Olympiad problems

2005 Bundeswettbewerb Mathematik, 2

Let $a$ be such an integer, that $3a$ can be written in the form $x^2 + 2y^2$, with integers $x$ and $y$. Prove that the number $a$ can also be written in this form. [b]Additional problems:[/b] [b]a)[/b] Find a general (necessary and sufficent) criterion for an integer $n$ to be of that form. [b]b)[/b] In how many ways can the integer $n$ be represented in that way?

1969 IMO, 2

Tags: trigonometric equation , function , trigonometry

Let $f(x)=\cos(a_1+x)+{1\over2}\cos(a_2+x)+{1\over4}\cos(a_3+x)+\ldots+{1\over2^{n-1}}\cos(a_n+x)$, where $a_i$ are real constants and $x$ is a real variable. If $f(x_1)=f(x_2)=0$, prove that $x_1-x_2$ is a multiple of $\pi$.

1998 Akdeniz University MO, 4

Tags: geometry

Let $ABC$ be an equilateral triangle with side lenght is $1$ $cm$.Let $D \in [AB]$ is a point. Perpendiculars from $D$ to $[AC]$ and $[BC]$ intersects with $[AC]$ and $[BC]$ at points $E$ and $F$ respectively. Perpendiculars from $E$ and $F$ to $[AB]$ intersects with $[AB]$ at points $E_1$ and $F_1$. Prove that $$[E_1F_1]=\frac{3}{4}$$

1979 Czech And Slovak Olympiad IIIA, 2

Tags: geometry , 3d geometry , geometric inequality

Given a cuboid $Q$ with dimensions $a, b, c$, $a < b < c$. Find the length of the edge of a cube $K$ , which has parallel faces and a common center with the given cuboid so that the volume of the difference of the sets $Q \cup K$ and $Q \cap K$ is minimal.

1998 Taiwan National Olympiad, 3

Tags: symmetry , combinatorics

Let $ m,n$ be positive integers, and let $ F$ be a family of $ m$-element subsets of $ \{1,2,...,n\}$ satisfying $ A\cap B \not \equal{} \emptyset$ for all $ A,B\in F$. Determine the maximum possible number of elements in $ F$.

2021 AIME Problems, 4

Tags:

Find the number of ways $66$ identical coins can be separated into three nonempty piles so that there are fewer coins in the first pile than in the second pile and fewer coins in the second pile than in the third pile.

2012 Indonesia TST, 3

Tags: quadratic , rectangle , vector , geometry

The [i]cross[/i] of a convex $n$-gon is the quadratic mean of the lengths between the possible pairs of vertices. For example, the cross of a $3 \times 4$ rectangle is $\sqrt{ \dfrac{3^2 + 3^2 + 4^2 + 4^2 + 5^2 + 5^2}{6} } = \dfrac{5}{3} \sqrt{6}$. Suppose $S$ is a dodecagon ($12$-gon) inscribed in a unit circle. Find the greatest possible cross of $S$.

2007 Iran MO (2nd Round), 3

Tags: analytic geometry , combinatorics

In a city, there are some buildings. We say the building $A$ is dominant to the building $B$ if the line that connects upside of $A$ to upside of $B$ makes an angle more than $45^{\circ}$ with earth. We want to make a building in a given location. Suppose none of the buildings are dominant to each other. Prove that we can make the building with a height such that again, none of the buildings are dominant to each other. (Suppose the city as a horizontal plain and each building as a perpendicular line to the plain.)

2019 Regional Olympiad of Mexico Center Zone, 4

Tags: geometry , collinear

Let $ABC$ be a triangle with $\angle BAC> 90 ^ \circ$ and $D$ a point on $BC$. Let $E$ and $F$be the reflections of the point $D$ about $AB$ and $AC$, respectively. Suppose that $BE$ and $CF$ intersect at $P$. Show that $AP$ passes through the circumcenter of triangle $ABC$.

1994 AMC 12/AHSME, 21

Tags: algebra , binomial theorem

Find the number of counter examples to the statement: \[``\text{If N is an odd positive integer the sum of whose digits is 4 and none of whose digits is 0, then N is prime}."\] $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 $

2018 Saudi Arabia IMO TST, 3

Tags: fibonacci number , algebra , inequalities , max

Consider the function $f (x) = (x - F_1)(x - F_2) ...(x -F_{3030})$ with $(F_n)$ is the Fibonacci sequence, which defined as $F_1 = 1, F_2 = 2$, $F_{n+2 }=F_{n+1} + F_n$, $n \ge 1$. Suppose that on the range $(F_1, F_{3030})$, the function $|f (x)|$ takes on the maximum value at $x = x_0$. Prove that $x_0 > 2^{2018}$.

2003 District Olympiad, 3

Tags: superior algebra , algebra , polynomial , quadratic

Let $\displaystyle \mathcal K$ be a finite field such that the polynomial $\displaystyle X^2-5$ is irreducible over $\displaystyle \mathcal K$. Prove that: (a) $1+1 \neq 0$; (b) for all $\displaystyle a \in \mathcal K$, the polynomial $\displaystyle X^5+a$ is reducible over $\displaystyle \mathcal K$. [i]Marian Andronache[/i] [Edit $1^\circ$] I wanted to post it in "Superior Algebra - Groups, Fields, Rings, Ideals", but I accidentally put it here :blush: Can any mod move it? I'd be very grateful. [Edit $2^\circ$] OK, thanks.

