Olympiad problems

2011 Uzbekistan National Olympiad, 4

Does existes a function $f:N->N$ and for all positeve integer n $f(f(n)+2011)=f(n)+f(f(n))$

2006 Estonia National Olympiad, 4

Tags: floor function , equation , algebra

Solve the equation $\left[\frac{x}{3}\right]+\left [\frac{2x}{3}\right]=x $

2016 Ukraine Team Selection Test, 9

Tags: combinatorics , game

Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are: (i) A player cannot choose a number that has been chosen by either player on any previous turn. (ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn. (iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game. The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies. [i]Proposed by Finland[/i]

2001 AMC 12/AHSME, 5

Tags:

What is the product of all odd positive integers less than 10000? $ \textbf{(A)} \ \frac {10000!}{(5000!)^2} \qquad \textbf{(B)} \ \frac {10000!}{2^{5000}} \ \qquad \textbf{(C)} \ \frac {9999!}{2^{5000}} \qquad \textbf{(D)} \ \frac {10000!}{2^{5000} \cdot 5000!} \qquad \textbf{(E)} \ \frac {5000!}{2^{5000}}$

2012 Paraguay Mathematical Olympiad, 5

Tags: circumcircle , trigonometry , geometry

Let $ABC$ be an equilateral triangle. Let $Q$ be a random point on $BC$, and let $P$ be the meeting point of $AQ$ and the circumscribed circle of $\triangle ABC$. Prove that $\frac{1}{PQ}=\frac{1}{PB}+\frac{1}{PC}$.

2010 HMNT, 7

Tags: combinatorics

George has two coins, one of which is fair and the other of which always comes up heads. Jacob takes one of them at random and flips it twice. Given that it came up heads both times, what is the probability that it is the coin that always comes up heads?

2013-2014 SDML (High School), 10

Tags: quadratic

The sum $$\frac{1}{1+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{7}}+\cdots+\frac{1}{\sqrt{2n-1}+\sqrt{2n+1}}$$ is a root of the quadratic $x^2+x+c$. What is $c$ in terms of $n$? $\text{(A) }-\frac{n}{2}\qquad\text{(B) }2n\qquad\text{(C) }-2n\qquad\text{(D) }n+\frac{1}{2}\qquad\text{(E) }n-2$

2012 Math Prize For Girls Problems, 12

Tags: trigonometry

What is the sum of all positive integer values of $n$ that satisfy the equation \[ \cos \Bigl( \frac{\pi}{n} \Bigr) \cos \Bigl( \frac{2\pi}{n} \Bigr) \cos \Bigl( \frac{4\pi}{n} \Bigr) \cos \Bigl( \frac{8\pi}{n} \Bigr) \cos \Bigl( \frac{16\pi}{n} \Bigr) = \frac{1}{32} \, ? \]

2019 BMT Spring, 4

Tags:

The area of right triangle $ ABC $ is 4, and the length of hypotenuse $ AB $ is 12. Compute the perimeter of $ \triangle ABC $.

2005 Georgia Team Selection Test, 2

Tags: trigonometry , geometric transformation , reflection , geometry

In triangle $ ABC$ we have $ \angle{ACB} \equal{} 2\angle{ABC}$ and there exists the point $ D$ inside the triangle such that $ AD \equal{} AC$ and $ DB \equal{} DC$. Prove that $ \angle{BAC} \equal{} 3\angle{BAD}$.

1994 IberoAmerican, 2

Tags: function , ceiling function , combinatorics

Let $n$ and $r$ two positive integers. It is wanted to make $r$ subsets $A_1,\ A_2,\dots,A_r$ from the set $\{0,1,\cdots,n-1\}$ such that all those subsets contain exactly $k$ elements and such that, for all integer $x$ with $0\leq{x}\leq{n-1}$ there exist $x_1\in{}A_1,\ x_2\in{}A_2 \dots,x_r\in{}A_r$ (an element of each set) with $x=x_1+x_2+\cdots+x_r$. Find the minimum value of $k$ in terms of $n$ and $r$.

2023 CUBRMC, 7

Tags: algebra , inequalities

Among all ordered pairs of real numbers $(a, b)$ satisfying $a^4 + 2a^2b + 2ab + b^2 = 960$, find the smallest possible value for $a$.

