Olympiad problems

2010 Iran MO (3rd Round), 3

prove that for each natural number $n$ there exist a polynomial with degree $2n+1$ with coefficients in $\mathbb{Q}[x]$ such that it has exactly $2$ complex zeros and it's irreducible in $\mathbb{Q}[x]$.(20 points)

2011 Sharygin Geometry Olympiad, 22

Tags: geometry , euler circle , tangent , circumcircle , concurrency

Let $CX, CY$ be the tangents from vertex $C$ of triangle $ABC$ to the circle passing through the midpoints of its sides. Prove that lines $XY , AB$ and the tangent to the circumcircle of $ABC$ at point $C$ concur.

1982 IMO Shortlist, 3

Tags: limit , geometric series , inequality , minimization , geometric sequence

Consider infinite sequences $\{x_n\}$ of positive reals such that $x_0=1$ and $x_0\ge x_1\ge x_2\ge\ldots$. [b]a)[/b] Prove that for every such sequence there is an $n\ge1$ such that: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}\ge3.999. \] [b]b)[/b] Find such a sequence such that for all $n$: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}<4. \]

1999 Irish Math Olympiad, 4

Tags: number theory

Find all positive integers $ m$ with the property that the fourth power of the number of (positive) divisors of $ m$ equals $ m$.

2022 China Team Selection Test, 2

Tags: prime number , number theory , combinatorics

Let $p$ be a prime, $A$ is an infinite set of integers. Prove that there is a subset $B$ of $A$ with $2p-2$ elements, such that the arithmetic mean of any pairwise distinct $p$ elements in $B$ does not belong to $A$.

2016 Swedish Mathematical Competition, 1

Tags: geometry , max , area

In a garden there is an $L$-shaped fence, see figure. You also have at your disposal two finished straight fence sections that are $13$ m and $14$ m long respectively. From point $A$ you want to delimit a part of the garden with an area of at least $200$ m$^2$ . Is it possible to do this? [img]https://1.bp.blogspot.com/-VLWIImY7HBA/X0yZq5BrkTI/AAAAAAAAMbg/8CyP6DzfZTE5iX01Qab3HVrTmaUQ7PvcwCK4BGAYYCw/s400/sweden%2B16p1.png[/img]

1985 Austrian-Polish Competition, 8

Tags: triangulation , polygon , combinatorics

A convex $n$-gon $A_0A_1\dots A_{n-1}$ has been partitioned into $n-2$ triangles by certain diagonals not intersecting inside the $n$-gon. Prove that these triangles can be labeled $\triangle_1,\triangle_2,\dots,\triangle_{n-2}$ in such a way that $A_i$ is a vertex of $\triangle_i$, for $i=1,2,\dots,n-2$. Find the number of all such labellings.

2000 Croatia National Olympiad, Problem 3

Tags: geometry , 3d geometry

A plane intersects a rectangular parallelepiped in a regular hexagon. Prove that the rectangular parallelepiped is a cube.

2011 AIME Problems, 4

Tags: geometry , trigonometry , geometric transformation , trig identities , law of sines , homothety , angle bisector

In triangle $ABC$, $AB=125,AC=117$, and $BC=120$. The angle bisector of angle $A$ intersects $\overline{BC}$ at point $L$, and the angle bisector of angle $B$ intersects $\overline{AC}$ at point $K$. Let $M$ and $N$ be the feet of the perpendiculars from $C$ to $\overline{BK}$ and $\overline{AL}$, respectively. Find $MN$.

2023 MOAA, 12

Tags:

Let $N$ be the number of $105$-digit positive integers that contain the digit 1 an odd number of times. Find the remainder when $N$ is divided by $1000$. [i]Proposed by Harry Kim[/i]

Kvant 2019, M2573

Tags: polyhedron , edge , combinatorics

Two ants are moving along the edges of a convex polyhedron. The route of every ant ends in its starting point, so that one ant does not pass through the same point twice along its way. On every face $F$ of the polyhedron are written the number of edges of $F$ belonging to the route of the first ant and the number of edges of $F$ belonging to the route of the second ant. Is there a polyhedron and a pair of routes described as above, such that only one face contains a pair of distinct numbers? [i]Proposed by Nikolai Beluhov[/i]

2006 Austrian-Polish Competition, 6

Tags: geometry

Let $D$ be an interior point of the triangle $ABC$. $CD$ and $AB$ intersect at $D_{c}$, $BD$ and $AC$ intersect at $D_{b}$, $AD$ and $BC$ intersect at $D_{a}$. Prove that there exists a triangle $KLM$ with orthocenter $H$ and the feet of altitudes $H_{k}\in LM, H_{l}\in KM, H_{m}\in KL$, so that $(AD_{c}D) = (KH_{m}H)$ $(BD_{c}D) = (LH_{m}H)$ $(BD_{a}D) = (LH_{k}H)$ $(CD_{a}D) = (MH_{k}H)$ $(CD_{b}D) = (MH_{l}H)$ $(AD_{b}D) = (KH_{l}H)$ where $(PQR)$ denotes the area of the triangle $PQR$

