Olympiad problems

2021 Moldova Team Selection Test, 12

Tags: number theory

Prove that $n!\cdot(n+1)!\cdot(n+2)!$ divides $(3n)!$ for every integer $n \geq 3$.

1953 AMC 12/AHSME, 6

Tags:

Charles has $ 5q \plus{} 1$ quarters and Richard has $ q \plus{} 5$ quarters. The difference in their money in dimes is: $ \textbf{(A)}\ 10(q \minus{} 1) \qquad\textbf{(B)}\ \frac {2}{5}(4q \minus{} 4) \qquad\textbf{(C)}\ \frac {2}{5}(q \minus{} 1) \\ \textbf{(D)}\ \frac {5}{2}(q \minus{} 1) \qquad\textbf{(E)}\ \text{none of these}$

1966 Swedish Mathematical Competition, 3

Tags: Sum of Squares , number theory , Perfect Square

Show that an integer $= 7 \mod 8$ cannot be sum of three squares.

1989 IMO Longlists, 33

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Find all square numbers $ S_1$ and $ S_2$ such that $ S_1 \minus{} S_2 \equal{} 1989.$

1992 China Team Selection Test, 1

Tags: combinatorics , combinatorics solved , Probabilistic Method , double counting , China , China TST , maximum

16 students took part in a competition. All problems were multiple choice style. Each problem had four choices. It was said that any two students had at most one answer in common, find the maximum number of problems.

1957 AMC 12/AHSME, 24

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If the square of a number of two digits is decreased by the square of the number formed by reversing the digits, then the result is not always divisible by: $ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ \text{the product of the digits}\qquad \textbf{(C)}\ \text{the sum of the digits}\qquad \textbf{(D)}\ \text{the difference of the digits}\qquad \textbf{(E)}\ 11$

MOAA Individual Speed General Rounds, 2023.2

Tags: MOAA 2023

In the coordinate plane, the line passing through points $(2023,0)$ and $(-2021,2024)$ also passes through $(1,c)$ for a constant $c$. Find $c$. [i]Proposed by Andy Xu[/i]

2015 IFYM, Sozopol, 6

Tags: number theory

A natural number is called [i]“sozopolian”[/i], if it has exactly two prime divisors. Does there exist 12 consecutive [i]“sozopolian”[/i] numbers?

1961 AMC 12/AHSME, 14

Tags: geometry , rhombus , parallelogram , AMC

A rhombus is given with one diagonal twice the length of the other diagonal. Express the side of the rhombus is terms of $K$, where $K$ is the area of the rhombus in square inches. ${{ \textbf{(A)}\ \sqrt{K} \qquad\textbf{(B)}\ \frac{1}{2}\sqrt{2K} \qquad\textbf{(C)}\ \frac{1}{3}\sqrt{3K} \qquad\textbf{(D)}\ \frac{1}{4}\sqrt{4K} }\qquad\textbf{(E)}\ \text{None of these are correct} } $

2019 ELMO Shortlist, N3

Tags: number theory

Let $S$ be a nonempty set of positive integers such that, for any (not necessarily distinct) integers $a$ and $b$ in $S$, the number $ab+1$ is also in $S$. Show that the set of primes that do not divide any element of $S$ is finite. [i]Proposed by Carl Schildkraut[/i]

1999 India Regional Mathematical Olympiad, 3

Tags: geometry , circumcircle

Let $ABCD$ be a square and $M,N$ points on sides $AB, BC$ respectively such that $\angle MDN = 45^{\circ}$. If $R$ is the midpoint of $MN$ show that $RP =RQ$ where $P,Q$ are points of intersection of $AC$ with the lines $MD, ND$.

2016 BMT Spring, 1

Tags: number theory

A bag is filled with quarters and nickels. The average value when pulling out a coin is $10$ cents. What is the least number of nickels in the bag possible?

2010 Romania Team Selection Test, 2

Tags: geometry , circumcircle , incenter , romania , TST

Let $ABC$ be a scalene triangle, let $I$ be its incentre, and let $A_1$, $B_1$ and $C_1$ be the points of contact of the excircles with the sides $BC$, $CA$ and $AB$, respectively. Prove that the circumcircles of the triangles $AIA_1$, $BIB_1$ and $CIC_1$ have a common point different from $I$. [i]Cezar Lupu & Vlad Matei[/i]

Kvant 2019, M2563

Tags: Combinatorial games , game strategy , arithmetic , combinatorics

Pasha and Vova play the following game, making moves in turn; Pasha moves first. Initially, they have a large piece of plasticine. By a move, Pasha cuts one of the existing pieces into three(of arbitrary sizes), and Vova merges two existing pieces into one. Pasha wins if at some point there appear to be $100$ pieces of equal weights. Can Vova prevent Pasha's win?

