Olympiad problems

Estonia Open Junior - geometry, 1996.1.4

Tags: geometry , trapezoid

In a trapezoid, the two non parallel sides and a base have length $1$, while the other base and both the diagonals have length $a$. Find the value of $a$.

2016 NIMO Problems, 6

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Consider a sequence $a_0$, $a_1$, $\ldots$, $a_9$ of distinct positive integers such that $a_0=1$, $a_i < 512$ for all $i$, and for every $1 \le k \le 9$ there exists $0 \le m \le k-1$ such that \[(a_k-2a_m)(a_k-2a_m-1) = 0.\] Let $N$ be the number of these sequences. Find the remainder when $N$ is divided by $1000$. [i]Based on a proposal by Gyumin Roh[/i]

2019 Stars of Mathematics, 2

Tags: number theory

If $n\geqslant 3$ is an integer and $a_1,a_2,\dotsc ,a_n$ are non-zero integers such that $$a_1a_2\cdots a_n\left( \frac{1}{a_1^2}+\frac{1}{a_2^2} +\cdots +\frac{1}{a_n^2}\right)$$is an integer, does it follow that the product $a_1a_2\cdots a_n$ is divisible by each $a_i^2$?

2014 Belarus Team Selection Test, 4

Tags: combinatorial geometry , combinatorics , acute

Thirty rays with the origin at the same point are constructed on a plane. Consider all angles between any two of these rays. Let $N$ be the number of acute angles among these angles. Find the smallest possible value of $N$. (E. Barabanov)

2009 VJIMC, Problem 3

Tags: combinatorics

Let $k$ and $n$ be positive integers such that $k\le n-1$. Let $S:=\{1,2,\ldots,n\}$ and let $A_1,A_2,\ldots,A_k$ be nonempty subsets of $S$. Prove that it is possible to color some elements of $S$ using two colors, red and blue, such that the following conditions are satisfied: (i) Each element of $S$ is either left uncolored or is colored red or blue. (ii) At least one element of $S$ is colored. (iii) Each set $A_i~(i=1,2,\ldots,k)$ is either completely uncolored or it contains at least one red and at least one blue element.

2019 Brazil Team Selection Test, 5

Tags: wrapped , fe , algebra

Determine all the functions $f : \mathbb{R} \to \mathbb{R}$ such that \[ f(x^2 + f(y)) = f(f(x)) + f(y^2) + 2f(xy) \] for all real numbers $x$ and $y$.

1992 All Soviet Union Mathematical Olympiad, 570

Tags: sequence , integer sequence , perfect square , algebra , number theory

Define the sequence $a_1 = 1, a_2, a_3, ...$ by $$a_{n+1} = a_1^2 + a_2 ^2 + a_3^2 + ... + a_n^2 + n$$ Show that $1$ is the only square in the sequence.

Russian TST 2017, P2

Tags: combinatorics , chess rook

A regular hexagon is divided by straight lines parallel to its sides into $6n^2$ equilateral triangles. On them, there are $2n$ rooks, no two of which attack each other (a rook attacks in directions parallel to the sides of the hexagon). Prove that if we color the triangles black and white such that no two adjacent triangles have the same color, there will be as many rooks on the black triangles as on the white ones.

1960 AMC 12/AHSME, 33

Tags: number theory , prime number

You are given a sequence of $58$ terms; each term has the form $P+n$ where $P$ stands for the product $2 \times 3 \times 5 \times... \times 61$ of all prime numbers less than or equal to $61$, and $n$ takes, successively, the values $2, 3, 4, ...., 59$. let $N$ be the number of primes appearing in this sequence. Then $N$ is: $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 17\qquad\textbf{(D)}\ 57\qquad\textbf{(E)}\ 58 $

2004 May Olympiad, 1

Tags: number theory , digit

Javier multiplies four digits, not necessarily different, and obtains a number ending in $7$. Determine how much the sum of the four digits that Javier multiplies can be worth. Give all the possibilities.

2017 IFYM, Sozopol, 7

Tags: number theory

Find all pairs $(x,y)$, $x,y\in \mathbb{N}$ for which $gcd(n(x!-xy-x-y+2)+2,n(x!-xy-x-y+3)+3)>1$ for $\forall$ $n\in \mathbb{N}$.

2006 MOP Homework, 3

Tags: geometric transformation , rotation , geometry

Let $ABC$ be a triangle with $AB\neq AC$, and let $A_{1}B_{1}C_{1}$ be the image of triangle $ABC$ through a rotation $R$ centered at $C$. Let $M,E , F$ be the midpoints of the segments $BA_{1}, AC, BC_{1}$ respectively Given that $EM = FM$, compute $\angle EMF$.

