Olympiad problems

2018 May Olympiad, 3

Let $ABCDEFGHIJ$ be a regular $10$-sided polygon that has all its vertices in one circle with center $O$ and radius $5$. The diagonals $AD$ and $BE$ intersect at $P$ and the diagonals $AH$ and $BI$ intersect at $Q$. Calculate the measure of the segment $PQ$.

2011 Princeton University Math Competition, B4

Tags: algebra

Let $f$ be an invertible function defined on the complex numbers such that \[z^2 = f(z + f(iz + f(-z + f(-iz + f(z + \ldots)))))\] for all complex numbers $z$. Suppose $z_0 \neq 0$ satisfies $f(z_0) = z_0$. Find $1/z_0$. (Note: an invertible function is one that has an inverse).

1994 Bundeswettbewerb Mathematik, 2

Tags: sequence , number theory , divisibility

Let $k$ be an integer and define a sequence $a_0 , a_1 ,a_2 ,\ldots$ by $$ a_0 =0 , \;\; a_1 =k \;\;\text{and} \;\; a_{n+2} =k^{2}a_{n+1}-a_n \; \text{for} \; n\geq 0.$$ Prove that $a_{n+1} a_n +1$ divides $a_{n+1}^{2} +a_{n}^{2}$ for all $n$.

2007 May Olympiad, 5

Tags: geometry , paper , rectangle , pentagon

You have a paper pentagon, $ABCDE$, such that $AB = BC = 3$ cm, $CD = DE= 5$ cm, $EA = 4$ cm, $\angle ABC = 100^o$ ,$ \angle CDE = 80^o$. You have to divide the pentagon into four triangles, by three straight cuts, so that with the four triangles assemble a rectangle, without gaps or overlays. (The triangles can be rotated and / or turned around.)

2014-2015 SDML (High School), 9

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What is the smallest number of queens that can be placed on an $8\times8$ chess board so that every square is either occupied or can be reached in one move? (A queen can be moved any number of unoccupied squares in a straight line vertically, horizontally, or diagonally.) $\text{(A) }4\qquad\text{(B) }5\qquad\text{(C) }6\qquad\text{(D) }7\qquad\text{(E) }8$

2021 Math Prize for Girls Problems, 18

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Let $N$ be the set of square-free positive integers less than or equal to 50. (A [i]square-free[/i] number is an integer that is not divisible by a perfect square bigger than 1.) How many 3-element subsets $S$ of $N$ are there such that the greatest common divisor of all 3 numbers in $S$ is 1, but no pair of numbers in $S$ is relatively prime?

2010 Ukraine Team Selection Test, 12

Tags: algebra , rational , sum , reciprocal sum , number theory

Is there a positive integer $n$ for which the following holds: for an arbitrary rational $r$ there exists an integer $b$ and non-zero integers $a _1, a_2, ..., a_n$ such that $r=b+\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}$ ?

1978 AMC 12/AHSME, 19

Tags: probability

A positive integer $n$ not exceeding $100$ is chosen in such a way that if $n\le 50$, then the probability of choosing $n$ is $p$, and if $n > 50$, then the probability of choosing $n$ is $3p$. The probability that a perfect square is chosen is $\textbf{(A) }.05\qquad\textbf{(B) }.065\qquad\textbf{(C) }.08\qquad\textbf{(D) }.09\qquad \textbf{(E) }.1$

1993 National High School Mathematics League, 6

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$m,n$ are non-zero-real numbers, $z\in\mathbb{C}$. Then, the figure of equations $|z+n\text{i}|+|z-m\text{i}|=n$ and $|z+n\text{i}|-|z-m\text{i}|=-m$ in complex plane is ($F_1,F_2$ are focal points) [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMS84L2RkYWZjM2JmNTc0N2RmYjJlMGUwMGFmMWRkY2RkZTA4NTljZTUwLnBuZw==&rn=MTI0NTI0NTQucG5n[/img]

Geometry Mathley 2011-12, 10.2

Tags: collinear , circumcircle , incircle , radical axis , concurrency , geometry

Let $ABC$ be an acute triangle, not isoceles triangle and $(O), (I)$ be its circumcircle and incircle respectively. Let $A_1$ be the the intersection of the radical axis of $(O), (I)$ and the line $BC$. Let $A_2$ be the point of tangency (not on $BC$) of the tangent from $A_1$ to $(I)$. Points $B_1,B_2,C_1,C_2$ are defined in the same manner. Prove that (a) the lines $AA_2,BB_2,CC_2$ are concurrent. (b) the radical centers circles through triangles $BCA_2, CAB_2$ and $ABC_2$ are all on the line $OI$. Lê Phúc Lữ

2019 Moldova EGMO TST, 8

Tags: sequence

The sequence $(a_n)_{n\geq1}$ is defined as: $$a_1=2, a_2=20, a_3=56, a_{n+3}=7a_{n+2}-11a_{n+1}+5a_n-3\cdot2^n.$$ Prove that $a_n$ is positive for every positive integer $n{}$. Find the remainder of the divison of $a_{673}$ to $673$.

