Olympiad problems

IV Soros Olympiad 1997 - 98 (Russia), 11.9

Cut pyramid $ABCD$ into $8$ equal and similar pyramids, if: a) $AB = BC = CD$, $\angle ABC =\angle BCD = 90^o$, dihedral angle at edge $BC$ is right b) all plane angles at vertex $B$ are right and $AB = BC = BD\sqrt2$. Note. Whether there are other types of triangular pyramids that can be cut into any number similar to the original pyramids (their number is not necessarily $8$ and the pyramids are not necessarily equal to each other) is currently unknown

2001 Bundeswettbewerb Mathematik, 1

Tags: geometry , perimeter , combinatorics solved , combinatorics

10 vertices of a regular 100-gon are coloured red and ten other (distinct) vertices are coloured blue. Prove that there is at least one connection edge (segment) of two red which is as long as the connection edge of two blue points. [hide="Hint"]Possible approaches are pigeon hole principle, proof by contradiction, consider turns (bijective congruent mappings) which maps red in blue points. [/hide]

2006 Vietnam Team Selection Test, 2

Tags: number theory proposed , number theory

Find all pair of integer numbers $(n,k)$ such that $n$ is not negative and $k$ is greater than $1$, and satisfying that the number: \[ A=17^{2006n}+4.17^{2n}+7.19^{5n} \] can be represented as the product of $k$ consecutive positive integers.

2012 Grigore Moisil Intercounty, 4

Tags: vector , geometry proposed , geometry

Let $ \Delta ABC$ be a triangle with $M$ the middle of the side $[BC]$. On the line $BC$, to the left and to the right of the point $M,$ at the same distance from $M,$ let us consider $d_1$ and $d_2,$ which are perpendicular to the line BC. The perpendicular line from $M$ to $AB$ intersects $d_1$ in $P,$ and the perpendicular line from $M$ to $AC$ intersects $d_2$ in $Q.$ Prove that \[AM\perp PQ.\] [b]Author: Marin Bancoș Regional Mathematical Contest GRIGORE MOISIL, Romania, Baia Mare, 2012, 9th grade[/b]

1984 IMO Shortlist, 19

Tags: algebra , combinatorics , recurrence relation , Linear Recurrences , IMO Shortlist

The harmonic table is a triangular array: $1$ $\frac 12 \qquad \frac 12$ $\frac 13 \qquad \frac 16 \qquad \frac 13$ $\frac 14 \qquad \frac 1{12} \qquad \frac 1{12} \qquad \frac 14$ Where $a_{n,1} = \frac 1n$ and $a_{n,k+1} = a_{n-1,k} - a_{n,k}$ for $1 \leq k \leq n-1.$ Find the harmonic mean of the $1985^{th}$ row.

2006 Cuba MO, 5

Tags: number theory , Divisors , primes

The following sequence of positive integers $a_1, a_2, ..., a_{400}$ satisfies the relationship $a_{n+1} = \tau (a_n) + \tau (n)$ for all $1 \le n \le 399$, where $\tau (k) $ is the number of positive integer divisors that $k$ has. Prove that in the sequence there are no more than $210$ prime numbers.

1992 India Regional Mathematical Olympiad, 7

Tags: parameterization

Solve the system \begin{eqnarray*} \\ (x+y)(x+y+z) &=& 18 \\ (y+z)(x+y+z) &=& 30 \\ (x+z)(x+y+z) &=& 2A \end{eqnarray*} in terms of the parameter $A$.

2021 CMIMC, 1

Tags: geometry

Given a trapezoid with bases $AB$ and $CD$, there exists a point $E$ on $CD$ such that drawing the segments $AE$ and $BE$ partitions the trapezoid into $3$ similar isosceles triangles, each with long side twice the short side. What is the sum of all possible values of $\frac{CD}{AB}$? [i]Proposed by Adam Bertelli[/i]

2002 Moldova Team Selection Test, 1

Tags: inequalities , number theory proposed , number theory

Consider the triangular numbers $T_n = \frac{n(n+1)}{2} , n \in \mathbb N$. [list][b](a)[/b] If $a_n$ is the last digit of $T_n$, show that the sequence $(a_n)$ is periodic and find its basic period. [b](b)[/b] If $s_n$ is the sum of the first $n$ terms of the sequence $(T_n)$, prove that for every $n \geq 3$ there is at least one perfect square between $s_{n-1} and $s_n$.[/list]

2005 Germany Team Selection Test, 3

Tags: geometry , inradius , trigonometry , inequalities , function , triangle inequality , geometry solved

Let ABC be a triangle and let $r, r_a, r_b, r_c$ denote the inradius and ex-radii opposite to the vertices $A, B, C$, respectively. Suppose that $a>r_a, b>r_b, c>r_c$. Prove that [b](a)[/b] $\triangle ABC$ is acute. [b](b)[/b] $a+b+c > r+r_a+r_b+r_c$.

2018 Rio de Janeiro Mathematical Olympiad, 3

Tags: function , Perfect Square , combinatorics

Let $n$ be a positive integer. A function $f : \{1, 2, \dots, 2n\} \to \{1, 2, 3, 4, 5\}$ is [i]good[/i] if $f(j+2)$ and $f(j)$ have the same parity for every $j = 1, 2, \dots, 2n-2$. Prove that the number of good functions is a perfect square.

