Olympiad problems

2005 Harvard-MIT Mathematics Tournament, 10

Tags: calculus , function

Let $ f : \mathbf{R} \to \mathbf{R} $ be a smooth function such that $f'(x)=f(1-x)$ for all $x$ and $f(0)=1$. Find $f(1)$.

1978 All Soviet Union Mathematical Olympiad, 268

Tags: analysis , algebra , sequence

Consider a sequence $$x_n=(1+\sqrt2+\sqrt3)^n$$ Each member can be represented as $$x_n=q_n+r_n\sqrt2+s_n\sqrt3+t_n\sqrt6$$ where $q_n, r_n, s_n, t_n$ are integers. Find the limits of the fractions $r_n/q_n, s_n/q_n, t_n/q_n$.

2023 Thailand TSTST, 3

Tags: number theory

If $d$ is a positive integer such that $d \mid 5+2022^{2022}$, show that $d=2x^2+2xy+3y^2$ for some $x, y \in \mathbb{Z}$ iff $d \equiv 3,7 \pmod {20}$.

1966 IMO Shortlist, 34

Tags: number theory , equation

Find all pairs of positive integers $\left( x;\;y\right) $ satisfying the equation $2^{x}=3^{y}+5.$

2012 IMC, 3

Tags: logarithm

Is the set of positive integers $n$ such that $n!+1$ divides $(2012n)!$ finite or infinite? [i]Proposed by Fedor Petrov, St. Petersburg State University.[/i]

LMT Team Rounds 2010-20, 2020.S1

Tags:

Compute the smallest nonnegative integer that can be written as the sum of 2020 distinct integers.

2015 China Team Selection Test, 2

Tags: probabilistic method , combinatorics

Let $X$ be a non-empty and finite set, $A_1,...,A_k$ $k$ subsets of $X$, satisying: (1) $|A_i|\leq 3,i=1,2,...,k$ (2) Any element of $X$ is an element of at least $4$ sets among $A_1,....,A_k$. Show that one can select $[\frac{3k}{7}] $ sets from $A_1,...,A_k$ such that their union is $X$.

2023 Junior Balkan Team Selection Tests - Moldova, 5

Tags: geometry , number theory

The positive integers $ a, b, c $ are the lengths of the sides of a right triangle. Prove that $abc$ is divisible by $60$.

2019 India PRMO, 15

Tags: algebra

In base-$2$ notation, digits are $0$ and $1$ only and the places go up in powers of $-2$. For example, $11011$ stands for $(-2)^4+(-2)^3+(-2)^1+(-2)^0$ and equals number $7$ in base $10$. If the decimal number $2019$ is expressed in base $-2$ how many non-zero digits does it contain ?

2006 Sharygin Geometry Olympiad, 9.4

Tags: hexagon , cut , geometry , angle

In a non-convex hexagon, each angle is either $90$ or $270$ degrees. Is it true that for some lengths of the sides it can be cut into two hexagons similar to it and unequal to each other?

2017 Baltic Way, 1

Tags: algebra , inequality , sequence , convexity , easy sequence , induction

Let $a_0,a_1,a_2,...$ be an infinite sequence of real numbers satisfying $\frac{a_{n-1}+a_{n+1}}{2}\geq a_n$ for all positive integers $n$. Show that $$\frac{a_0+a_{n+1}}{2}\geq \frac{a_1+a_2+...+a_n}{n}$$ holds for all positive integers $n$.

1969 IMO Shortlist, 66

Tags: algebra , polynomial , inequality , taylor series

$(USS 3)$ $(a)$ Prove that if $0 \le a_0 \le a_1 \le a_2,$ then $(a_0 + a_1x - a_2x^2)^2 \le (a_0 + a_1 + a_2)^2\left(1 +\frac{1}{2}x+\frac{1}{3}x^2+\frac{1}{2}x^3+x^4\right)$ $(b)$ Formulate and prove the analogous result for polynomials of third degree.

2008 AMC 10, 3

Tags:

For the positive integer $n$, let $\left< n \right>$ denote the sum of all the positive divisors of $n$ with the exception of $n$ itself. For example, $\left<4\right> = 1+2=3$ and $\left<12\right>=1+2+3+4+6=16$ What is $\left< \left< \left< 6 \right>\right>\right>$? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 32 \qquad \textbf{(E)}\ 36$

2006 Oral Moscow Geometry Olympiad, 4

Tags: symmetric , distance , cyclic quadrilateral , geometry

The quadrangle $ABCD$ is inscribed in a circle, the center $O$ of which lies inside it. The tangents to the circle at points $A$ and $C$ and a straight line, symmetric to $BD$ wrt point $O$, intersect at one point. Prove that the products of the distances from $O$ to opposite sides of the quadrilateral are equal. (A. Zaslavsky)

2017 Irish Math Olympiad, 5

Tags: algebra , sequence , sum

Given a positive integer $m$, a sequence of real numbers $a= (a_1,a_2,a_3,...)$ is called $m$-powerful if it satisfies $$(\sum_{k=1}^{n} a_k )^{m} = \sum_{k=1}^{n} a_k^{m}$$for all positive integers $n$. (a) Show that a sequence is $30$-powerful if and only if at most one of its terms is non-zero. (b) Find a sequence none of whose terms are zero but which is $2017$-powerful.

