Olympiad problems

2024 China Girls Math Olympiad, 3

Tags: number theory

Find the smallest real $\lambda$, such that for any positive integers $n, a, b$, such that $n \nmid a+b$, there exists a positive integer $1 \leq k \leq n-1$, satisfying $$\{\frac{ak} {n}\}+\{\frac{bk} {n}\} \leq \lambda.$$

2020 LIMIT Category 1, 3

Tags: geometry

The diagnols $\overline{AC}$ and $\overline{BD}$ of a quaderilateral $ABCD$ meet at $O$. Let $s_1$ be the area of $\triangle{AOB}$ and $s_2$ be the area of $\triangle{OCD}$. Then show that $$\sqrt{s_1}+\sqrt{s_2} \leq \sqrt{s}$$ Also find a geometrical condition for equality to hold (By geometrical condition we mean something like parallel lines, perpendicular lines,bisecting lines etc.)

2013 Hong kong National Olympiad, 4

Tags: geometric transformation , reflection , graph theory , geometry , combinatorics

In a chess tournament there are $n>2$ players. Every two players play against each other exactly once. It is known that exactly $n$ games end as a tie. For any set $S$ of players, including $A$ and $B$, we say that $A$ [i]admires[/i] $B$ [i]in that set [/i]if i) $A$ does not beat $B$; or ii) there exists a sequence of players $C_1,C_2,\ldots,C_k$ in $S$, such that $A$ does not beat $C_1$, $C_k$ does not beat $B$, and $C_i$ does not beat $C_{i+1}$ for $1\le i\le k-1$. A set of four players is said to be [i]harmonic[/i] if each of the four players admires everyone else in the set. Find, in terms of $n$, the largest possible number of harmonic sets.

2012-2013 SDML (Middle School), 14

Tags:

Sammy and Tammy run laps around a circular track that has a radius of $1$ kilometer. They begin and end at the same point and at the same time. Sammy runs $3$ laps clockwise while Tammy runs $4$ laps counterclockwise. How many times during their run is the straight-line distance between Sammy and Tammy exactly $1$ kilometer? $\text{(A) }7\qquad\text{(B) }8\qquad\text{(C) }13\qquad\text{(D) }14\qquad\text{(E) }21$

1982 USAMO, 1

Tags: combinatorics

In a party with $1982$ persons, among any group of four there is at least one person who knows each of the other three. What is the minimum number of people in the party who know everyone else?

2024 CMIMC Geometry, 1

Tags: geometry

Let $ABCD$ be a rectangle with $AB=5$. Let $E$ be on $\overline{AB}$ and $F$ be on $\overline {CD}$ such that $AE=CF=4$. Let $P$ and $Q$ lie inside $ABCD$ such that triangles $AEP$ and $CFQ$ are equilateral. If $E$, $P$, $Q$, and $F$ lie on a single line, find $\overline{BC}$. [i]Proposed by Connor Gordon[/i]

1974 IMO Longlists, 32

Tags: inequalities

Let $a_1,a_2,\ldots ,a_n$ be $n$ real numbers such that $0<a\le a_k\le b$ for $k=1,2,\ldots ,n$. If $m_1=\frac{1}{n}(a_1+a_2+\cdots+a_n)$ and $m_2=\frac{1}{n}(a_1^2+a_2^2+\cdots + a_n^2)$, prove that $m_2\le\frac{(a+b)^2}{4ab}m_1^2$ and find a necessary and sufficient condition for equality.

VMEO III 2006, 12.4

Tags: combinatorics , symmetry

Given a binary serie $A=a_1a_2...a_k$ is called "symmetry" if $a_i=a_{k+1-i}$ for all $i=1,2,3,...,k$, and $k$ is the length of that binary serie. If $A=11...1$ or $A=00...0$ then it is called "special". Find all positive integers $m$ and $n$ such that there exist non "special" binary series $A$ (length $m$) and $B$ (length $n$) satisfying when we place them next to each other, we receive a "symmetry" binary serie $AB$

2005 Today's Calculation Of Integral, 56

Tags: limit , integration , calculus , calculus computations

Evaluate \[\lim_{n\to\infty} \sum_{k=1}^n \frac{[\sqrt{2n^2-k^2}\ ]}{n^2}\] $[x]$ is the greatest integer $\leq x$.

