Olympiad problems

2015 District Olympiad, 3

Tags: sequence , set

Consider the following sequence of sets: $ \{ 1,2\} ,\{ 3,4,5\}, \{ 6,7,8,9\} ,... $ [b]a)[/b] Find the samllest element of the $ 100\text{-th} $ term. [b]b)[/b] Is $ 2015 $ the largest element of one of these sets?

1994 Romania TST for IMO, 1:

Tags: combinatorics

Let $ X_n\equal{}\{1,2,...,n\}$,where $ n \geq 3$. We define the measure $ m(X)$ of $ X\subset X_n$ as the sum of its elements.(If $ |X|\equal{}0$,then $ m(X)\equal{}0$). A set $ X \subset X_n$ is said to be even(resp. odd) if $ m(X)$ is even(resp. odd). (a)Show that the number of even sets equals the number of odd sets. (b)Show that the sum of the measures of the even sets equals the sum of the measures of the odd sets. (c)Compute the sum of the measures of the odd sets.

1983 IMO Longlists, 33

Tags: coefficient , algebra , polynomial , sequence

Let $F(n)$ be the set of polynomials $P(x) = a_0+a_1x+\cdots+a_nx^n$, with $a_0, a_1, . . . , a_n \in \mathbb R$ and $0 \leq a_0 = a_n \leq a_1 = a_{n-1 } \leq \cdots \leq a_{[n/2] }= a_{[(n+1)/2]}.$ Prove that if $f \in F(m)$ and $g \in F(n)$, then $fg \in F(m + n).$

2014 IMO Shortlist, N2

Tags: number theory , equation

Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

2010 IMO Shortlist, 2

Tags: inequalities , algebra , polynomial

Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that \[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\] [i]Proposed by Nazar Serdyuk, Ukraine[/i]

2020 SIME, 11

Tags:

Let $d_1, d_2, \ldots , d_{k}$ be the distinct positive integer divisors of $6^8$. Find the number of ordered pairs $(i, j)$ such that $d_i - d_j$ is divisible by $11$.

1953 Miklós Schweitzer, 4

Tags: geometry , real analysis

[b]4.[/b] Show that every closed curve c of length less than $ 2\pi $ on the surface of the unit sphere lies entirely on the surface of some hemisphere of the unit sphere. [b](G. 8)[/b]

2010 Canadian Mathematical Olympiad Qualification Repechage, 6

Tags: geometry , combinatorics

There are $15$ magazines on a table, and they cover the surface of the table entirely. Prove that one can always take away $7$ magazines in such a way that the remaining ones cover at least $\dfrac{8}{15}$ of the area of the table surface

2012 AMC 10, 3

Tags: analytic geometry

The point in the $xy$-plane with coordinates $(1000,2012)$ is reflected across line $y=2000$. What are the coordinates of the reflected point? $ \textbf{(A)}\ (998,2012) \qquad\textbf{(B)}\ (1000,1988)\qquad\textbf{(C)}\ (1000,2024)\qquad\textbf{(D)}\ (1000,4012)\qquad\textbf{(E)}\ (1012,2012) $

1996 Iran MO (2nd round), 1

Tags: inequalities

Let $a, b, c$ be real numbers. Prove that there exists a triangle with side lengths $a, b, c$ if and only if \[2(a^4 + b^4 + c^4) < (a^2 + b^2 + c^2)^2.\]

LMT Speed Rounds, 2

Tags: speed , alg

Eddie has a study block that lasts $1$ hour. It takes Eddie $25$ minutes to do his homework and $5$ minutes to play a game of Clash Royale. He can’t do both at the same time. How many games can he play in this study block while still completing his homework? [i]Proposed by Edwin Zhao[/i] [hide=Solution] [i]Solution.[/i] $\boxed{7}$ Study block lasts 60 minutes, thus he has 35 minutes to play Clash Royale, during which he can play $\frac{35}{5}=\boxed{7}$ games. [/hide]

2022 Paraguay Mathematical Olympiad, 1

Tags: algebra

There are $13$ positive integers greater than $\sqrt{15}$ and less than $\sqrt[3]{B}$. What is the smallest integer value of $B$?

1981 Vietnam National Olympiad, 1

Tags: trigonometry , perimeter , circumcircle , inradius , geometry

Prove that a triangle $ABC$ is right-angled if and only if \[\sin A + \sin B + \sin C = \cos A + \cos B + \cos C + 1\]

1966 All Russian Mathematical Olympiad, 083

Tags: minimum , maximum , game strategy , combinatorics

$20$ numbers are written on the board $1, 2, ... ,20$. Two players are putting signs before the numbers in turn ($+$ or $-$). The first wants to obtain the minimal possible absolute value of the sum. What is the maximal value of the absolute value of the sum that can be achieved by the second player?

