Olympiad problems

1984 Vietnam National Olympiad, 1

$(a)$ Let $x, y$ be integers, not both zero. Find the minimum possible value of $|5x^2 + 11xy - 5y^2|$. $(b)$ Find all positive real numbers $t$ such that $\frac{9t}{10}=\frac{[t]}{t - [t]}$.

1988 AMC 8, 10

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Chris' birthday is on a Thursday this year. What day of the week will it be $60$ days after her birthday? $ \text{(A)}\ \text{Monday}\qquad\text{(B)}\ \text{Wednesday}\qquad\text{(C)}\ \text{Thursday}\qquad\text{(D)}\ \text{Friday}\qquad\text{(E)}\ \text{Saturday} $

2016 Korea USCM, 2

Tags: real analysis , sequence , series

Suppose $\{a_n\}$ is a decreasing sequence of reals and $\lim\limits_{n\to\infty} a_n = 0$. If $S_{2^k} - 2^k a_{2^k} \leq 1$ for any positive integer $k$, show that $$\sum_{n=1}^{\infty} a_n \leq 1$$ (At here, $S_m = \sum_{n=1}^m a_n$ is a partial sum of $\{a_n\}$.)

2008 Nordic, 2

Tags: combinatorics

Assume that $n\ge 3$ people with different names sit around a round table. We call any unordered pair of them, say $M,N$, dominating if 1) they do not sit in adjacent seats 2) on one or both arcs connecting $M,N$ along the table, all people have names coming alphabetically after $M,N$. Determine the minimal number of dominating pairs.

2025 Serbia Team Selection Test for the IMO 2025, 5

Tags: algebra

Determine the smallest positive real number $\alpha$ such that there exists a sequence of positive real numbers $(a_n)$, $n \in \mathbb{N}$, with the property that for every $n \in \mathbb{N}$ it holds that: \[ a_1 + \cdots + a_{n+1} < \alpha \cdot a_n. \] [i]Proposed by Pavle Martinović[/i]

1998 AMC 12/AHSME, 2

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Letters $A,B,C,$ and $D$ represent four different digits from 0,1,2,3...9. If $(A+B)/(C+D)$ is an integer that is as large as possible, what is the value of $A+B$? $\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 14 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 17$

1998 Tournament Of Towns, 1

Tags: combinatorics , weighing

Nineteen weights of mass $1$ gm, $2$ gm, $3$ gm, . . . , $19$ gm are given. Nine are made of iron, nine are of bronze and one is pure gold. It is known that the total mass of all the iron weights is $90$ gm more than the total mass of all the bronze ones. Find the mass of the gold weight . (V Proizvolov)

2010 IFYM, Sozopol, 2

Tags: inequalities

If $a,b,c>0$ and $abc=3$,find the biggest value of: $\frac{a^2b^2}{a^7+a^3b^3c+b^7}+\frac{b^2c^2}{b^7+b^3c^3a+c^7}+\frac{c^2a^2}{c^7+c^3a^3b+a^7}$

1975 Poland - Second Round, 3

Tags: combinatorics , probability

In a certain family, a husband and wife made the following agreement: If the wife washes the dishes one day, the husband washes the dishes the next day. However, if the husband washes the dishes one day, then who washes the dishes the next day is decided by drawing a coin. Let $ p_n $ denote the probability of the event that the husband washes the dishes on the $ n $-th day of the contract. Prove that there is a limit $ \lim_{n\to \infty} p_n $ and calculate it. We assume $ p_1 = \frac{1}{2} $.

2020-2021 OMMC, 7

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Derek fills a square $10$ by $10$ grid with $50$ $1$s and $50$ $2$s. He takes the product of the numbers in each of the $10$ rows. He takes the product of the numbers in each of the $10$ columns. He then sums these $20$ products up to get an integer $N.$ Find the minimum possible value of $N.$

2011 IMAC Arhimede, 6

Tags: inequalities

Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove that $\frac{a}{a^3+b^2c+c^2b} + \frac{b}{b^3+c^2a+a^2c} + \frac{c}{c^3+a^2b+b^2a} \le 1+\frac{8}{27abc}$

2019 AMC 12/AHSME, 25

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Let $ABCD$ be a convex quadrilateral with $BC=2$ and $CD=6.$ Suppose that the centroids of $\triangle ABC,\triangle BCD,$ and $\triangle ACD$ form the vertices of an equilateral triangle. What is the maximum possible value of the area of $ABCD$? $\textbf{(A) } 27 \qquad\textbf{(B) } 16\sqrt3 \qquad\textbf{(C) } 12+10\sqrt3 \qquad\textbf{(D) } 9+12\sqrt3 \qquad\textbf{(E) } 30$

2017 Kosovo National Mathematical Olympiad, 2

Tags: algebra

Prove that for every positive real $a,b,c$ the inequality holds : $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+1 \geq \frac{2\sqrt2}{3} (\sqrt{\frac{a+b}{c}}+\sqrt{\frac{b+c}{a}}+\sqrt{\frac{c+a}{b}})$ When does the equality hold?

