Olympiad problems

2019 Junior Balkan Team Selection Tests - Romania, 4

Ana and Bogdan play the following turn based game: Ana starts with a pile of $n$ ($n \ge 3$) stones. At his turn each player has to split one pile. The winner is the player who can make at his turn all the piles to have at most two stones. Depending on $n$, determine which player has a winning strategy.

2012 BAMO, 3

Tags: combinatorics

Two infinite rows of evenly-spaced dots are aligned as in the figure below. Arrows point from every dot in the top row to some dot in the lower row in such a way that: [list][*]No two arrows point at the same dot. [*]Now arrow can extend right or left by more than 1006 positions.[/list] [img]https://cdn.artofproblemsolving.com/attachments/7/6/47abf37771176fce21bce057edf0429d0181fb.png[/img] Show that at most 2012 dots in the lower row could have no arrow pointing to them.

2004 Federal Competition For Advanced Students, P2, 6

Tags: geometry

Over the sides of an equilateral triangle with area $ 1$ are triangles with the opposite angle $ 60^{\circ}$ to each side drawn outside of the triangle. The new corners are $ P$, $ Q$ and $ R$. (and the new triangles $ APB$, $ BQC$ and $ ARC$) 1)What is the highest possible area of the triangle $ PQR$? 2)What is the highest possible area of the triangle whose vertexes are the midpoints of the inscribed circles of the triangles $ APB$, $ BQC$ and $ ARC$?

2018 Romania National Olympiad, 2

Tags: calculus , inequalities

Let $x>0.$ Prove that $$2^{-x}+2^{-1/x} \leq 1.$$

1979 IMO Longlists, 80

Tags: algebra , functional equation

Prove that the functional equations \[f(x + y) = f(x) + f(y),\] \[ \text{and} \qquad f(x + y + xy) = f(x) + f(y) + f(xy) \quad (x, y \in \mathbb R)\] are equivalent.

2000 AIME Problems, 1

Tags:

Find the least positive integer $n$ such that no matter how $10^{n}$ is expressed as the product of any two positive integers, at least one of these two integers contains the digit $0.$

Swiss NMO - geometry, 2022.1

Tags: geometry , ratio

Let $k$ be a circle with centre $M$ and let $AB$ be a diameter of $k$. Furthermore, let $C$ be a point on $k$ such that $AC = AM$. Let $D$ be the point on the line $AC$ such that $CD = AB$ and $C$ lies between $A$ and $D$. Let $E$ be the second intersection of the circumcircle of $BCD$ with line $AB$ and $F$ be the intersection of the lines $ED$ and $BC$. The line $AF$ cuts the segment $BD$ in $X$. Determine the ratio $BX/XD$.

1958 Polish MO Finals, 4

Tags: algebra

Prove that if $ k $ is a natural number, then $$ (1 + x)(1 + x^2) (1 + x^4) \ldots (1 + x^{2^k}) =1 + x + x^2 + x^3+ \ldots + x^m$$ where $ m $ is a natural number dependent on $ k $; determine $ m $.

2004 Nicolae Coculescu, 2

Tags: primitive , real analysis , function

Consider a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that admits bounded primitives. Prove that the function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $$ f(x)=\left\{ \begin{matrix} x, & \quad x\le 0 \\ f(1/x)\cdot\ln x ,& \quad x>0 \end{matrix}\right. $$ admits primitives. [i]Florian Dumitrel[/i]

2022 VN Math Olympiad For High School Students, Problem 3

Tags: algebra , number theory

Given a positive integer $N$. Prove that: there are infinitely elements of the [i]Fibonacci[/i] sequence that are divisible by $N$.

1971 Putnam, A1

Tags:

Let there be given nine lattice points (points with integral coordinates) in three dimensional Euclidean space. Show that there is a lattice point on the interior of one of the line segments joining two of these points.

2023 Moldova EGMO TST, 11

Tags: number theory

Find all three digit positive integers that have distinct digits and after their greatest digit is switched to $1$ become multiples of $30$.

1985 Federal Competition For Advanced Students, P2, 6

Tags: function , algebra

Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying: $ x^2 f(x)\plus{}f(1\minus{}x)\equal{}2x\minus{}x^4$ for all $ x \in \mathbb{R}$.

