Olympiad problems

1996 Estonia National Olympiad, 3

An equilateral triangle of side$ 1$ is rotated around its center, yielding another equilareral triangle. Find the area of the intersection of these two triangles.

2009 Junior Balkan Team Selection Tests - Romania, 1

Tags: multiple , number theory , digit

For all positive integers $n$ define $a_n=2 \underbrace{33...3}_{n \, times}$, where digit $3$ occurs $n$ times. Show that the number $a_{2009}$ has infinitely many multiples in the set $\{a_n | n \in N*\}$.

PEN I Problems, 5

Tags: floor function

Find all real numbers $\alpha$ for which the equality \[\lfloor \sqrt{n}+\sqrt{n+\alpha}\rfloor =\lfloor \sqrt{4n+1}\rfloor\] holds for all positive integers $n$.

2010 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , divide

Let $p$ be a prime number, $p> 5$. Determine the non-zero natural numbers $x$ with the property that $5p + x$ divides $5p ^ n + x ^ n$, whatever $n \in N ^ {*} $.

MathLinks Contest 3rd, 2

Tags: number theory

Let $a_1, a_2, ..., a_{2004}$ be integer numbers such that for all positive integers $n$ the number $A_n = a^n_1 + a^n_2 + ...+ a^n_{2004}$ is a perfect square. What is the minimal number of zeros within the $2004$ numbers?

2024 Bulgarian Autumn Math Competition, 8.4

Tags: lattice points , equilateral triangle , combinatorics , function

Let $n$ be a positive integers. Equilateral triangle with sides of length $n$ is split into equilateral triangles with side lengths $1$, forming a triangular lattice. Call an equilateral triangle with vertices in the lattice "important". Let $p_k$ be the number of unordered pairs of vertices in the lattice which participate in exactly $k$ important triangles. Find (as a function of $n$) (a) $p_0+p_1+p_2$ (b) $p_1+2p_2$

2003 Oral Moscow Geometry Olympiad, 1

Tags: geometry , construction

Construct a triangle given an angle, the side opposite the angle and the median to the other side (researching the number of solutions is not required).

1997 Greece Junior Math Olympiad, 2

Tags: number theory

Determine all natural numbers n for which the number $A = n^4 + 4n^3 +5n^2 + 6n$ is a perfect square of a natural number.

2010 VJIMC, Problem 2

Tags: matrix , linear algebra

If $A,B\in M_2(C)$ such that $AB-BA=B^2$ then prove that \[AB=BA\]

2013 National Olympiad First Round, 23

Tags:

If the conditions \[\begin{array}{rcl} f(2x+1)+g(3-x) &=& x \\ f((3x+5)/(x+1))+2g((2x+1)/(x+1)) &=& x/(x+1) \end{array}\] hold for all real numbers $x\neq 1$, what is $f(2013)$? $ \textbf{(A)}\ 1007 \qquad\textbf{(B)}\ \dfrac {4021}{3} \qquad\textbf{(C)}\ \dfrac {6037}7 \qquad\textbf{(D)}\ \dfrac {4029}{5} \qquad\textbf{(E)}\ \text{None of above} $

2023 Indonesia TST, G

Tags: geometry

Incircle of triangle $ABC$ tangent to $AB$ and $AC$ on $E$ and $F$ respectively. If $X$ is the midpoint of $EF$, prove $\angle BXC > 90^{\circ}$

1996 India National Olympiad, 3

Tags: algebra

Solve the following system for real $a , b, c, d, e$: \[ \left\{ \begin{array}{ccc} 3a & = & ( b + c+ d)^3 \\ 3b & = & ( c + d +e ) ^3 \\ 3c & = & ( d + e +a )^3 \\ 3d & = & ( e + a +b )^3 \\ 3e &=& ( a + b +c)^3. \end{array}\right. \]

CIME I 2018, 13

Tags:

Find the number of positive integers $n<2017$ such that $n^2+n^0+n^1+n^7$ is not divisible by the square of any prime. [i]Proposed by [b]illogical_21[/b][/i]

1950 AMC 12/AHSME, 11

Tags:

If in the formula $ C \equal{} \frac {en}{R\plus{}nr}$, $n$ is increased while $ e$, $R$ and $r$ are kept constant, then $C$: $\textbf{(A)}\ \text{Increases} \qquad \textbf{(B)}\ \text{Decreases} \qquad \textbf{(C)}\ \text{Remains constant} \qquad \textbf{(D)}\ \text{Increases and then decreases} \qquad\\ \textbf{(E)}\ \text{Decreases and then increases}$

2003 Moldova Team Selection Test, 3

Tags: angle bisector , incenter , geometry

The sides $ [AB]$ and $ [AC]$ of the triangle $ ABC$ are tangent to the incircle with center $ I$ of the $ \triangle ABC$ at the points $ M$ and $ N$, respectively. The internal bisectors of the $ \triangle ABC$ drawn form $ B$ and $ C$ intersect the line $ MN$ at the points $ P$ and $ Q$, respectively. Suppose that $ F$ is the intersection point of the lines $ CP$ and $ BQ$. Prove that $ FI\perp BC$.

