Olympiad problems

2024 USAMTS Problems, 4

Tags:

Let $x_1 \le x_2 \le \dots < x_n$ (with $n \ge 2$) and let $S$ be the set of all the $x_i$. Let $T$ be a randomly chosen subset of $S$. What is the expected value of the indexed alternating sum of $T$ ? Express your answer in terms of the $x_i$. Note: We define the indexed alternating sum of $T$ as \[ \sum_{i=1}^{|T|} (-1)^{i+1}(i) T[i], \] where $T[i]$ is the ith element of $T$ when listed in increasing order. For example, if $T = \{1, 3, 5\}$ then the indexed alternating sum of $T$ is \[ 1 \cdot 1 - 2 \cdot 3 + 3 \cdot 5 = 10. \] Alternating sums of empty sets are defined to be $0$.

2000 Singapore Team Selection Test, 3

There are $n$ blue points and $n$ red points on a straight line. Prove that the sum of all distances between pairs of points of the same colour is less than or equal to the sum of all distances between pairs of points of different colours

1983 IMO Longlists, 69

Tags: geometry , circles , midpoint , angle

Let $A$ be one of the two distinct points of intersection of two unequal coplanar circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ respectively. One of the common tangents to the circles touches $C_1$ at $P_1$ and $C_2$ at $P_2$, while the other touches $C_1$ at $Q_1$ and $C_2$ at $Q_2$. Let $M_1$ be the midpoint of $P_1Q_1$ and $M_2$ the midpoint of $P_2Q_2$. Prove that $\angle O_1AO_2=\angle M_1AM_2$.

2007 Iran MO (3rd Round), 3

Tags: inequalities

Find the largest real $ T$ such that for each non-negative real numbers $ a,b,c,d,e$ such that $ a\plus{}b\equal{}c\plus{}d\plus{}e$: \[ \sqrt{a^{2}\plus{}b^{2}\plus{}c^{2}\plus{}d^{2}\plus{}e^{2}}\geq T(\sqrt a\plus{}\sqrt b\plus{}\sqrt c\plus{}\sqrt d\plus{}\sqrt e)^{2}\]

2019 Pan-African, 1

Tags: linear recurrence , recurrence relation , algebra , induction

Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers defined as follows: [list] [*] $a_0 = 3$, $a_1 = 2$, and $a_2 = 12$; and [*] $2a_{n + 3} - a_{n + 2} - 8a_{n + 1} + 4a_n = 0$ for $n \geq 0$. [/list] Show that $a_n$ is always a strictly positive integer.

1994 Irish Math Olympiad, 3

Tags: polynomial , induction , algebra

Find all real polynomials $ f(x)$ satisfying $ f(x^2)\equal{}f(x)f(x\minus{}1)$ for all $ x$.

2024 Harvard-MIT Mathematics Tournament, 1

Tags: geometry , rectangle , combinatorics

Compute the number of ways to divide a $20 \times 24 $ rectangle into $4 \times 5$ rectangles. (Rotations and reflections are considered distinct.)

2004 Peru MO (ONEM), 1

Tags: number theory , digit

Let $a$ be number of $n$ digits ($ n > 1$). A number $b$ of $2n$ digits is obtained by writing two copies of $a$ one after the other. If $\frac{b}{a^2}$ is an integer $k$, find the possible values values of $k$.

2023 Olimphíada, 3

Tags: combinatorics , operation , algorithm

Let $n$ be a positive integer. On a blackboard is a circle, and around it $n$ non-negative integers are written. Raphinha plays a game in which an operation consists of erasing a number $a$ neighboring $b$ and $c$, with $b \geq c$, and writing in its place $b + c$ if $b + c \leq 5a/4$ and $b - c$ otherwise. Your goal is to make all the numbers on the board equal $0$. Find all $n$ such that Raphinha always manages to reach her goal.

2023 Korea National Olympiad, 3

Tags: polynomial , number theory

For a given positive integer $n(\ge 2)$, find maximum positive integer $A$ such that there exists $P \in \mathbb{Z}[x]$ with degree $n$ that satisfies the following two conditions. [list] [*] For any $1 \le k \le A$, it satisfies that $A \mid P(k)$, and [*] $P(0)= 0$ and the coefficient of the first term of $P$ is $1$, which means that $P(x)$ is in the following form where $c_2, c_3, \cdots, c_n$ are all integers and $c_n \neq 0$. $$P(x) = c_nx^n + c_{n-1}x^{n-1}+\dots+c_2x^2+x$$ [/list]

2006 National Olympiad First Round, 3

Tags:

$a_1=-1$, $a_2=2$, and $a_n=\frac {a_{n-1}}{a_{n-2}}$ for $n\geq 3$. What is $a_{2006}$? $ \textbf{(A)}\ -2 \qquad\textbf{(B)}\ -1 \qquad\textbf{(C)}\ -\frac 12 \qquad\textbf{(D)}\ \frac 12 \qquad\textbf{(E)}\ 2 $

2014 AMC 12/AHSME, 8

Tags:

In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$? \[\begin{array}{lr} &ABBCB \\ +& BCADA \\ \hline & DBDDD \end{array}\] $\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

2024 IFYM, Sozopol, 4

Tags: combinatorics

Let $ n \geq 4 $ be a positive integer. Initially, each of $ n $ girls knows one piece of gossip that no one else knows, and they want to share them. For greater security, to avoid being spied, they only talk in pairs, and in a conversation, each girl shares all the gossip she knows so far with the other one. What is the minimum number of conversations needed so that every girl knows all the gossip?