III Soros Olympiad 1996 - 97 (Russia), 10.9

Tags: system of equations , algebra

For any positive $a$ and $b$, find positive solutions of the system $$\begin{cases} \dfrac{a^2}{x^2}- \dfrac{b^2}{y^2}=8(y^4-x^4) \\ ax-by=x^4-y^4 \end{cases}$$

2020 JBMO Shortlist, 3

Tags: algebra , inequality , minimum , shortlist

Find all triples of positive real numbers $(a, b, c)$ so that the expression $M = \frac{(a + b)(b + c)(a + b + c)}{abc}$ gets its least value.

2013 Thailand Mathematical Olympiad, 7

Tags: combinatorial geometry , combinatorics

Let $P_1, ... , P_{2556}$ be distinct points in a regular hexagon $ABCDEF$ with unit side length. Suppose that no three points in the set $S = \{A, B, C, D, E, F, P_1, ... , P_{2556}\}$ are collinear. Show that there is a triangle whose vertices are in $S$ and whose area is less than $\frac{1}{1700}$ .

1976 IMO Longlists, 11

Tags: algebra , recurrence relation , root , chebyshev polynomial , polynomial

Let $P_{1}(x)=x^{2}-2$ and $P_{j}(x)=P_{1}(P_{j-1}(x))$ for j$=2,\ldots$ Prove that for any positive integer n the roots of the equation $P_{n}(x)=x$ are all real and distinct.

2006 Estonia Team Selection Test, 2

Tags: projection , circumcenter , concyclic , cyclic , independent , geometry

The center of the circumcircle of the acute triangle $ABC$ is $O$. The line $AO$ intersects $BC$ at $D$. On the sides $AB$ and $AC$ of the triangle, choose points $E$ and $F$, respectively, so that the points $A, E, D, F$ lie on the same circle. Let $E'$ and $F'$ projections of points $E$ and $F$ on side $BC$ respectively. Prove that length of the segment $E'F'$ does not depend on the position of points $E$ and $F$.

2014 Singapore Senior Math Olympiad, 27

Tags:

Determine the number of ways of colouring a $10\times 10$ square board using two colours black and white such that each $2\times 2$ subsquare contains 2 black squares and 2 white squares.

1988 IMO Longlists, 82

Tags: pythagorean theorem , geometry , inradius

The triangle $ABC$ has a right angle at $C.$ The point $P$ is located on segment $AC$ such that triangles $PBA$ and $PBC$ have congruent inscribed circles. Express the length $x = PC$ in terms of $a = BC, b = CA$ and $c = AB.$

2016 PUMaC Number Theory B, 4

Tags: number theory

For a positive integer $n$, let $P(n)$ be the product of the factors of $n$ (including $n$ itself). A positive integer $n$ is called [i]deplorable [/i] if $n > 1$ and $\log_n P(n)$ is an odd integer. How many factors of $2016$ are [i]deplorable[/i]?

2014 Taiwan TST Round 2, 2

Tags: functional equation , function , number theory

Determine all functions $f: \mathbb{Q} \rightarrow \mathbb{Z} $ satisfying \[ f \left( \frac{f(x)+a} {b}\right) = f \left( \frac{x+a}{b} \right) \] for all $x \in \mathbb{Q}$, $a \in \mathbb{Z}$, and $b \in \mathbb{Z}_{>0}$. (Here, $\mathbb{Z}_{>0}$ denotes the set of positive integers.)

2013 Swedish Mathematical Competition, 6

Tags: inequalities , algebra

Let $a, b, c$, be real numbers such that $$a^2b^2 + 18 abc > 4b^3+4a^3c+27c^2 .$$ Prove that $a^2>3b$.

1991 Irish Math Olympiad, 1

Tags: geometry

Three points $X,Y$ and $Z$ are given that are, respectively, the circumcenter of a triangle $ABC$, the mid-point of $BC$, and the foot of the altitude from $B$ on $AC$. Show how to reconstruct the triangle $ABC$.

2018 Purple Comet Problems, 15

Tags: polynomial , algebra

There are integers $a_1, a_2, a_3,...,a_{240}$ such that $x(x + 1)(x + 2)(x + 3) ... (x + 239) =\sum_{n=1}^{240}a_nx^n$. Find the number of integers $k$ with $1\le k \le 240$ such that ak is a multiple of $3$.

2010 Romania Team Selection Test, 2

Tags: euler , geometry

Let $ABC$ be a scalene triangle. The tangents at the perpendicular foot dropped from $A$ on the line $BC$ and the midpoint of the side $BC$ to the nine-point circle meet at the point $A'$\,; the points $B'$ and $C'$ are defined similarly. Prove that the lines $AA'$, $BB'$ and $CC'$ are concurrent. [i]Gazeta Matematica[/i]