Olympiad problems

PEN Q Problems, 9

Tags: function , algebra , polynomial , calculus , integration , rational function , Polynomials

For non-negative integers $n$ and $k$, let $P_{n, k}(x)$ denote the rational function \[\frac{(x^{n}-1)(x^{n}-x) \cdots (x^{n}-x^{k-1})}{(x^{k}-1)(x^{k}-x) \cdots (x^{k}-x^{k-1})}.\] Show that $P_{n, k}(x)$ is actually a polynomial for all $n, k \in \mathbb{N}$.

2018 Iranian Geometry Olympiad, 4

Tags: tangential quadrilateral , geometry , cyclic quadrilateral , angle bisector

Quadrilateral $ABCD$ is circumscribed around a circle. Diagonals $AC,BD$ are not perpendicular to each other. The angle bisectors of angles between these diagonals, intersect the segments $AB,BC,CD$ and $DA$ at points $K,L,M$ and $N$. Given that $KLMN$ is cyclic, prove that so is $ABCD$. Proposed by Nikolai Beluhov (Bulgaria)

1987 Greece National Olympiad, 3

Tags: parameterization , inequalities , algebra

Solve for real values of parameter $a$, the inequality : $$\sqrt{a+x}+ \sqrt{a-x}>a , \ \ x\in\mathbb{R}$$

2020 Novosibirsk Oral Olympiad in Geometry, 3

Tags: geometry , combinatorial geometry

Cut an arbitrary triangle into $2019$ pieces so that one of them turns out to be a triangle, one is a quadrilateral, ... one is a $2019$-gon and one is a $2020$-gon. Polygons do not have to be convex.

2014 Saudi Arabia BMO TST, 2

Tags: Asymptote , geometry unsolved , geometry

Circles $\omega_1$ and $\omega_2$ meet at $P$ and $Q$. Segments $AC$ and $BD$ are chords of $\omega_1$ and $\omega_2$ respectively, such that segment $AB$ and ray $CD$ meet at $P$. Ray $BD$ and segment $AC$ meet at $X$. Point $Y$ lies on $\omega_1$ such that $P Y \parallel BD$. Point $Z$ lies on $\omega_2$ such that $P Z \parallel AC$. Prove that points $Q,~ X,~ Y,~ Z$ are collinear.

2022 Estonia Team Selection Test, 2

Tags: number theory , First digit

Let $d_i$ be the first decimal digit of $2^i$ for every non-negative integer $i$. Prove that for each positive integer $n$ there exists a decimal digit other than $0$ which can be found in the sequence $d_0, d_1, \dots, d_{n-1}$ strictly less than $\frac{n}{17}$ times.

2020 CCA Math Bonanza, I1

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An ant is crawling along the coordinate plane. Each move, it moves one unit up, down, left, or right with equal probability. If it starts at $(0,0)$, what is the probability that it will be at either $(2,1)$ or $(1,2)$ after $6$ moves? [i]2020 CCA Math Bonanza Individual Round #1[/i]

2002 AMC 10, 12

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Mr. Earl E. Bird leaves his house for work at exactly 8:00 A.M. every morning. When he averages $ 40$ miles per hour, he arrives at his workplace three minutes late. When he averages $ 60$ miles per hour, he arrives three minutes early. At what average speed, in miles per hour, should Mr. Bird drive to arrive at his workplace precisely on time? $ \textbf{(A)}\ 45 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 55 \qquad \textbf{(E)}\ 58$

2014 Spain Mathematical Olympiad, 1

Tags: geometry , number theory unsolved , number theory

Let $(x_n)$ be a sequence of positive integers defined by $x_1=2$ and $x_{n+1}=2x_n^3+x_n$ for all integers $n\ge1$. Determine the largest power of $5$ that divides $x_{2014}^2+1$.

1990 National High School Mathematics League, 3

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There are $n$ schools in a city. $i$th school dispatches $C_i(1\leq C_i\leq39,1\leq i\leq n)$ students to watch a football match. The number of all students $\sum_{i=1}^{n}C_{i}=1990$. In each line, there are $199$ seats, but students from the same school must sit in the same line. So, how many lines of seats we need to have to make sure all students have a seat.

2014 ELMO Shortlist, 3

Tags: function , algebra , geometry , reflection

We say a finite set $S$ of points in the plane is [i]very[/i] if for every point $X$ in $S$, there exists an inversion with center $X$ mapping every point in $S$ other than $X$ to another point in $S$ (possibly the same point). (a) Fix an integer $n$. Prove that if $n \ge 2$, then any line segment $\overline{AB}$ contains a unique very set $S$ of size $n$ such that $A, B \in S$. (b) Find the largest possible size of a very set not contained in any line. (Here, an [i]inversion[/i] with center $O$ and radius $r$ sends every point $P$ other than $O$ to the point $P'$ along ray $OP$ such that $OP\cdot OP' = r^2$.) [i]Proposed by Sammy Luo[/i]

1976 Putnam, 3

Tags: college contests

Suppose that we have $n$ events $A_1,\dots, A_n,$ each of which has probability at least $1-a$ of occuring, where $a<1/4.$ Further suppose that $A_i$ and $A_j$ are mutually independent if $|i-j|>1.$ Assume as known that the recurrence $u_{k+1}=u_k-au_{k-1}, u_0=1, u_1=1-a,$ defines positive real numb $u_k$ for $k=0,1,\dots.$ Show that the probability of all of $A_1,\dots, A_n$ occuring is at least $u_n.$

2021 HMNT, 2

Tags: number theory

Suppose $a$ and $b$ are positive integers for which $8a^ab^b = 27a^bb^a$. Find $a^2 + b^2$.

