Olympiad problems

Ukrainian TYM Qualifying - geometry, IV.7

Tags: min , geometric inequality , inscribed , max , perimeter , geometry

Let $ABCD$ be the quadrilateral whose area is the largest among the quadrilaterals with given sides $a, b, c, d$, and let $PORS$ be the quadrilateral inscribed in $ABCD$ with the smallest perimeter. Find this perimeter.

2020 Bulgaria Team Selection Test, 2

Tags: algebra , number theory

Given two odd natural numbers $ a,b$ prove that for each $ n\in\mathbb{N}$ there exists $ m\in\mathbb{N}$ such that either $ a^mb^2-1$ or $ b^ma^2-1$ is multiple of $ 2^n.$

2012 Kyrgyzstan National Olympiad, 5

Tags: number theory , modular arithmetic

The sequence of natural numbers is defined as follows: for any $ k\geq 1 $,$ a_{k+2}= a_{k+1}\cdot a_k+1 $. Prove that for $ k\geq 9 $ the number $ a_k-22 $ is composite.

2018 India National Olympiad, 4

Tags: divisibility , roots of a polynomial , polynomial , algebra

Find all polynomials with real coefficients $P(x)$ such that $P(x^2+x+1)$ divides $P(x^3-1)$.

2009 Kazakhstan National Olympiad, 5

Tags: geometry

Quadrilateral $ABCD$ inscribed in circle with center $O$. Let lines $AD$ and $BC$ intersects at $M$, lines $AB$ and $CD$- at $N$, lines $AC$ and $BD$ -at $P$, lines $OP$ and $MN$ at $K$. Proved that $ \angle AKP = \angle PKC$. As I know, this problem was very short solution by polars, but in olympiad for this solution gives maximum 4 balls (in marking schemes written, that needs to prove all theorems about properties of polars)

2018 Stanford Mathematics Tournament, 2

Tags: geometry , 3d geometry

What is the largest possible height of a right cylinder with radius $3$ that can fit in a cube with side length $12$?

2012 Irish Math Olympiad, 2

Tags: circumcircle , geometry , angle bisector

Consider a triangle $ABC$ with $|AB|\neq |AC|$. The angle bisector of the angle $CAB$ intersects the circumcircle of $\triangle ABC$ at two points $A$ and $D$. The circle of center $D$ and radius $|DC|$ intersects the line $AC$ at two points $C$ and $B’$. The line $BB’$ intersects the circumcircle of $\triangle ABC$ at $B$ and $E$. Prove that $B’$ is the orthocenter of $\triangle AED$.

2011 South East Mathematical Olympiad, 4

Tags: circumcircle , projective geometry , geometry

Let $O$ be the circumcenter of triangle $ABC$ , a line passes through $O$ intersects sides $AB,AC$ at points $M,N$ , $E$ is the midpoint of $MC$ , $F$ is the midpoint of $NB$ , prove that : $\angle FOE= \angle BAC$

2001 AIME Problems, 9

Tags: probability , rectangle , geometry , relatively prime , number theory

Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2015 AMC 8, 10

Tags:

How many integers between $1000$ and $9999$ have four distinct digits? $\textbf{(A) }3024\qquad\textbf{(B) }4536\qquad\textbf{(C) }5040\qquad\textbf{(D) }6480\qquad \textbf{(E) }6561$

2023 Simon Marais Mathematical Competition, B4

Tags: number theory

[i](The following problem is open in the sense that the answer to part (b) is not currently known.)[/i] [list=a] [*] Let $n$ be a positive integer that is not a perfect square. Find all pairs $(a,b)$ of positive integers for which there exists a positive real number $r$, such that $$r^a+\sqrt{n} \ \ \text{and} \ \ r^b+\sqrt{n}$$ are both rational numbers. [*] Let $n$ be a positive integer that is not a perfect square. Find all pairs $(a,b)$ of positive integers for which there exists a real number $r$, such that $$r^a+\sqrt{n} \ \ \text{and} \ \ r^b+\sqrt{n}$$ are both rational numbers. [/list]

2020 BMT Fall, 10

Tags: number theory , floor function

How many integers $100 \le x \le 999$ have the property that, among the six digits in $\lfloor 280 + \frac{x}{100} \rfloor$ and $x$, exactly two are identical?

2000 Turkey Junior National Olympiad, 1

Tags: angle bisector , geometry

Let $ABC$ be a triangle with $\angle BAC = 90^\circ$. Construct the square $BDEC$ such as $A$ and the square are at opposite sides of $BC$. Let the angle bisector of $\angle BAC$ cut the sides $[BC]$ and $[DE]$ at $F$ and $G$, respectively. If $|AB|=24$ and $|AC|=10$, calculate the area of quadrilateral $BDGF$.