2017 South East Mathematical Olympiad, 3

Tags: inequalities , algebra

Let $a_1,a_2,\cdots,a_{n+1}>0$. Prove that$$\sum_{i-1}^{n}a_i\sum_{i=1}^{n}a_{i+1}\geq \sum_{i=1}^{n}\frac{a_i a_{i+1}}{a_i+a_{i+1}}\cdot \sum_{i=1}^{n}(a_i+a_{i+1})$$

2012 Belarus Team Selection Test, 3

Tags: algebra , functional , functional equation

Find all functions $f : Q \to Q$, such that $$f(x + f (y + f(z))) = y + f(x + z)$$ for all $x ,y ,z \in Q$ . (I. Voronovich)

1999 Denmark MO - Mohr Contest, 5

Tags: number theory , digit , divide , divisible

Is there a number whose digits are only $1$'s and which is divided by $1999$?

1979 IMO Shortlist, 1

Tags: polygon , dissection , rhombus , geometry

Prove that in the Euclidean plane every regular polygon having an even number of sides can be dissected into lozenges. (A lozenge is a quadrilateral whose four sides are all of equal length).

1980 Miklós Schweitzer, 4

Tags: linear algebra , matrix

Let $ T \in \textsl{SL}(n,\mathbb{Z})$, let $ G$ be a nonsingular $ n \times n$ matrix with integer elements, and put $ S\equal{}G^{\minus{}1}TG$. Prove that there is a natural number $ k$ such that $ S^k \in \textsl{SL}(n,\mathbb{Z})$. [i]Gy. Szekeres[/i]

1966 IMO Shortlist, 29

Tags: number theory , summation , equation

A given natural number $N$ is being decomposed in a sum of some consecutive integers. [b]a.)[/b] Find all such decompositions for $N=500.$ [b]b.)[/b] How many such decompositions does the number $N=2^{\alpha }3^{\beta }5^{\gamma }$ (where $\alpha ,$ $\beta $ and $\gamma $ are natural numbers) have? Which of these decompositions contain natural summands only? [b]c.)[/b] Determine the number of such decompositions (= decompositions in a sum of consecutive integers; these integers are not necessarily natural) for an arbitrary natural $N.$ [b]Note by Darij:[/b] The $0$ is not considered as a natural number.

2017 Simon Marais Mathematical Competition, A4

Tags: vector

Let $A_1,A_2,\ldots,A_{2017}$ be the vertices of a regular polygon with $2017$ sides.Prove that there exists a point $P$ in the plane of the polygon such that the vector $$\sum_{k=1}^{2017}k\frac{\overrightarrow{PA}_k}{\left\lVert\overrightarrow{PA}_k\right\rVert^5}$$ is the zero vector. (The notation $\left\lVert\overrightarrow{XY}\right\rVert$ represents the length of the vector $\overrightarrow{XY}$.)

2016 PUMaC Algebra Individual B, B7

Tags:

Define a sequence $a_i$ as follows: $a_1 = 181$ and for $i \ge 2$, $a_i = a_{i-1}^2-1$ if $a_{i-1}$ is odd and $a_i = a_{i-1}/2$ if $a_{i-1}$ is even. Find the least $i$ such that $a_i = 0$.

1978 All Soviet Union Mathematical Olympiad, 257

Tags: sequence , inequalities , algebra

Prove that there exists such an infinite sequence $\{x_i\}$, that for all $m$ and all $k$ ($m\ne k$) holds the inequality $$|x_m-x_k|>1/|m-k|$$

2004 239 Open Mathematical Olympiad, 6

Tags: number theory , least common multiple

Given distinct positive integers $a_1,\,a_2,\,\dots,a_n$. Let $b_i = (a_i - a_1) (a_i-a_2) \dots (a_i-a_{i-1}) (a_i-a_{i+1})\dots(a_i-a_n)$. Prove that the least common multiple $[b_1,b_2,\dots,b_n]$ is divisible by $(n-1)!.$

2009 AIME Problems, 13

Tags: trigonometry , modular arithmetic , search , algebra , polynomial , vieta

Let $ A$ and $ B$ be the endpoints of a semicircular arc of radius $ 2$. The arc is divided into seven congruent arcs by six equally spaced points $ C_1,C_2,\ldots,C_6$. All chords of the form $ \overline{AC_i}$ or $ \overline{BC_i}$ are drawn. Let $ n$ be the product of the lengths of these twelve chords. Find the remainder when $ n$ is divided by $ 1000$.

2013 Tuymaada Olympiad, 4

Tags: induction , algorithm , graph theory , combinatorics

The vertices of a connected graph cannot be coloured with less than $n+1$ colours (so that adjacent vertices have different colours). Prove that $\dfrac{n(n-1)}{2}$ edges can be removed from the graph so that it remains connected. [i]V. Dolnikov[/i] [b]EDIT.[/b] It is confirmed by the official solution that the graph is tacitly assumed to be [b]finite[/b].

2012-2013 SDML (High School), 10

Tags: geometry

Pentagon $ABCDE$ is inscribed in a circle such that $ACDE$ is a square with area $12$. What is the largest possible area of pentagon $ABCDE$? $\text{(A) }9+3\sqrt{2}\qquad\text{(B) }13\qquad\text{(C) }12+\sqrt{2}\qquad\text{(D) }14\qquad\text{(E) }12+\sqrt{6}-\sqrt{3}$