Olympiad problems

2014 German National Olympiad, 1

For which non-negative integers $n$ is \[K=5^{2n+3} + 3^{n+3} \cdot 2^n\] prime?

2003 District Olympiad, 1

Tags: 3d geometry , equilateral , plane , angle , geometry

Let $ABC$ be an equilateral triangle. On the plane $(ABC)$ rise the perpendiculars $AA'$ and $BB'$ on the same side of the plane, so that $AA' = AB$ and $BB' =\frac12 AB$. Determine the measure the angle between the planes $(ABC)$ and $(A'B'C')$.

2016 Germany Team Selection Test, 3

Tags: geometry

Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.

2002 China Team Selection Test, 2

Tags: number theory

Find all non-negative integers $m$ and $n$, such that $(2^n-1) \cdot (3^n-1)=m^2$.

2019 Math Prize for Girls Olympiad, 1

Tags:

Let $A_1$, $A_2$, $\ldots\,$, $A_n$ be finite sets. Prove that \[ \Bigl| \bigcup_{1 \le i \le n} A_i \Bigr| \ge \frac{1}{2} \sum_{1 \le i \le n} \left| A_i \right| - \frac{1}{6} \sum_{1 \le i < j \le n} \left| A_i \cap A_j \right| \, . \] Recall that if $S$ is a finite set, then its cardinality $|S|$ is the number of elements of $S$.

2024 Canadian Mathematical Olympiad Qualification, 4

Tags: algebra , recurrence relation , inequalities

A sequence $\{a_i\}$ is given such that $a_1 = \frac13$ and for all positive integers $n$ $$a_{n+1} =\frac{a^2_n}{a^2_n - a_n + 1}.$$ Prove that $$\frac12 - \frac{1}{3^{2^{n-1}}} < a_1 + a_2 +... + a_n <\frac12 - \frac{1}{3^{2^n}} ,$$ for all positive integers $n$.

1984 AIME Problems, 2

Tags: number theory , divisibility tests

The integer $n$ is the smallest positive multiple of 15 such that every digit of $n$ is either 8 or 0. Compute $\frac{n}{15}$.

Croatia MO (HMO) - geometry, 2013.7

Tags: angle bisector , geometry , angle

In triangle $ABC$, the angle at vertex $B$ is $120^o$. Let $A_1, B_1, C_1$ be points on the sides $BC, CA, AB$ respectively such that $AA_1, BB_1, CC_1$ are bisectors of the angles of triangle $ABC$. Determine the angle $\angle A_1B_1C_1$.

2006 IMO Shortlist, 1

Tags: floor function , algebra , sequence , recurrence relation , periodic sequence

A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula \[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0; \]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large. [i]Proposed by Harmel Nestra, Estionia[/i]

2013 Junior Balkan Team Selection Tests - Romania, 2

Tags: sum of digits , digit , mutliple , number theory

Call the number $\overline{a_1a_2... a_m}$ ($a_1 \ne 0,a_m \ne 0$) the reverse of the number $\overline{a_m...a_2a_1}$. Prove that the sum between a number $n$ and its reverse is a multiple of $81$ if and only if the sum of the digits of $n$ is a multiple of $81$.

2010 AMC 12/AHSME, 25

Tags: function

For every integer $ n\ge 2$, let $ \text{pow}(n)$ be the largest power of the largest prime that divides $ n$. For example $ \text{pow}(144)\equal{}\text{pow}(2^4\cdot 3^2)\equal{}3^2$. What is the largest integer $ m$ such that $ 2010^m$ divides \[ \prod_{n\equal{}2}^{5300}\text{pow}(n)\text{?}\] $ \textbf{(A)}\ 74 \qquad \textbf{(B)}\ 75 \qquad \textbf{(C)}\ 76 \qquad \textbf{(D)}\ 77 \qquad \textbf{(E)}\ 78$

2022 Bosnia and Herzegovina Junior BMO TST, 4

Tags: combinatorics

Some people know each other in a group of people, where "knowing" is a symmetric relation. For a person, we say that it is $social$ if it knows at least $20$ other persons and at least $2$ of those $20$ know each other. For a person, we say that it is $shy$ if it doesn't know at least $20$ other persons and at least $2$ of those $20$ don't know each other. Find the maximal number of people in that group, if we know that group doesn't have any $social$ nor $shy$ persons.

2005 Estonia National Olympiad, 4

Tags: radical , algebra

Represent the number $\sqrt[3]{1342\sqrt{167}+2005}$ in the form where it contains only addition, subtraction, multiplication, division and square roots.