1992 Baltic Way, 20

Tags: incenter , geometry

Let $ a\le b\le c$ be the sides of a right triangle, and let $ 2p$ be its perimeter. Show that \[ p(p \minus{} c) \equal{} (p \minus{} a)(p \minus{} b) \equal{} S, \] where $ S$ is the area of the triangle.

2022 China Girls Math Olympiad, 1

Tags: sequence , inequalities

Consider all the real sequences $x_0,x_1,\cdots,x_{100}$ satisfying the following two requirements: (1)$x_0=0$; (2)For any integer $i,1\leq i\leq 100$,we have $1\leq x_i-x_{i-1}\leq 2$. Find the greatest positive integer $k\leq 100$,so that for any sequence $x_0,x_1,\cdots,x_{100}$ like this,we have \[x_k+x_{k+1}+\cdots+x_{100}\geq x_0+x_1+\cdots+x_{k-1}.\]

2019 Durer Math Competition Finals, 3

Tags: combinatorics

On a piece of paper we have $2019$ statements numbered from $1$ to $2019$. The $n^{th}$ statement is the following: "On this piece of paper there are at most $n$ true statements". How many of the statements are true?

1985 Greece National Olympiad, 3

Tags: geometry , angle

Interior in alake there are two points $A,B$ from which we can see every other point of the lake. Prove that also from any other point of the segment $AB$, we can see all points of the lake.

2019 China Northern MO, 7

Tags: combinatorics

There are $n$ cities in Qingqiu Country. The distance between any two cities are different. The king of the country plans to number the cities and set up two-way air lines in such ways: The first time, set up a two-way air line between city 1 and the city nearest to it. The second time, set up a two-way air line between city 2 and the city second nearest to it. ... The $n-1$th time, set up a two-way air line between city $n-1$ and the city farthest to it. Prove: The king can number the cities in a proper way so that he can go to any other city from any city by plane.

1996 China Team Selection Test, 3

Tags: complex number , algebra

Does there exist non-zero complex numbers $a, b, c$ and natural number $h$ such that if integers $k, l, m$ satisfy $|k| + |l| + |m| \geq 1996$, then $|ka + lb + mc| > \frac {1}{h}$ is true?

2017 Pan-African Shortlist, N1

Tags: number theory , greatest common divisor , floor function , function , algebra , divisibility theory

Prove that the expression \[\frac{\gcd(m, n)}{n}{n \choose m}\] is an integer for all pairs of positive integers $(m, n)$ with $n \ge m \ge 1$.

2010 BAMO, 3

Tags: algebra , inequalities

Suppose $a,b,c$ are real numbers such that $a+b \ge 0, b+c \ge 0$, and $c+a \ge 0$. Prove that $a+b+c \ge \frac{|a|+|b|+|c|}{3}$ . (Note: $|x|$ is called the absolute value of $x$ and is defined as follows. If $x \ge 0$ then $|x|= x$, and if $x < 0$ then $|x| = -x$. For example, $|6|= 6, |0| = 0$ and $|-6| = 6$.)

1999 Tuymaada Olympiad, 2

Tags: algebra , polynomial , function , calculus , derivative

Can the graphs of a polynomial of degree 20 and the function $\displaystyle y={1\over x^{40}}$ have exactly 30 points of intersection? [i]Proposed by K. Kokhas[/i]

2009 Princeton University Math Competition, 4

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Find the sum of all integers $x$ for which there is an integer $y$, such that $x^3-y^3=xy+61$.

2024 Argentina National Olympiad Level 2, 5

Tags: combinatorics

Let $A_1A_2\cdots A_n$ be a regular polygon with $n$ sides, $n \geqslant 3$. Initially, there are three ants standing at vertex $A_1$. Every minute, two ants simultaneously move to an adjacent vertex, but in different directions (one clockwise and the other counterclockwise), and the third stays at its current vertex. Determine all the values of $n$ for which, after some time, the three ants can meet at the same vertex of the polygon, different from $A_1$.

2022 DIME, 6

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In $\triangle ABC$ with $AC>AB$, let $D$ be the foot of the altitude from $A$ to side $\overline{BC}$, and let $M$ be the midpoint of side $\overline{AC}$. Let lines $AB$ and $DM$ intersect at a point $E$. If $AC=8$, $AE=5$, and $EM=6$, find the square of the area of $\triangle ABC$. [i]Proposed by [b]DeToasty3[/b][/i]