1994 India Regional Mathematical Olympiad, 8

Tags: inequalities

If $a,b,c$ are positive real numbers such that $a+b+c = 1$, prove that \[ (1+a)(1+b)(1+c) \geq 8 (1-a)(1-b)(1-c) . \]

2023 Switzerland - Final Round, 3

Tags: greatest common divisor , number theory

Let $x,y$ and $a_0, a_1, a_2, \cdots $ be integers satisfying $a_0 = a_1 = 0$, and $$a_{n+2} = xa_{n+1}+ya_n+1$$for all integers $n \geq 0$. Let $p$ be any prime number. Show that $\gcd(a_p,a_{p+1})$ is either equal to $1$ or greater than $\sqrt{p}$.

2008 Thailand Mathematical Olympiad, 8

Tags: perfect square , number theory

Prove that $2551 \cdot 543^n -2008\cdot 7^n$ is never a perfect square, where $n$ varies over the set of positive integers

2014 Danube Mathematical Competition, 4

Tags: combinatorics , cardinality , lattice points , maximal , parallelogram , geometry , combinatorial geometry

Let $n$ be a positive integer and let $\triangle$ be the closed triangular domain with vertices at the lattice points $(0, 0), (n, 0)$ and $(0, n)$. Determine the maximal cardinality a set $S$ of lattice points in $\triangle$ may have, if the line through every pair of distinct points in $S$ is parallel to no side of $\triangle$.

2010 Germany Team Selection Test, 2

Tags: reflection , geometry , incenter , inequalities

Let $ABC$ be a triangle with incenter $I$ and let $X$, $Y$ and $Z$ be the incenters of the triangles $BIC$, $CIA$ and $AIB$, respectively. Let the triangle $XYZ$ be equilateral. Prove that $ABC$ is equilateral too. [i]Proposed by Mirsaleh Bahavarnia, Iran[/i]

1990 India National Olympiad, 1

Tags: algebra , equation

Given the equation \[ x^4 \plus{} px^3 \plus{} qx^2 \plus{} rx \plus{} s \equal{} 0\] has four real, positive roots, prove that (a) $ pr \minus{} 16s \geq 0$ (b) $ q^2 \minus{} 36s \geq 0$ with equality in each case holding if and only if the four roots are equal.

2011 NIMO Summer Contest, 10

Tags:

The edges and diagonals of convex pentagon $ABCDE$ are all colored either red or blue. How many ways are there to color the segments such that there is exactly one monochromatic triangle with vertices among $A$, $B$, $C$, $D$, $E$; that is, triangles, whose edges are all the same color? [i]Proposed by Eugene Chen [/i]

1954 Moscow Mathematical Olympiad, 278

Tags: combinatorics , combinatorial geometry , parallel , square table

A $17 \times 17$ square is cut out of a sheet of graph paper. Each cell of this square has one of thenumbers from $1$ to $70$. Prove that there are $4$ distinct squares whose centers $A, B, C, D$ are the vertices of a parallelogramsuch that $AB // CD$, moreover, the sum of the numbers in the squares with centers $A$ and $C$ is equal to that in the squares with centers $B$ and $D$.

2017 Sharygin Geometry Olympiad, 1

Tags: circumcircle , geometry

If two circles intersect at $A,B$ and common tangents of them intesrsect circles at $C,D$if $O_a$is circumcentre of $ACD$ and $O_b$ is circumcentre of $BCD$ prove $AB$ intersects $O_aO_b$ at its midpoint

2009 All-Russian Olympiad, 3

Tags: symmetry , 3d geometry , pyramid , perpendicular bisector , sphere , geometry , circumcircle

Let $ ABCD$ be a triangular pyramid such that no face of the pyramid is a right triangle and the orthocenters of triangles $ ABC$, $ ABD$, and $ ACD$ are collinear. Prove that the center of the sphere circumscribed to the pyramid lies on the plane passing through the midpoints of $ AB$, $ AC$ and $ AD$.

2018 Polish Junior MO First Round, 6

Tags: equation , number theory , algebra

Positive integers $k, m, n$ satisfy the equation $m^2 + n = k^2 + k$. Show that $m \le n$.

2024 Caucasus Mathematical Olympiad, 3

Tags: number theory

Given $10$ positive integers with a sum equal to $1000$. The product of their factorials is a $10$-th power of an integer. Prove that all these numbers are equal.