2004 Silk Road, 4

Tags: group theory , abstract algebra , combinatorics

Natural $n \geq 2$ is given. Group of people calls $n-compact$, if for any men from group, we can found $n$ people (without he), each two of there are familiar. Find maximum $N$ such that for any $n-compact$ group, consisting $N$ people contains subgroup from $n+1$ people, each of two of there are familiar.

2011 Bogdan Stan, 3

Tags: unique solution , function , algebra

Solve in $ \mathbb{R} $ the equation $ 4^{x^2-x}=\log_2 x+\sqrt{x-1} +14. $ [i]Marin Tolosi[/i]

2013 Saudi Arabia Pre-TST, 3.2

Tags: divide , divisible , number theory

Let $a_1, a_2,..., a_9$ be integers. Prove that if $19$ divides $a_1^9+a_2^9+...+a_9^9$ then $19$ divides the product $a_1a_2...a_9$.

2023 IFYM, Sozopol, 3

Tags: number theory

Let $n \geq 2$ be an integer such that $6^n + 11^n$ is divisible by $n$. Prove that $n^{100} + 6^n + 11^n$ is divisible by $17n$ and not divisible by $289n$.

1993 Tournament Of Towns, (380) 2

Tags: geometry , collinear

Vertices $A$, $B$ and $C$ of a triangle are connected with points $A'$ , $B'$ and $C'$ lying in the opposite sides of the triangle (not at vertices). Can the midpoints of the segments $AA'$, $BB'$ and $CC'$ lie in a straight line? (Folklore)

2023 Belarus Team Selection Test, 3.1

Tags: algebra

Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

2022 New Zealand MO, 2

Tags: perfect power , number theory , power of number

Is it possible to pair up the numbers $0, 1, 2, 3,... , 61$ in such a way that when we sum each pair, the product of the $31$ numbers we get is a perfect f ifth power?

2022 Taiwan TST Round 1, N

Tags: number theory

Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2021 CCA Math Bonanza, I5

Tags:

If digits $A$, $B$, and $C$ (between $0$ and $9$ inclusive) satisfy \begin{tabular}{c@{\,}c@{\,}c@{\,}c} & $C$ & $C$ & $A$ \\ + & $B$ & $2$ & $B$ \\\hline & $A$ & $8$ & $8$ \\ \end{tabular} what is $A \cdot B \cdot C$? [i]2021 CCA Math Bonanza Individual Round #5[/i]

2014 NIMO Problems, 9

Tags: geometry , rectangle , induction

Two players play a game involving an $n \times n$ grid of chocolate. Each turn, a player may either eat a piece of chocolate (of any size), or split an existing piece of chocolate into two rectangles along a grid-line. The player who moves last loses. For how many positive integers $n$ less than $1000$ does the second player win? (Splitting a piece of chocolate refers to taking an $a \times b$ piece, and breaking it into an $(a-c) \times b$ and a $c \times b$ piece, or an $a \times (b-d)$ and an $a \times d$ piece.) [i]Proposed by Lewis Chen[/i]

2022 AMC 10, 19

Tags: least common multiple

Define $L_n$ as the least common multiple of all the integers from $1$ to $n$ inclusive. There is a unique integer $h$ such that $\frac{1}{1}+\frac{1}{2}+\frac{1}{3} \ldots +\frac{1}{17}=\frac{h}{L_{17}}$. What is the remainder when $h$ is divided by $17?$ $\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9$

1984 AMC 12/AHSME, 28

Tags:

The number of distinct pairs of integers $(x,y)$ such that \[0 < x < y\quad \text{and}\quad \sqrt{1984} = \sqrt{x} + \sqrt{y}\] is $\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }2\qquad \textbf{(D) }3\qquad \textbf{(E) }7$

1991 National High School Mathematics League, 11

Tags: complex number

For two complex numbers $z_1,z_2$ satisfy that $|z_1|=|z_1+z_2|=3,|z_1-z_2|=3\sqrt3$, then $\log_3|(z_1\overline{z_2})^{2000}+(\overline{z_1}z_2)^{2000}|=$________.

2022 Saudi Arabia IMO TST, 2

Tags: geometry , cyclic quadrilateral , concurrency , angle chasing

Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.

2023 CCA Math Bonanza, L1.1

Tags: probability

If 100 dice are rolled, what is the probability that the sum of the numbers rolled is even? [i]Lightning 1.1[/i]

2013 India Regional Mathematical Olympiad, 2

Tags: divisor , prime

Determine the smallest prime that does not divide any five-digit number whose digits are in a strictly increasing order.