2016 Singapore Senior Math Olympiad, 1

Tags: geometry , angle bisector

Let $\triangle ABC$ be a triangle with $AB < AC$. Let the angle bisector of $\angle BAC$ meet $BC$ at $D$ , and let $M$ be the midpoint of $BC$ . Let $P$ be the foot of the perpendicular from $B$ to $AD$ . $Q$ the intersection of $BP$ and $AM$ . Show that : $(DQ) // (AB) $ .

2010 Gheorghe Vranceanu, 4

Tags: number theory , algebra , seuqences , Periodicity

Let be two real numbers $ \alpha ,\beta $ and two sequences $ \left(x_n \right)_{n\ge 1} ,\left(y_n \right)_{n\ge 1} $ whose smallest periods are $ p,q, $ respectively. Prove that the sequence $ \left( \alpha x_n+\beta y_n\right)_{n\ge 1} $ is periodic if $ \text{gcd}^2 (p,q) | \text{lcm} (p,q) , $ and in this case find its smallest period.

1982 USAMO, 3

Tags: AMC , USA(J)MO , USAMO , geometry , perimeter , function , trigonometry

If a point $A_1$ is in the interior of an equilateral triangle $ABC$ and point $A_2$ is in the interior of $\triangle{A_1BC}$, prove that \[\operatorname{I.Q.} (A_1BC) > \operatorname{I.Q.} (A_2BC),\] where the [i]isoperrimetric quotient[/i] of a figure $F$ is defined by \[\operatorname{I.Q.}(F) = \frac{\operatorname{Area}(F)}{[\operatorname{Perimeter}(F)]^2}.\]

2020 Latvia Baltic Way TST, 5

Tags: combinatorics

Natural numbers $1,2,...,500$ are written on a blackboard. Two players $A$ and $B$ consecutively make moves, $A$ starts. Each move a player chooses two numbers $n$ and $2n$ and erases them from the blackboard. If a player cannot perform a valid move, he loses. Which player can guarantee a win?

2006 IMS, 2

Tags: induction , number theory proposed , number theory

For each subset $C$ of $\mathbb N$, Suppose $C\oplus C=\{x+y|x,y\in C, x\neq y\}$. Prove that there exist a unique partition of $\mathbb N$ to sets $A$, $B$ that $A\oplus A$ and $B\oplus B$ do not have any prime numbers.

2018 MIG, 15

Tags: 2018 Individual

Gordon has the least number of coins (half-dollars, quarters, dimes, nickels, pennies) needed to make $99\cent$. He randomly chooses one. What is the probability that it is a penny? $\textbf{(A) } \dfrac15\qquad\textbf{(B) } \dfrac13\qquad\textbf{(C) } \dfrac12\qquad\textbf{(D) } \dfrac23\qquad\textbf{(E) } \dfrac34$

2020 USMCA, 21

Tags:

The sequence $a_1,a_2,\ldots$ is defined by $a_1=2019$, $a_2=2020$, $a_3=2021$, $a_{n+3}=a_n(a_{n+1}a_{n+2}+1)$ for $n\ge 1$. Determine the value of the infinite sum \[\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\cdots \]

2016 Oral Moscow Geometry Olympiad, 6

Tags: paper , Segment , geometry , folding , square

Given a square sheet of paper with a side of $2016$. Is it possible to bend its not more than ten times, construct a segment of length $1$?

2009 Hong Kong TST, 3

Tags: number theory proposed , number theory

Prove that there are infinitely many primes $ p$ such that the total number of solutions mod $ p$ to the equation $ 3x^{3}\plus{}4y^{4}\plus{}5z^{3}\minus{}y^{4}z \equiv 0$ is $ p^2$

2017 Miklós Schweitzer, 7

Tags: measure , Measure theory , interval , real analysis , college contests , Miklos Schweitzer

Characterize all increasing sequences $(s_n)$ of positive reals for which there exists a set $A\subset \mathbb{R}$ with positive measure such that $\lambda(A\cap I)<\frac{s_n}{n}$ holds for every interval $I$ with length $1/n$, where $\lambda$ denotes the Lebesgue measure.

2007 Purple Comet Problems, 8

Tags: probability , number theory , relatively prime

You know that the Jones family has five children, and the Smith family has three children. Of the eight children you know that there are five girls and three boys. Let $\dfrac{m}{n}$ be the probability that at least one of the families has only girls for children. Given that $m$ and $n$ are relatively prime positive integers, find $m+ n$.

2011 Hanoi Open Mathematics Competitions, 1

Tags: number theory , divisible

An integer is called "octal" if it is divisible by $8$ or if at least one of its digits is $8$. How many integers between $1$ and $100$ are octal? (A): $22$, (B): $24$, (C): $27$, (D): $30$, (E): $33$

2014 Dutch IMO TST, 2

Tags: number theory unsolved , number theory

The sets $A$ and $B$ are subsets of the positive integers. The sum of any two distinct elements of $A$ is an element of $B$. The quotient of any two distinct elements of $B$ (where we divide the largest by the smallest of the two) is an element of $A$. Determine the maximum number of elements in $A\cup B$.

2018 CMIMC Algebra, 9

Tags: 2018 , algebra

Suppose $a_0,a_1,\ldots, a_{2018}$ are integers such that \[(x^2-3x+1)^{1009} = \sum_{k=0}^{2018}a_kx^k\] for all real numbers $x$. Compute the remainder when $a_0^2 + a_1^2 + \cdots + a_{2018}^2$ is divided by $2017$.