2019 Gulf Math Olympiad, 2

Tags: number theory , Digits , multiple

1. Find $N$, the smallest positive multiple of $45$ such that all of its digits are either $7$ or $0$. 2. Find $M$, the smallest positive multiple of $32$ such that all of its digits are either $6$ or $1$. 3. How many elements of the set $\{1,2,3,...,1441\}$ have a positive multiple such that all of its digits are either $5$ or $2$?

1999 Slovenia National Olympiad, Problem 1

Tags: algebra , floor function , function , optimization

Let $r_1,r_2,\ldots,r_m$ be positive rational numbers with a sum of $1$. Find the maximum values of the function $f:\mathbb N\to\mathbb Z$ defined by $$f(n)=n-\lfloor r_1n\rfloor-\lfloor r_2n\rfloor-\ldots-\lfloor r_mn\rfloor$$

2021 Saudi Arabia Training Tests, 21

Tags: cyclic quadrilateral , geometry , tangent

Let $ABCD$ be a cyclic quadrilateral with $O$ is circumcenter and $AC$ meets $BD$ at $I$ Suppose that rays $DA,CD$ meet at $E$ and rays $BA,CD$ meet at $F$. The Gauss line of $ABCD$ meets $AB,BC,CD,DA$ at points $M,N,P,Q$ respectively. Prove that the circle of diameter $OI$ is tangent to two circles $(ENQ), (FMP)$

2000 Federal Competition For Advanced Students, Part 2, 1

Tags: algebra , binomial theorem , algebra proposed

The sequence an is defined by $a_0 = 4, a_1 = 1$ and the recurrence formula $a_{n+1} = a_n + 6a_{n-1}$. The sequence $b_n$ is given by \[b_n=\sum_{k=0}^n \binom nk a_k.\] Find the coefficients $\alpha,\beta$ so that $b_n$ satisfies the recurrence formula $b_{n+1} = \alpha b_n + \beta b_{n-1}$. Find the explicit form of $b_n$.

2018 Vietnam Team Selection Test, 4

Tags: algebra , Sequence , inequalities

Let $a\in\left[ \tfrac{1}{2},\ \tfrac{3}{2}\right]$ be a real number. Sequences $(u_n),\ (v_n)$ are defined as follows: $$u_n=\frac{3}{2^{n+1}}\cdot (-1)^{\lfloor2^{n+1}a\rfloor},\ v_n=\frac{3}{2^{n+1}}\cdot (-1)^{n+\lfloor 2^{n+1}a\rfloor}.$$ a. Prove that $${{({{u}_{0}}+{{u}_{1}}+\cdots +{{u}_{2018}})}^{2}}+{{({{v}_{0}}+{{v}_{1}}+\cdots +{{v}_{2018}})}^{2}}\le 72{{a}^{2}}-48a+10+\frac{2}{{{4}^{2019}}}.$$ b. Find all values of $a$ in the equality case.

1999 Slovenia National Olympiad, Problem 3

Tags: geometry

Let $O$ be the circumcenter of a triangle $ABC$, $P$ be the midpoint of $AO$, and $Q$ be the midpoint of $BC$. If $\angle ABC=4\angle OPQ$ and $\angle ACB=6\angle OPQ$, compute $\angle OPQ$.

2017 AIME Problems, 3

Tags: 2017 AIME I

For a positive integer $n$, let $d_n$ be the units digit of $1 + 2 + \dots + n$. Find the remainder when \[\sum_{n=1}^{2017} d_n\] is divided by $1000$.

2003 France Team Selection Test, 1

Tags: geometry , circumcircle , homothety , incenter , IMO Shortlist , geometry solved , tangent circles

Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.

2012 NIMO Problems, 5

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If $w = a + bi$, where $a$ and $b$ are real numbers, then $\Re(w) = a$ and $\Im(w) = b$. Let $z=c+di$, where $c, d \ge 0$. If \begin{align*} \Re(z) + \Im (z) & = 7, \\ \Re(z^2) + \Im(z^2) & = 17, \end{align*} then compute $\left | \Re\left (z^3 \right ) + \Im \left (z^3 \right ) \right |$. [i]Proposed by Lewis Chen[/i]

2018 AMC 12/AHSME, 9

Tags: Gauss , 2018 AMC 12B , 2018 AMC , AMC 12

What is \[ \sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) ? \] $ \textbf{(A) }100,100 \qquad \textbf{(B) }500,500\qquad \textbf{(C) }505,000 \qquad \textbf{(D) }1,001,000 \qquad \textbf{(E) }1,010,000 \qquad $

1990 Romania Team Selection Test, 6

Tags: partition , Subsets , combinatorics

Prove that there are infinitely many n’s for which there exists a partition of $\{1,2,...,3n\}$ into subsets $\{a_1,...,a_n\}, \{b_1,...,b_n\}, \{c_1,...,c_n\}$ such that $a_i +b_i = c_i$ for all $i$, and prove that there are infinitely many $n$’s for which there is no such partition.