2022 Novosibirsk Oral Olympiad in Geometry, 4

Tags: incenter , circumcircle , geometry

A point $D$ is marked on the side $AC$ of triangle $ABC$. The circumscribed circle of triangle $ABD$ passes through the center of the inscribed circle of triangle $BCD$. Find $\angle ACB$ if $\angle ABC = 40^o$.

2019 District Olympiad, 1

Tags: endomorphism , superior algebra , group , abstract algebra

Let $n$ be a positive integer and $G$ be a finite group of order $n.$ A function $f:G \to G$ has the $(P)$ property if $f(xyz)=f(x)f(y)f(z)~\forall~x,y,z \in G.$ $\textbf{(a)}$ If $n$ is odd, prove that every function having the $(P)$ property is an endomorphism. $\textbf{(b)}$ If $n$ is even, is the conclusion from $\textbf{(a)}$ still true?

1996 Spain Mathematical Olympiad, 6

Tags: geometry , regular polygon , volume , 3-dimensional geometry

A regular pentagon is constructed externally on each side of a regular pentagon of side $1$. The figure is then folded and the two edges of the external pentagons meeting at each vertex of the original pentagon are glued together. Find the volume of water that can be poured into the obtained container.

2024 Mongolian Mathematical Olympiad, 2

Tags: combinatorics , geometry , extremal

We call a triangle consisting of three vertices of a pentagon [i]big[/i] if it's area is larger than half of the pentagon's area. Find the maximum number of [i]big[/i] triangles that can be in a convex pentagon. [i]Proposed by Gonchigdorj Sandag[/i]

2018 IFYM, Sozopol, 7

Tags: recurrence relation , recursion , algebra

The rows $x_n$ and $y_n$ of positive real numbers are such that: $x_{n+1}=x_n+\frac{1}{2y_n}$ and $y_{n+1}=y_n+\frac{1}{2x_n}$ for each positive integer $n$. Prove that at least one of the numbers $x_{2018}$ and $y_{2018}$ is bigger than 44,9

2018 Mexico National Olympiad, 3

Tags: number theory , relatively prime

A sequence $a_2, a_3, \dots, a_n$ of positive integers is said to be [i]campechana[/i], if for each $i$ such that $2 \leq i \leq n$ it holds that exactly $a_i$ terms of the sequence are relatively prime to $i$. We say that the [i]size[/i] of such a sequence is $n - 1$. Let $m = p_1p_2 \dots p_k$, where $p_1, p_2, \dots, p_k$ are pairwise distinct primes and $k \geq 2$. Show that there exist at least two different campechana sequences of size $m$.

2014 Contests, 2

Tags:

How many pairs of integers $(m,n)$ are there such that $mn+n+14=\left (m-1 \right)^2$? $ \textbf{a)}\ 16 \qquad\textbf{b)}\ 12 \qquad\textbf{c)}\ 8 \qquad\textbf{d)}\ 6 \qquad\textbf{e)}\ 2 $

2019 District Olympiad, 1

Tags: limit , convergence , calculus , sequence

Let $(a_n)_{n \ge 1}$ be a sequence of positive real numbers such that the sequence $(a_{n+1}-a_n)_{n \ge 1}$ is convergent to a non-zero real number. Evaluate the limit $$ \lim_{n \to \infty} \left( \frac{a_{n+1}}{a_n} \right)^n.$$

Estonia Open Junior - geometry, 2005.1.3

Tags: geometry , angle bisector , concurrency

In triangle $ABC$, the midpoints of sides $AB$ and $AC$ are $D$ and $E$, respectively. Prove that the bisectors of the angles $BDE$ and $CED$ intersect at the side $BC$ if the length of side $BC$ is the arithmetic mean of the lengths of sides $AB$ and $AC$.

1962 IMO Shortlist, 2

Tags:

Determine all real numbers $x$ which satisfy the inequality: \[ \sqrt{3-x}-\sqrt{x+1}>\dfrac{1}{2} \]

2020 OMpD, 4

Tags: function , real number , positive real , algebra

Let $\mathbb{R}^+$ the set of positive real numbers. Determine all the functions $f, g: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that, for all positive real numbers $x, y$ we have that $$f(x + g(y)) = f(x + y) + g(y) \text{ and } g(x + f(y)) = g(x + y) + f(y)$$

2003 Moldova National Olympiad, 12.5

Tags: polynomial , algebra

Consider the polynomial $P(x)=X^{2n}-X^{2n-1}+\dots-x+1$, where $n\in{N^*}$. Find the remainder of the division of polynomial $P(x^{2n+1})$ by $P(x)$.