2016 Hanoi Open Mathematics Competitions, 6

Tags: interval , minimum , algebra , number theory

Determine the smallest positive number $a$ such that the number of all integers belonging to $(a, 2016a]$ is $2016$.

2003 China Western Mathematical Olympiad, 2

Tags: vector , inequalities

Let $ a_1, a_2, \ldots, a_{2n}$ be $ 2n$ real numbers satisfying the condition $ \sum_{i \equal{} 1}^{2n \minus{} 1} (a_{i \plus{} 1} \minus{} a_i)^2 \equal{} 1$. Find the greatest possible value of $ (a_{n \plus{} 1} \plus{} a_{n \plus{} 2} \plus{} \ldots \plus{} a_{2n}) \minus{} (a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n)$.

1962 AMC 12/AHSME, 24

Tags:

Three machines $ \text{P, Q, and R,}$ working together, can do a job in $ x$ hours. When working alone, $ \text{P}$ needs an additional $ 6$ hours to do the job; $ \text{Q}$, one additional hour; and $ R$, $ x$ additional hours. The value of $ x$ is: $ \textbf{(A)}\ \frac23 \qquad \textbf{(B)}\ \frac{11}{12} \qquad \textbf{(C)}\ \frac32 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 3$

2008 Romania Team Selection Test, 1

Tags: modular arithmetic , number theory , relatively prime , combinatorics

Let $ n$ be an integer, $ n\geq 2$. Find all sets $ A$ with $ n$ integer elements such that the sum of any nonempty subset of $ A$ is not divisible by $ n\plus{}1$.

2001 Denmark MO - Mohr Contest, 3

Tags: geometry , min , square

In the square $ABCD$ of side length $2$ the point $M$ is the midpoint of $BC$ and $P$ a point on $DC$. Determine the smallest value of $AP+PM$. [img]https://1.bp.blogspot.com/-WD8WXIE6DK4/XzcC9GYsa6I/AAAAAAAAMXg/vl2OrbAdChEYrRpemYmj6DiOrdOSqj_IgCLcBGAsYHQ/s178/2001%2BMohr%2Bp3.png[/img]

2014 NIMO Summer Contest, 12

Tags: search

Find the sum of all positive integers $n$ such that \[ \frac{2n+1}{n(n-1)} \] has a terminating decimal representation. [i]Proposed by Evan Chen[/i]

2016 China Team Selection Test, 2

Tags: geometry , inequalities , geometric inequality

Find the smallest positive number $\lambda$, such that for any $12$ points on the plane $P_1,P_2,\ldots,P_{12}$(can overlap), if the distance between any two of them does not exceed $1$, then $\sum_{1\le i<j\le 12} |P_iP_j|^2\le \lambda$.

2011 Saudi Arabia BMO TST, 1

Tags: algebra , polynomial

Find all polynomials $P$ with real coefficients such that for all $x, y ,z \in R$, $$P(x)+P(y)+P(z)+P(x+y+z)=P(x+y)+P(y+z)+P(z+x)$$

2012 Princeton University Math Competition, A6

Tags: geometry

Consider a pool table with the shape of an equilateral triangle. A ball of negligible size is initially placed at the center of the table. After it has been hit, it will keep moving in the direction it was hit towards and bounce off any edges with perfect symmetry. If it eventually reaches the midpoint of any edge, we mark the midpoint of the entire route that the ball has travelled through. Repeating this experiment, how many points can we mark at most? [img]https://cdn.artofproblemsolving.com/attachments/5/d/3ae7aad4271b9a417826f3bc8dd7b43aeca5d4.png[/img]

2014 Dutch BxMO/EGMO TST, 2

Tags: function , algebra

Find all functions $f:\mathbb{R}\backslash\{0\}\rightarrow\mathbb{R}$ for which $xf(xy) + f(-y) = xf(x)$ for all non-zero real numbers $x, y$.

2016 Kyiv Mathematical Festival, P3

Tags: game , combinatorics

Two players in turn paint cells of the $7\times7$ table each using own color. A player can't paint a cell if its row or its column contains a cell painted by the other player. The game stops when one of the players can't make his turn. What maximal number of the cells can remain unpainted when the game stops?