2019 SAFEST Olympiad, 1

Tags: geometry , midpoint , circumcircle , isosceles

Let $ABC$ be an isosceles triangle with $AB = AC$. Let $AD$ be the diameter of the circumcircle of $ABC$ and let $P$ be a point on the smaller arc $BD$. The line $DP$ intersects the rays $AB$ and $AC$ at points $M$ and $N$, respectively. The line $AD$ intersects the lines $BP$ and $CP$ at points $Q$ and $R$, respectively. Prove that the midpoint of $MN$ lies on the circumcircle of $PQR$

2002 Kurschak Competition, 2

Tags: number theory

The Fibonacci sequence is defined as $f_1=f_2=1$, $f_{n+2}=f_{n+1}+f_n$ ($n\in\mathbb{N}$). Suppose that $a$ and $b$ are positive integers such that $\frac ab$ lies between the two fractions $\frac{f_n}{f_{n-1}}$ and $\frac{f_{n+1}}{f_{n}}$. Show that $b\ge f_{n+1}$.

2008 F = Ma, 19

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A car has an engine which delivers a constant power. It accelerates from rest at time $t = 0$, and at $t = t_\text{0}$ its acceleration is $a_\text{0}$. What is its acceleration at $t = 2t_\text{0}$? Ignore energy loss due to friction. (a) $\frac{1}{2}a_\text{0}$ (b) $\frac{1}{\sqrt{2}}a_\text{0}$ (c) $a_\text{0}$ (d) $\sqrt{2}a_\text{0}$ (e) $2a_\text{0}$

2022 Durer Math Competition Finals, 5

Tags: ratio , geometry , angle

On a circle $k$, we marked four points $(A, B, C, D)$ and drew pairwise their connecting segments. We denoted angles as seen on the diagram. We know that $\alpha_1 : \alpha_2 = 2 : 5$, $\beta_1 : \beta_2 = 7 : 11$, and $\gamma_1 : \gamma_2 = 10 : 3$. If $\delta_1 : \delta_2 = p : q$, where $p$ and $q$ are coprime positive integers, then what is $p$? [img]https://cdn.artofproblemsolving.com/attachments/c/e/b532dd164a7cf99cea7b3b7d98f81796622da5.png[/img]

1970 IMO Longlists, 1

Tags: inequalities

Prove that $\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\le \frac{a+b+c}{2}$, where $a,b,c\in\mathbb{R}^{+}$.

LMT Speed Rounds, 2016.23

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Call a positive integer $n\geq 2$ [i]junk[/i] if there exist two distinct $n$ digit binary strings $a_1a_2\cdots a_n$ and $b_1b_2\cdots b_n$ such that [list] [*] $a_1+a_2=b_1+b_2,$ [*] $a_{i-1}+a_i+a_{i+1}=b_{i-1}+b_i+b_{i+1}$ for all $2\leq i\leq n-1,$ and [*] $a_{n-1}+a_n=b_{n-1}+b_n$. [/list] Find the number of junk positive integers less than or equal to $2016$. [i]Proposed by Nathan Ramesh

2019 Harvard-MIT Mathematics Tournament, 6

Tags: hmmt , geometry

Six unit disks $C_1$, $C_2$, $C_3$, $C_4$, $C_5$, $C_6$ are in the plane such that they don't intersect each other and $C_i$ is tangent to $C_{i+1}$ for $1 \le i \le 6$ (where $C_7 = C_1$). Let $C$ be the smallest circle that contains all six disks. Let $r$ be the smallest possible radius of $C$, and $R$ the largest possible radius. Find $R - r$.

2015 May Olympiad, 2

Tags: combinatorics , board

We have a 7x7 board. We want to color some 1x1 squares such that any 3x3 sub-board have more painted 1x1 than no painted 1x1. What is the smallest number of 1x1 that we need to color?

1950 Polish MO Finals, 5

Tags: trigonometry , geometry , right triangle , right angle

Prove that if for angles $A,B,C$ of a triangle holds $$\sin^2 A+\sin^2 B +\sin^2 C=2$$ iff the triangle $ABC$ is right.

2017 Estonia Team Selection Test, 6

Tags: algebra , functional equation

Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$

BIMO 2022, 1

Tags: algebra , functional equation

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x,y$, we have $$f(xf(x)+2y)=f(x)^2+x+2f(y)$$

1980 IMO Shortlist, 7

Tags: function , algebra , functional equation

The function $f$ is defined on the set $\mathbb{Q}$ of all rational numbers and has values in $\mathbb{Q}$. It satisfies the conditions $f(1) = 2$ and $f(xy) = f(x)f(y) - f(x+y) + 1$ for all $x,y \in \mathbb{Q}$. Determine $f$.

2020 Italy National Olympiad, #4

Tags: geometry , circumcircle

Let $ABC$ be an acute-angled triangle with $AB=AC$, let $D$ be the foot of perpendicular, of the point $C$, to the line $AB$ and the point $M$ is the midpoint of $AC$. Finally, the point $E$ is the second intersection of the line $BC$ and the circumcircle of $\triangle CDM$. Prove that the lines $AE, BM$ and $CD$ are concurrents if and only if $CE=CM$.