1986 AIME Problems, 2

Tags: polynomial , algebra

Evaluate the product \[(\sqrt 5+\sqrt6+\sqrt7)(-\sqrt 5+\sqrt6+\sqrt7)(\sqrt 5-\sqrt6+\sqrt7)(\sqrt 5+\sqrt6-\sqrt7).\]

2019 Purple Comet Problems, 10

Tags: combinatorics , number theory

Find the number of positive integers less than $2019$ that are neither multiples of $3$ nor have any digits that are multiples of $3$.

1996 Iran MO (3rd Round), 4

Tags: combinatorics

Show that there doesn't exist two infinite and separate sets $A,B$ of points such that [b](i)[/b] There are no three collinear points in $A \cup B$, [b](ii)[/b] The distance between every two points in $A \cup B$ is at least $1$, and [b](iii)[/b] There exists at least one point belonging to set $B$ in interior of each triangle which all of its vertices are chosen from the set $A$, and there exists at least one point belonging to set $A$ in interior of each triangle which all of its vertices are chosen from the set $B$.

2005 Tournament of Towns, 2

Tags: geometry

The altitudes $AD$ and $BE$ of triangle $ABC$ meet at its orthocentre $H$. The midpoints of $AB$ and $CH$ are $X$ and $Y$, respectively. Prove that $XY$ is perpendicular to $DE$. [i](5 points)[/i]

2020 Thailand TST, 5

Tags: functional equation , number theory

Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.

1998 Bulgaria National Olympiad, 2

Tags: polynomial , algebra

The polynomials $P_n(x,y), n=1,2,... $ are defined by \[P_1(x,y)=1, P_{n+1}(x,y)=(x+y-1)(y+1)P_n(x,y+2)+(y-y^2)P_n(x,y)\] Prove that $P_{n}(x,y)=P_{n}(y,x)$ for all $x,y \in \mathbb{R}$ and $n $.

2017 ASDAN Math Tournament, 27

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How many primes between $2$ and $2^{30}$ are $1$ more than a multiple of $2017$? If $C$ is the correct answer and $A$ is your answer, then your score will be rounded up from $\max(0,25-15|\ln\tfrac{A}{C}|)$.

1999 Tournament Of Towns, 4

Tags: combinatorial geometry , combinatorics , tlies , tiling , equilateral , equilateral triangle

A black unit equilateral triangle is drawn on the plane. How can we place nine tiles, each a unit equilateral triangle, on the plane so that they do not overlap, and each tile covers at least one interior point of the black triangle? (Folklore)

2004 Peru MO (ONEM), 2

Tags: combinatorics

There are $100$ apparently identical coins, where at least one of them is counterfeit . The real ones coins are of equal weight and counterfeit coins are also of equal weight, but lighter than the real ones. Explain how the number of counterfeit coins can be found, using a pan balance, at most $51$ times.

2020 CCA Math Bonanza, TB2

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Shayan is playing a game by himself. He picks [b]relatively prime[/b] integers $a$ and $b$ such that $1<a<b<2020$. He wins if every integer $m\geq\frac{ab}{2}$ can be expressed in the form $ax+by$ for nonnegative integers $x$ and $y$. He hasn't been winning often, so he decides to write down all winning pairs $(a,b)$, from $(a_1,b_1)$ to $(a_n,b_n)$. What is $b_1+b_2+\ldots+b_n$? [i]2020 CCA Math Bonanza Tiebreaker Round #2[/i]

2016 Harvard-MIT Mathematics Tournament, 6

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Call a positive integer $N \ge 2$ ``special'' if for every $k$ such that $2 \leq k \leq N$, $N$ can be expressed as a sum of $k$ positive integers that are relatively prime to $N$ (although not necessarily relatively prime to each other). How many special integers are there less than $100$?

1966 IMO Shortlist, 15

Tags: geometry , euler , perpendicular , circles

Given four points $A,$ $B,$ $C,$ $D$ on a circle such that $AB$ is a diameter and $CD$ is not a diameter. Show that the line joining the point of intersection of the tangents to the circle at the points $C$ and $D$ with the point of intersection of the lines $AC$ and $BD$ is perpendicular to the line $AB.$

1997 Israel Grosman Mathematical Olympiad, 4

Tags: altitude , tetrahedron , 3d geometry , geometry

Prove that if two altitudes of a tetrahedron intersect, then so do the other two altitudes.

2001 Grosman Memorial Mathematical Olympiad, 2

Tags: sum , inequalities , algebra

If $x_1,x_2,...,x_{2001}$ are real numbers with $0 \le x_n \le 1$ for $n = 1,2,...,2001$, find the maximum value of $$\left(\frac{1}{2001}\sum_{n=1}^{2001}x_n^2\right)-\left(\frac{1}{2001}\sum_{n=1}^{2001}x_n\right)^2$$ Where is this maximum attained?