2012 Danube Mathematical Competition, 3

Tags: geometry , equal angles

Let $ABC$ be a triangle with $\angle BAC = 90^o$. Angle bisector of the $\angle CBA$ intersects the segment $(AB)$ at point $E$. If there exists $D \in (CE)$ so that $\angle DAC = \angle BDE =x^o$ , calculate $x$.

2022 AMC 12/AHSME, 9

Tags:

The sequence $a_0,a_1,a_2,\cdots$ is a strictly increasing arithmetic sequence of positive integers such that \[2^{a_7}=2^{27} \cdot a_7.\] What is the minimum possible value of $a_2$? $\textbf{(A)}8~\textbf{(B)}12~\textbf{(C)}16~\textbf{(D)}17~\textbf{(E)}22$

2024 Belarus - Iran Friendly Competition, 2.2

Tags: geometry

The circle $\Omega$ centered at $O$ is the circumcircle of the triangle $ABC$. Point $D$ is chosen so that $BD \perp BC$ and points $A$ and $D$ lie in different half-planes with respect to the line $BC$. Let $E$ be a point such that $\angle ADB=\angle BDE$ and $\angle EBD+\angle ACB=90$. Point $P$ is chosen on the line $AD$ so that $OP \perp BC$. Let $Q$ be an arbitrary point on $\Omega$, and $R$ be a point on the line $BQ$ such that $PQ \parallel DR$. Prove that $\angle ARB=\angle BRE$. (All angles are oriented in the same way)

1999 Baltic Way, 20

Tags: prime number , number theory

Let $a,b,c$ and $d$ be prime numbers such that $a>3b>6c>12d$ and $a^2-b^2+c^2-d^2=1749$. Determine all possible values of $a^2+b^2+c^2+d^2$ .

2023 AMC 12/AHSME, 12

Tags: pattern

What is the value of \[ 2^3 - 1^2 + 4^3 - 3^3 + 6^3 - 5^3 + \dots + 18^3 - 17^3?\] $\textbf{(A) } 2023 \qquad\textbf{(B) } 2679 \qquad\textbf{(C) } 2941 \qquad\textbf{(D) } 3159 \qquad\textbf{(E) } 3235$

1997 Bosnia and Herzegovina Team Selection Test, 4

Tags: incircle , algebra , inequalities , geometry

$a)$ In triangle $ABC$ let $A_1$, $B_1$ and $C_1$ be touching points of incircle $ABC$ with $BA$, $CA$ and $AB$, respectively. Let $l_1$, $l_2$ and $l_3$ be lenghts of arcs $ B_1C_1$, $A_1C_1$, $B_1A_1$ of incircle $ABC$, respectively, which does not contain points $A_1$, $B_1$ and $C_1$, respectively. Does the following inequality hold: $$ \frac{a}{l_1}+\frac{b}{l_2}+\frac{c}{l_3} \geq \frac{9\sqrt{3}}{\pi}$$ $b)$ Tetrahedron $ABCD$ has three pairs of equal opposing sides. Find length of height of tetrahedron in function od lengths of sides

2009 Portugal MO, 2

Tags: perimeter , power of a point , radical axis , geometry

Points $N$ and $M$ are on the sides $CD$ and $BC$ of square $ABCD$, respectively. The perimeter of triangle $MCN$ is equal to the double of the length of the square's side. Find $\angle MAN$.

2018 AMC 12/AHSME, 13

Tags:

How many nonnegative integers can be written in the form $$a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0,$$ where $a_i\in \{-1,0,1\}$ for $0\le i \le 7$? $\textbf{(A) } 512 \qquad \textbf{(B) } 729 \qquad \textbf{(C) } 1094 \qquad \textbf{(D) } 3281 \qquad \textbf{(E) } 59,048 $

1949 Putnam, B3

Tags: curve

Let $K$ be a closed plane curve such that the distance between any two points of $K$ is always less than $1.$ Show that $K$ lies in a circle of radius $\frac{1}{\sqrt{3}}.$

2013 Ukraine Team Selection Test, 9

Tags: function , algebra

Determine all functions $f:\Bbb{R}\to\Bbb{R}$ such that \[ f^2(x+y)=f^2(x)+2f(xy)+f^2(y), \] for all $x,y\in \Bbb{R}.$