2003 Tournament Of Towns, 6

Tags: geometry

An ant crawls on the outer surface of the box in a shape of rectangular parallelepiped. From ant’s point of view, the distance between two points on a surface is defined by the length of the shortest path ant need to crawl to reach one point from the other. Is it true that if ant is at vertex then from ant’s point of view the opposite vertex be the most distant point on the surface?

2025 Romania EGMO TST, P2

Tags: divisible , number theory , inequalities , subset

Let $m$ and $n$ be positive integers with $m > n \ge 2$. Set $S =\{1,2,...,m\}$, and set $T = \{a_1,a_2,...,a_n\}$ is a subset of $S$ such that every element of $S$ is not divisible by any pair of distinct elements of $T$. Prove that $$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$

2017 Online Math Open Problems, 10

Tags:

Determine the value of $-1+2+3+4-5-6-7-8-9+...+10000$, where the signs change after each perfect square. [i]Proposed by Michael Ren

1988 Tournament Of Towns, (175) 1

Tags: number theory , digit

Is it possible to select two natural numbers $m$ and $n$ so that the number $n$ results from a permutation of the digits of $m$, and $m+n =999 . . . 9$ ?

1996 Miklós Schweitzer, 5

Tags: real analysis

Let K and D be the set of convergent and divergent series of positive terms respectively. Does there exist a bijection between K and D such that for all $\sum a_n,\sum b_n\in K$ and $\sum a_n',\sum b_n'\in D$ , $\frac{a_n}{b_n}\to 0\iff \frac{a_n'}{b_n'}\to 0$ ? Under the bijection, $\sum a_n\leftrightarrow\sum a_n'$ and $\sum b_n\leftrightarrow\sum b_n'$.

2012 Stars of Mathematics, 2

Tags: geometry , geometric transformation

Let $\ell$ be a line in the plane, and a point $A \not \in \ell$. Also let $\alpha \in (0, \pi/2)$ be fixed. Determine the locus of the points $Q$ in the plane, for which there exists a point $P\in \ell$ such that $AQ=PQ$ and $\angle PAQ = \alpha$. ([i]Dan Schwarz[/i])

1997 Turkey MO (2nd round), 2

Tags: circumcircle , geometric transformation , geometry

In a triangle $ABC$, the inner and outer bisectors of the $\angle A$ meet the line $BC$ at $D$ and $E$, respectively. Let $d$ be a common tangent of the circumcircle $(O)$ of $\triangle ABC$ and the circle with diameter $DE$ and center $F$. The projections of the tangency points onto $FO$ are denoted by $P$ and $Q$, and the length of their common chord is denoted by $m$. Prove that $PQ = m$

1995 Tuymaada Olympiad, 2

Tags: limit , limit of sequence , recurrence relation , maximum , algebra

Let $x_1=a, x_2=a^{x_1}, ..., x_n=a^{x_{n-1}}$ where $a>1$. What is the maximum value of $a$ for which lim exists $\lim_{n\to \infty} x_n$ and what is this limit?

1951 Putnam, B3

Tags:

Show that if $x$ is positive, then \[ \log_e (1 + 1/x) > 1 / (1 + x).\]

2017 Harvard-MIT Mathematics Tournament, 15

Tags: combinatorics

Start by writing the integers $1, 2, 4, 6$ on the blackboard. At each step, write the smallest positive integer $n$ that satisfies both of the following properties on the board. [list] [*] $n$ is larger than any integer on the board currently. [*] $n$ cannot be written as the sum of $2$ distinct integers on the board. [/list] Find the $100$-th integer that you write on the board. Recall that at the beginning, there are already $4$ integers on the board.

1956 Miklós Schweitzer, 9

Tags:

[b]9.[/b] Show that if the trigonometric polynomial $f(\theta)= \sum_{v=1}^{n} a_v \cos v\theta$ monotonically decreases over the closed interval $[0,\pi]$, then the trigonometric polynomial $g(\theta)=\sum_{v=1}^{n}a_v \sin v\theta$ is non negative in the same interval. [b](S. 26)[/b]

2013 F = Ma, 20

Tags: trigonometry

A simple pendulum experiment is constructed from a point mass $m$ attached to a pivot by a massless rod of length $L$ in a constant gravitational field. The rod is released from an angle $\theta_0 < \frac{\pi}{2}$ at rest and the period of motion is found to be $T_0$. Ignore air resistance and friction. What is the maximum value of the tension in the rod? $\textbf{(A) } mg\\ \textbf{(B) } 2mg\\ \textbf{(C) } mL\theta_0/T_0^2\\ \textbf{(D) } mg \sin \theta_0\\ \textbf{(E) } mg(3 - 2 \cos \theta_0)$