Olympiad problems

1976 AMC 12/AHSME, 5

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How many integers greater than $10$ and less than $100$, written in base-$10$ notation, are increased by $9$ when their digits are reversed? $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 10$

2005 Postal Coaching, 26

Tags: number theory

Let $a_1,a_2,\ldots a_n$ be real numbers such that their sum is equal to zero. Find the value of \[ \sum_{j=1}^{n} \frac{1}{a_j (a_j +a _{j+1}) (a_j + a_{j+1} + a_{j+2}) \ldots (a_j + \ldots a_{j+n-2})}. \] where the subscripts are taken modulo $n$ assuming none of the denominators is zero.

2024 Romania National Olympiad, 3

Tags: matrix , function , linear algebra

Let $A,B \in \mathcal{M}_n(\mathbb{R}).$ We consider the function $f:\mathcal{M}_n(\mathbb{C}) \to \mathcal{M}_n(\mathbb{C}),$ defined by $f(Z)=AZ+B\overline{Z},$ $Z \in \mathcal{M}_n(\mathbb{C}),$ where $\overline{Z}$ is the matrix whose entries are the conjugates of the corresponding entries of $Z.$ Prove that the following statements are equivalent: $(1)$ the function $f$ is injective; $(2)$ the function $f$ is surjective; $(3)$ the matrices $A+B$ and $A-B$ are invertible.

PEN B Problems, 3

Tags: primitive root , modular arithmetic

Show that for each odd prime $p$, there is an integer $g$ such that $1<g<p$ and $g$ is a primitive root modulo $p^n$ for every positive integer $n$.

2004 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: polynomial , algebra

Find all polynomials $P(x)$ with real coefficients such that \[xP\bigg(\frac{y}{x}\bigg)+yP\bigg(\frac{x}{y}\bigg)=x+y\] for all nonzero real numbers $x$ and $y$.

1990 Baltic Way, 3

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Given $a_0 > 0$ and $c > 0$, the sequence $(a_n)$ is defined by \[a_{n+1}=\frac{a_n+c}{1-ca_n}\quad\text{for }n=1,2,\dots\] Is it possible that $a_0, a_1, \dots , a_{1989}$ are all positive but $a_{1990}$ is negative?

1987 Traian Lălescu, 2.2

Tags: real analysis , darboux s property , darboux , continuity , pathological , function

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} ,f(x)=\left\{\begin{matrix} \sin x , & x\not\in\mathbb{Q} \\ 0, & x\in\mathbb{Q}\end{matrix}\right. . $ [b]a)[/b] Determine the maximum length of an interval $ I\subset\mathbb{R} $ such that $ f|_I $ is discontinuous everywhere, yet has the intermediate value property. [b]b)[/b] Study the convergence of the sequence $ \left( x_n\right)_{n\in\mathbb{N}\cup\{ 0\}} $ defined by $ x_0\in (0,\pi /2),x_{n+1}=f\left( x_n\right),\forall n\ge 0. $

2017 Kyiv Mathematical Festival, 5

Tags: number theory , diophantine equation

Find all the pairs of integers $(x,y)$ for which $(x^2+y)(y^2+x)=(x+1)(y+1).$

2020 Simon Marais Mathematics Competition, A4

Tags: linear algebra , geometry

A [i]regular spatial pentagon[/i] consists of five points $P_1,P_2,P_3,P_4$ and $P_5$ in $\mathbb{R}^3$ such that $|P_iP_{i+1}|=|P_jP_{j+1}|$ and $\angle P_{i-1}P_iP_{i+1}=\angle P_{j-1}P_jP_{j+1}$ for all $1\leq i,\leq 5$, where $P_0=P_5$ and $P_{6}=P_{1}$. A regular spatial pentagon is [i]planar[/i] if there is a plane passing through all five points $P_1,P_2,P_3,P_4$ and $P_5$. Show that every regular spatial pentagon is planar.

V Soros Olympiad 1998 - 99 (Russia), 10.2

Tags: algebra

In $1748$, the great Russian mathematician Leonhard Euler published one of his most important works, Introduction to the Analysis of Infinites. In this work, in particular, Euler finds the values of two infinite sums $1 +\frac14 +\frac19+ \frac{1}{16}+...$ and $1 +\frac19+ \frac{1}{16}+...$ (the terms in the first sum are the inverses of the squares of the natural numbers, and in the second are the inverses of the squares of the odd numbers of the natural series). The value of the first sum, as Euler proved, equals $\frac{\pi^2}{6}$. Given this result, find the value of the second sum.

2014 Middle European Mathematical Olympiad, 6

Tags: geometric transformation , reflection , geometry , homothety , perpendicular bisector

Let the incircle $k$ of the triangle $ABC$ touch its side $BC$ at $D$. Let the line $AD$ intersect $k$ at $L \neq D$ and denote the excentre of $ABC$ opposite to $A$ by $K$. Let $M$ and $N$ be the midpoints of $BC$ and $KM$ respectively. Prove that the points $B, C, N,$ and $L$ are concyclic.

1990 National High School Mathematics League, 8

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Point $A(2,0)$. $P(\sin(2t-\frac{\pi}{3}),\cos(2t-\frac{\pi}{3}))$ is a moving point. When $t$ changes from $\frac{\pi}{12}$ to $\frac{\pi}{4}$, area swept by segment $AP$ is________.