2012 France Team Selection Test, 1

Tags: graph theory , combinatorics , group theory , abstract algebra

Let $n$ and $k$ be two positive integers. Consider a group of $k$ people such that, for each group of $n$ people, there is a $(n+1)$-th person that knows them all (if $A$ knows $B$ then $B$ knows $A$). 1) If $k=2n+1$, prove that there exists a person who knows all others. 2) If $k=2n+2$, give an example of such a group in which no-one knows all others.

2018 PUMaC Live Round, 7.3

Tags:

Kite $ABCD$ has right angles at $B$ and $D$, and $AB<BC$. Points $E\in AB$ and $F\in AD$ satisfy $AE=4$, $EF=7$, and $FA=5$. If $AB=8$ and points $X$ lies outside $ABCD$ while satisfying $XE-XF=1$ and $XE+XF+2XA=23$, then $XA$ may be written as $\tfrac{a-b\sqrt{c}}{d}$ for $a,b,c,d$ positive integers with $\gcd(a^2,b^2,c,d^2)=1$ and $c$ squarefree. Find $a+b+c+d$.

2011 Purple Comet Problems, 8

Tags: geometry

A square measuring $15$ by $15$ is partitioned into five rows of five congruent squares as shown below. The small squares are alternately colored black and white as shown. Find the total area of the part colored black. [asy] size(150); defaultpen(linewidth(0.8)); int i,j; for(i=1;i<=5;i=i+1) { for(j=1;j<=5;j=j+1) { if (floor((i+j)/2)==((i+j)/2)) { filldraw(shift((i-1,j-1))*unitsquare,gray); } else { draw(shift((i-1,j-1))*unitsquare); } } } [/asy]

2010 Danube Mathematical Olympiad, 5

Tags: function , algebra , calculus , n-variable inequality , inequalities , polynomial

Let $n\ge3$ be a positive integer. Find the real numbers $x_1\ge0,\ldots,x_n\ge 0$, with $x_1+x_2+\ldots +x_n=n$, for which the expression \[(n-1)(x_1^2+x_2^2+\ldots+x_n^2)+nx_1x_2\ldots x_n\] takes a minimal value.

1992 India National Olympiad, 7

Tags: counting , chessboard , arithmetic progression , combinatorics

Let $n\geq 3$ be an integer. Find the number of ways in which one can place the numbers $1, 2, 3, \ldots, n^2$ in the $n^2$ squares of a $n \times n$ chesboard, one on each, such that the numbers in each row and in each column are in arithmetic progression.

2004 Serbia Team Selection Test, 2

Tags: three variable inequality , inequalities

Let $a$, $b$ and $c$ be real numbers such that $abc=1$. Prove that the most two of numbers $$2a-\frac{1}{b},\ 2b-\frac{1}{c},\ 2c-\frac{1}{a}$$ are greater than $1$.

2019 Online Math Open Problems, 21

Tags:

Define a sequence by $a_0=2019$ and $a_n=a_{n-1}^{2019}$ for all positive integers $n$. Compute the remainder when \[a_0+a_1+a_2+\dots+a_{51}\] is divided by $856$. [i]Proposed by Tristan Shin[/i]

2013 Baltic Way, 2

Tags: polynomial , algebra

Let $k$ and $n$ be positive integers and let $x_1, x_2, \cdots, x_k, y_1, y_2, \cdots, y_n$ be distinct integers. A polynomial $P$ with integer coefficients satisfies \[P(x_1)=P(x_2)= \cdots = P(x_k)=54\] \[P(y_1)=P(y_2)= \cdots = P(y_n)=2013.\] Determine the maximal value of $kn$.

2014 AMC 10, 5

Tags:

On an algebra quiz, $10\%$ of the students scored $70$ points, $35\%$ scored $80$ points, $30\%$ scored $90$ points, and the rest scored $100$ points. What is the difference between the mean and median score of the students' scores on this quiz? ${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}}\ 4\qquad\textbf{(E)}\ 5$

2022 District Olympiad, P2

Tags: matrix

Let $A,B\in\mathcal{M}_3(\mathbb{R})$ de matrices such that $A^2+B^2=O_3.$ Prove that $\det(aA+bB)=0$ for any real numbers $a$ and $b.$