is the incenter and $D,E, F$ are the points of tangency of the incircle with $BC,CA,AB$, respectively. The bisector of angle $BIC$ meets $BC$ at $M$, and the line $AM$ intersects $EF$ at $P$. Prove that $DP$ bisects the angle $FDE$.

2014 Math Prize For Girls Problems, 13

Tags: probability

Deepali has a bag containing 10 red marbles and 10 blue marbles (and nothing else). She removes a random marble from the bag. She keeps doing so until all of the marbles remaining in the bag have the same color. Compute the probability that Deepali ends with exactly 3 marbles remaining in the bag.