2000 Finnish National High School Mathematics Competition, 2

Tags: calculus , integration , number theory unsolved , number theory

Prove that the integral part of the decimal representation of the number $(3+\sqrt{5})^n$ is odd, for every positive integer $n.$

Ukrainian From Tasks to Tasks - geometry, 2012.4

Tags: geometry , trapezoid , angles

Let $ABCD$ be an isosceles trapezoid ($AD\parallel BC$), $\angle BAD = 80^o$, $\angle BDA = 60^o$. Point $P$ lies on $CD$ and $\angle PAD = 50^o$. Find $\angle PBC$

2019 Teodor Topan, 2

Tags: arithmetic sequence , ratio , real analysis , seqeunces , Newton s binom , Cesaro-Stolz , squeeze theorem

Let $ \left( a_n \right)_{n\ge 1} $ be an arithmetic progression with $ a_1=1 $ and natural ratio. [b]a)[/b] Prove that $$ a_n^{1/a_k} <1+\sqrt{\frac{2\left( a_n-1 \right)}{a_k\left( a_k -1 \right)}} , $$ for any natural numbers $ 2\le k\le n. $ [b]b)[/b] Calculate $ \lim_{n\to\infty } \frac{1}{a_n}\sum_{k=1}^n a_n^{1/a_k} . $ [i]Nicolae Bourbăcuț[/i]

2016 Hong Kong TST, 3

Tags: number theory , prime numbers

Let $p$ be a prime number greater than 5. Suppose there is an integer $k$ satisfying that $k^2+5$ is divisible by $p$. Prove that there are positive integers $m$ and $n$ such that $p^2=m^2+5n^2$

1994 All-Russian Olympiad Regional Round, 11.4

Tags: combinatorics unsolved , combinatorics

On the vertices of a convex $ n$-gon are put $ m$ stones, $ m > n$. In each move we can choose two stones standing at the same vertex and move them to the two distinct adjacent vertices. After $ N$ moves the number of stones at each vertex was the same as at the beginning. Prove that $ N$ is divisible by $ n$.

2017 Federal Competition For Advanced Students, 3

Tags: combinatorics

Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. At the beginning of a turn there are n ≥ 1 marbles on the table, then the player whose turn is removes k marbles, where k ≥ 1 either is an even number with $k \le \frac{n}{2}$ or an odd number with $ \frac{n}{2}\le k \le n$. A player wins the game if she removes the last marble from the table. Determine the smallest number $N\ge100000$ which Berta has wining strategy. [i]proposed by Gerhard Woeginger[/i]

2022 German National Olympiad, 2

Tags: planar geometry , planametry , geometry proposed , lengths , geometry

As everyone knows, the people of [i]Plane Land[/i] love Planimetrics. Therefore, they imagine their country as completely planar, every city in the country as a geometric point and every road as the line segment connecting two points. Additionally to the existing cities, it is possible to build [i]roundabouts[/i], i.e. points in the road network from where at least two roads emanate. All road crossings or junctions are build as roundabouts. Via this route network, every two cities should be connected by a sequence of roads and possibly roundabouts. In Plane Land, the length of a road is taken as the geometric length of the corresponding line segment. The ingenious road engineer Armin Asphalt presents a new road map, of which it is known that there is no road network with a smaller total length of all roads. Moreover, there is no road map with the same total length of all roads and fewer roundabouts. Prove that in the road map of Armin Asphalt, at most three roads emanate from each city, and exactly three from each roundabout.

2016 Saudi Arabia GMO TST, 2

Tags: algebra , polynomial

Let $c$ be a given real number. Find all polynomials $P$ with real coefficients such that: $(x + 1)P(x - 1) - (x - 1)P(x) = c$ for all $x \in R$

2014 JBMO TST - Macedonia, 3

Tags: number theory , Digits , Divisibility

Find all positive integers $n$ which are divisible by 11 and satisfy the following condition: all the numbers which are generated by an arbitrary rearrangement of the digits of $n$, are also divisible by 11.

2004 AIME Problems, 6

Tags: LaTeX , AMC , AIME , AIME II

An integer is called snakelike if its decimal representation $a_1a_2a_3\cdots a_k$ satisfies $a_i<a_{i+1}$ if $i$ is odd and $a_i>a_{i+1}$ if $i$ is even. How many snakelike integers between 1000 and 9999 have four distinct digits?

LMT Accuracy Rounds, 2021 F3

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Two circles with radius $2$, $\omega_1$ and $\omega_2$, are centered at $O_1$ and $O_2$ respectively. The circles $\omega_1$ and $\omega_2$ are externally tangent to each other and internally tangent to a larger circle $\omega$ centered at $O$ at points $A$ and $B$, respectively. Let $M$ be the midpoint of minor arc $AB$. Let $P$ be the intersection of $\omega_1$ and $O_1M$, and let $Q$ be the intersection of $\omega_2$ and $O_2M$. Given that there is a point $R$ on $\omega$ such that $\triangle PQR$ is equilateral, the radius of $\omega$ can be written as $\frac{a+\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers and $a$ and $c$ are relatively prime. Find $a+b+c$.

2022 Abelkonkurransen Finale, 2a

Tags: geometry , Angle Chasing , harmonic division

A triangle $ABC$ with circumcircle $\omega$ satisfies $|AB| > |AC|$. Points $X$ and $Y$ on $\omega$ are different from $A$, such that the line $AX$ passes through the midpoint of $BC$, $AY$ is perpendicular to $BC$, and $XY$ is parallel to $BC$. Find $\angle BAC$.