2019 South East Mathematical Olympiad, 5

Tags: sequence , algebra

For positive integer n, define $a_n$ as the number of the triangles with integer length of every side and the length of the longest side being $2n.$ (1) Find $a_n$ in terms of $n;$ (2)If the sequence $\{ b_n\}$ satisfying for any positive integer $n,$ $\sum_{k=1}^n(-1)^{n-k}\binom {n}{k} b_k=a_n.$ Find the number of positive integer $n$ satisfying that $b_n\leq 2019a_n.$

2008 ITest, 76

Tags: quadratic , calculus

During the car ride home, Michael looks back at his recent math exams. A problem on Michael's calculus mid-term gets him starting thinking about a particular quadratic, \[x^2-sx+p,\] with roots $r_1$ and $r_2$. He notices that \[r_1+r_2=r_1^2+r_2^2=r_1^3+r_2^3=\cdots=r_1^{2007}+r_2^{2007}.\] He wonders how often this is the case, and begins exploring other quantities associated with the roots of such a quadratic. He sets out to compute the greatest possible value of \[\dfrac1{r_1^{2008}}+\dfrac1{r_2^{2008}}.\] Help Michael by computing this maximum.

2011 Math Prize For Girls Problems, 15

Tags: 3d geometry , probability , geometry

The game of backgammon has a "doubling" cube, which is like a standard 6-faced die except that its faces are inscribed with the numbers 2, 4, 8, 16, 32, and 64, respectively. After rolling the doubling cube four times at random, we let $a$ be the value of the first roll, $b$ be the value of the second roll, $c$ be the value of the third roll, and $d$ be the value of the fourth roll. What is the probability that $\frac{a + b}{c + d}$ is the average of $\frac{a}{c}$ and $\frac{b}{d}$ ?

1994 Romania TST for IMO, 3:

Tags: number theory

Prove that the sequence $a_n = 3^n- 2^n$ contains no three numbers in geometric progression.

1986 Putnam, A4

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A [i]transversal[/i] of an $n\times n$ matrix $A$ consists of $n$ entries of $A$, no two in the same row or column. Let $f(n)$ be the number of $n \times n$ matrices $A$ satisfying the following two conditions: (a) Each entry $\alpha_{i,j}$ of $A$ is in the set $\{-1,0,1\}$. (b) The sum of the $n$ entries of a transversal is the same for all transversals of $A$. An example of such a matrix $A$ is \[ A = \left( \begin{array}{ccc} -1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \end{array} \right). \] Determine with proof a formula for $f(n)$ of the form \[ f(n) = a_1 b_1^n + a_2 b_2^n + a_3 b_3^n + a_4, \] where the $a_i$'s and $b_i$'s are rational numbers.

2011 Harvard-MIT Mathematics Tournament, 2

Tags: hmmt

Let $a \star b = ab + a + b$ for all integers $a$ and $b$. Evaluate $1 \star ( 2 \star ( 3 \star (4 \star \ldots ( 99 \star 100 ) \ldots )))$.

1991 AIME Problems, 11

Tags: trigonometry , geometry

Twelve congruent disks are placed on a circle $C$ of radius 1 in such a way that the twelve disks cover $C$, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The sum of the areas of the twelve disks can be written in the from $\pi(a-b\sqrt{c})$, where $a,b,c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a+b+c$. [asy] real r=2-sqrt(3); draw(Circle(origin, 1)); int i; for(i=0; i<12; i=i+1) { draw(Circle(dir(30*i), r)); dot(dir(30*i)); } draw(origin--(1,0)--dir(30)--cycle); label("1", (0.5,0), S);[/asy]

2024 China Girls Math Olympiad, 4

Tags: circle inversion , geometry

Let $ABC$ be a triangle with $AB<BC<CA$ and let $D$ be a variable point on $BC$. The point $E$ on the circumcircle of $ABC$ is such that $\angle BAD=\angle BED$. The line through $D$ perpendicular to $AB$ meets $AC$ at $F$. Show that the measure of $\angle BEF$ is constant as $D$ varies.

the 14th XMO, P3

Tags: geometry

In quadrilateral $ABCD$, $E$ and $F$ are midpoints of $AB$ and $CD$, and $G$ is the intersection of $AD$ with $BC$. $P$ is a point within the quadrilateral, such that $PA=PB$, $PC=PD$, and $\angle APB+\angle CPD=180^{\circ}$. Prove that $PG$ and $EF$ are parallel.

2023 Durer Math Competition Finals, 7

Tags: geometry

The area of a rectangle is $64$ cm$^2$, and the radius of its circumscribed circle is $7$ cm. What is the perimeter of the rectangle in centimetres?

1962 AMC 12/AHSME, 5

Tags: ratio

If the radius of a circle is increased by $ 1$ unit, the ratio of the new circumference to the new diameter is: $ \textbf{(A)}\ \pi \plus{} 2 \qquad \textbf{(B)}\ \frac{2 \pi \plus{} 1}{2} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{2 \pi \minus{} 1}{2} \qquad \textbf{(E)}\ \pi \minus{} 2$

2023 Azerbaijan IMO TST, 4

Tags: number theory

A number is called [i]Norwegian[/i] if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number. (Note: The total number of positive divisors of a Norwegian number is allowed to be larger than $3$.)

2017 Czech-Polish-Slovak Match, 2

Tags: board , maximum , square , combinatorics

Each of the ${4n^2}$ unit squares of a ${2n \times 2n}$ board ${(n \ge 1) }$ has been colored blue or red. A set of four different unit squares of the board is called [i]pretty [/i]if these squares can be labeled ${A,B,C,D}$ in such a way that ${A}$ and ${B}$ lie in the same row, ${C}$ and ${D}$ lie in the same row, ${A}$ and ${C}$ lie in the same column, ${B}$ and ${D}$ lie in the same column, ${A}$ and ${D}$ are blue, and ${B}$ and ${C}$ are red. Determine the largest possible number of different [i]pretty [/i]sets on such a board. (Poland)