2023 BMT, 1

Tags: geometry

A semicircle of radius $2$ is inscribed inside of a rectangle, as shown in the diagram below. The diameter of the semicircle coincides with the bottom side of the rectangle, and the semicircle is tangent to the rectangle at all points of intersection. Compute the length of the diagonal of the rectangle. [img]https://cdn.artofproblemsolving.com/attachments/c/7/81fcfb759188eae7dcb82fa5d58fb9525d85de.png[/img]

2013 Today's Calculation Of Integral, 876

Tags: calculus , function , definite integral , integration , calculus computations

Suppose a function $f(x)$ is continuous on $[-1,\ 1]$ and satisfies the condition : 1) $f(-1)\geq f(1).$ 2) $x+f(x)$ is non decreasing function. 3) $\int_{-1}^ 1 f(x)\ dx=0.$ Show that $\int_{-1}^1 f(x)^2dx\leq \frac 23.$

2020 BMT Fall, 5

Tags: number theory

Call a positive integer [i]prime-simple[/i] if it can be expressed as the sum of the squares of two distinct prime numbers. How many positive integers less than or equal to $100$ are prime-simple?

2021 Abels Math Contest (Norwegian MO) Final, 4b

Tags: equal segments , geometry

The tangent at $C$ to the circumcircle of triangle $ABC$ intersects the line through $A$ and $B$ in a point $D$. Two distinct points $E$ and $F$ on the line through $B$ and $C$ satisfy $|BE| = |BF | =\frac{||CD|^2 - |BD|^2|}{|BC|}$. Show that either $|ED| = |CD|$ or $|FD| = |CD|$.

2012 CHMMC Fall, 1

Tags: number theory

Find the remainder when $5^{2012}$ is divided by $3$.

2019 Flanders Math Olympiad, 4

Tags: combinatorics

The Knights of the Round Table are gathering. Around the table are $34 $ chairs, numbered from 1 to $34$. When everyone has sat down, it turns out that between every two knights there is a maximum of $r$ places, which can be either empty or occupied by another knight. (a) For each $r \le 15$, determine the maximum number of knights present. (b) Determine for each $r \le 15$ how many sets of occupied seats there are that match meet the given and where the maximum number of knights is present.

2019 IFYM, Sozopol, 3

Tags: algorithm , combinatorics

There are 365 cards with 365 different numbers. Each step, we can choose 3 cards $a_{i},a_{j},a_{k}$ and we know the order of them (examble: $a_{i}<a_{j}<a_{k}$). With 2000 steps, can we order 365 cards from smallest to biggest??

1985 Bundeswettbewerb Mathematik, 2

Tags: 3d geometry , tetrahedron , sphere , inradius , geometry

The insphere of any tetrahedron has radius $r$. The four tangential planes parallel to the side faces of the tetrahedron cut from the tetrahedron four smaller tetrahedrons whose in-sphere radii are $r_1, r_2, r_3$ and $r_4$. Prove that $$r_1 + r_2 + r_3 + r_4 = 2r$$

2018 Danube Mathematical Competition, 3

Tags: parallel , perpendicular bisector , angle bisector , circumcircle , geometry

Let $ABC$ be an acute non isosceles triangle. The angle bisector of angle $A$ meets again the circumcircle of the triangle $ABC$ in $D$. Let $O$ be the circumcenter of the triangle $ABC$. The angle bisectors of $\angle AOB$, and $\angle AOC$ meet the circle $\gamma$ of diameter $AD$ in $P$ and $Q$ respectively. The line $PQ$ meets the perpendicular bisector of $AD$ in $R$. Prove that $AR // BC$.

2020 AMC 12/AHSME, 12

Tags: analytic geometry , rotation , geometry

Line $\ell$ in the coordinate plane has the equation $3x - 5y + 40 = 0$. This line is rotated $45^{\circ}$ counterclockwise about the point $(20, 20)$ to obtain line $k$. What is the $x$-coordinate of the $x$-intercept of line $k?$ $\textbf{(A) } 10 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 20 \qquad \textbf{(D) } 25 \qquad \textbf{(E) } 30$

1967 Kurschak Competition, 1

Tags: number theory

$A$ is a set of integers which is closed under addition and contains both positive and negative numbers. Show that the difference of any two elements of $A$ also belongs to $A$.

2010 China Team Selection Test, 1

Tags: inequalities , induction , function , combinatorics

Let $G=G(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. Suppose $|V|=n$. A map $f:\,V\rightarrow\mathbb{Z}$ is called good, if $f$ satisfies the followings: (1) $\sum_{v\in V} f(v)=|E|$; (2) color arbitarily some vertices into red, one can always find a red vertex $v$ such that $f(v)$ is no more than the number of uncolored vertices adjacent to $v$. Let $m(G)$ be the number of good maps. Prove that if every vertex in $G$ is adjacent to at least one another vertex, then $n\leq m(G)\leq n!$.