1987 IMO Shortlist, 7

Tags: algebra , inequality system

Given five real numbers $u_0, u_1, u_2, u_3, u_4$, prove that it is always possible to find five real numbers $v0, v_1, v_2, v_3, v_4$ that satisfy the following conditions: $(i)$ $u_i-v_i \in \mathbb N, \quad 0 \leq i \leq 4$ $(ii)$ $\sum_{0 \leq i<j \leq 4} (v_i - v_j)^2 < 4.$ [i]Proposed by Netherlands.[/i]

1990 Iran MO (2nd round), 3

Tags: number theory

[b](a)[/b] For every positive integer $n$ prove that \[1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2} <2\] [b](b)[/b] Let $X=\{1, 2, 3 ,\ldots, n\} \ ( n \geq 1)$ and let $A_k$ be non-empty subsets of $X \ (k=1,2,3, \ldots , 2^n -1).$ If $a_k$ be the product of all elements of the set $A_k,$ prove that \[\sum_{i=1}^{m} \sum_{j=1}^m \frac{1}{a_i \cdot j^2} <2n+1\]

2014 Cezar Ivănescu, 1

Tags: abstract algebra , group theory

Let $ S $ be a nonempty subset of a finite group $ G, $ and $ \left( S^j \right)_{j\ge 1} $ be a sequence of sets defined as $ S^j=\left.\left\{\underbrace{xy\cdots z}_{\text{j terms}} \right| \underbrace{x,y,\cdots ,z}_{\text{j terms}} \in S \right\} . $ Prove that: [b]a)[/b] $ \exists i_0\in\mathbb{N}^*\quad i\ge i_0\implies \left| S^i\right| =\left| S^{1+i}\right| $ [b]b)[/b] $ S^{|G|}\le G $

2002 Tuymaada Olympiad, 2

Tags: geometry , ratio , equilateral , equilateral triangle

Points on the sides $ BC $, $ CA $ and $ AB $ of the triangle $ ABC $ are respectively $ A_1 $, $ B_1 $ and $ C_1 $ such that $ AC_1: C_1B = BA_1: A_1C = CB_1: B_1A = 2: 1 $. Prove that if triangle $ A_1B_1C_1 $ is equilateral, then triangle $ ABC $ is also equilateral.

2017 Flanders Math Olympiad, 2

Tags: geometry , circumcircle , angle chasing , angle

In triangle $\vartriangle ABC$, $\angle A = 50^o, \angle B = 60^o$ and $\angle C = 70^o$. The point $P$ is on the side $[AB]$ (with $P \ne A$ and $P \ne B$). The inscribed circle of $\vartriangle ABC$ intersects the inscribed circle of $\vartriangle ACP$ at points $U$ and $V$ and intersects the inscribed circle of $\vartriangle BCP$ at points $X$ and $Y$. The rights $UV$ and $XY$ intersect in $K$. Calculate the $\angle UKX$.

2010 ELMO Shortlist, 6

Tags: circumcircle , geometry

Let $ABC$ be a triangle with circumcircle $\Omega$. $X$ and $Y$ are points on $\Omega$ such that $XY$ meets $AB$ and $AC$ at $D$ and $E$, respectively. Show that the midpoints of $XY$, $BE$, $CD$, and $DE$ are concyclic. [i]Carl Lian.[/i]

2002 Tournament Of Towns, 2

Tags: 3d geometry , inequalities , geometry

A cube is cut by a plane such that the cross section is a pentagon. Show there is a side of the pentagon of length $\ell$ such that the inequality holds: \[ |\ell-1|>\frac{1}{5} \]

2010 Today's Calculation Of Integral, 536

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $ \int_0^\frac{\pi}{4} \frac{x\plus{}\sin x}{1\plus{}\cos x}\ dx$.

Durer Math Competition CD Finals - geometry, 2023.D2

Tags: trapezoid , geometry

Let $ABCD$ be a isosceles trapezoid. Base $AD$ is $11$ cm long while the other three sides are each $5$ cm long. We draw the line that is perpendicular to $BD$ and contains $C$ and the line that is perpendicular to $AC$ and contains$ B$. We mark the intersection of these two lines with $E$. What is the distance between point $E$ and line $AD$?

2023 Kyiv City MO Round 1, Problem 1

Tags: algebra , expression

Find the integer which is closest to the value of the following expression: $$((7 + \sqrt{48})^{2023} + (7 - \sqrt{48})^{2023})^2 - ((7 + \sqrt{48})^{2023} - (7 - \sqrt{48})^{2023})^2$$

1947 Putnam, A6

Tags: linear algebra , matrix , cofactor

A $3\times 3$ matrix has determinant $0$ and the cofactor of any element is equal to the square of that element. Show that every element in the matrix is $0.$

2021 Silk Road, 3

Tags: geometry

In a triangle $ABC$, $M$ is the midpoint of the $AB$. A point $B_1$ is marked on $AC$ such that $CB=CB_1$. Circle $\omega$ and $\omega_1$, the circumcircles of triangles $ABC$ and $BMB_1$, respectively, intersect again at $K$. Let $Q$ be the midpoint of the arc $ACB$ on $\omega$. Let $B_1Q$ and $BC$ intersect at $E$. Prove that $KC$ bisects $B_1E$. [i]M. Kungozhin[/i]

2010 Bosnia Herzegovina Team Selection Test, 3

Tags: function , algebra

Find all functions $ f :\mathbb{Z}\mapsto\mathbb{Z} $ such that following conditions holds: $a)$ $f(n) \cdot f(-n)=f(n^2)$ for all $n\in\mathbb{Z}$ $b)$ $f(m+n)=f(m)+f(n)+2mn$ for all $m,n\in\mathbb{Z}$