Olympiad problems

2008 Princeton University Math Competition, 1

Tags: algebra

Calculate $$\sqrt{6 + \sqrt{6 + \sqrt{6 +... }}}+\frac{6}{1+ \frac{6}{1+...}}$$

2022 CCA Math Bonanza, L4.4

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Lukas Nepomuceno draws 5 congruent circles equally spaced around a 6th, and colors all of them 1 of 3 colors. Assume that rotations and reflections of colorings are indistinguishable. How many distinct colorings are there? [i]2022 CCA Math Bonanza Lightning Round 4.4[/i]

2005 May Olympiad, 5

Tags: combinatorics

a) In each box of a $7\times 7$ board one of the numbers is written: $1, 2, 3, 4, 5, 6$ or $7$ of so that each number is written in seven different boxes. Is it possible that in no row and no column are consecutive numbers written? b) In each box of a $5\times 5$ board one of the numbers is written: $1, 2, 3, 4$ or $5$ of so that each one is written in five different boxes. Is it possible that in no row and in no column are consecutive numbers written?

2008 Grigore Moisil Intercounty, 1

Tags: function , monotone , find all functions , trigonometry , algebra

Find all monotonic functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ with the property that $$ (f(\sin x))^2-3f(x)=-2, $$ for any real numbers $ x. $ [i]Dorin Andrica[/i] and [i]Mihai Piticari[/i]

2022 Romania Team Selection Test, 2

Tags: geometry

Let $ABC$ be an acute triangle and let $B'$ and $C'$ be the feet of the heights $B$ and $C$ of triangle $ABC$ respectively. Let $B_A'$ and $B_C'$ be reflections of $B'$ with respect to the lines $BC$ and $AB$, respectively. The circle $BB_A'B_C'$, centered in $O_B$, intersects the line $AB$ in $X_B$ for the second time. The points $C_A', C_B', O_C, X_C$ are defined analogously, by replacing the pair $(B, B')$ with the pair $(C, C')$. Show that $O_BX_B$ and $O_CX_C$ are parallel.

2012 Today's Calculation Of Integral, 844

Tags: calculus computations , integration , limit , calculus

Let $\alpha$ be a solution satisfying the equation $|x|=e^{-x}.$ Let $I_n=\int_0^{\alpha} (xe^{-nx}+\alpha x^{n-1})dx\ (n=1,\ 2,\ \cdots).$ Find $\lim_{n\to\infty} n^2I_n.$

2014 HMNT, 2

Tags: hmmt , difference of squares , special factorizations , algebra

Let $f(x) = x^2 + 6x + 7$. Determine the smallest possible value of $f(f(f(f(x))))$ over all real numbers $x.$

2011 AIME Problems, 6

Tags: probability

Define an ordered quadruple of integers $(a, b, c, d)$ as interesting if $1 \le a<b<c<d \le 10$, and $a+d>b+c$. How many ordered quadruples are there?

1988 China Team Selection Test, 3

Tags: combinatorics , lattice , geometry , geometric transformation , vector , analytic geometry

A polygon $\prod$ is given in the $OXY$ plane and its area exceeds $n.$ Prove that there exist $n+1$ points $P_{1}(x_1, y_1), P_{2}(x_2, y_2), \ldots, P_{n+1}(x_{n+1}, y_{n+1})$ in $\prod$ such that $\forall i,j \in \{1, 2, \ldots, n+1\}$, $x_j - x_i$ and $y_j - y_i$ are all integers.

2012 NIMO Summer Contest, 10

Tags: relatively prime , expected value , number theory , probability

A [i]triangulation[/i] of a polygon is a subdivision of the polygon into triangles meeting edge to edge, with the property that the set of triangle vertices coincides with the set of vertices of the polygon. Adam randomly selects a triangulation of a regular $180$-gon. Then, Bob selects one of the $178$ triangles in this triangulation. The expected number of $1^\circ$ angles in this triangle can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$. [i]Proposed by Lewis Chen[/i]

2016 BMT Spring, 7

Tags: graph theory , combinatorics

Consider the graph on $1000$ vertices $v_1, v_2, ...v_{1000}$ such that for all $1 \le i < j \le 1000$, $v_i$ is connected to $v_j$ if and only if $i$ divides $j$. Determine the minimum number of colors that must be used to color the vertices of this graph such that no two vertices sharing an edge are the same color.

2023 Princeton University Math Competition, A2 / B4

Tags: number theory

A number is called [i]good[/i] if it can be written as the sum of the squares of three consecutive positive integers. A number is called excellent if it can be written as the sum of the squares of four consecutive positive integers. (For instance, $14 = 1^2 + 2^2 + 3^2$ is good and $30 =1^2 +2^2 +3^2+4^2$ is excellent.) A good number $G$ is called [i]splendid[/i] if there exists an excellent number $E$ such that $3G-E = 2025.$ If the sum of all splendid numbers is $S,$ find the remainder when $S$ is divided by $1000.$

2018 Azerbaijan IZhO TST, 3

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Problem 5. Consider the integer number n>2. Let a_1,a_2,…,a_n and b_1,b_2,…,b_n be two permutations of 0,1,2,…,n-1. Prove that there exist some i≠j such that: n|a_i b_i-a_j b_j [color=#00f]Moved to HSO. ~ oVlad[/color]

1986 IMO, 1

Tags: number theory , system of equations , perfect square , quadratic , modular arithmetic

Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.

1972 Canada National Olympiad, 1

Tags: ratio , trigonometry

Given three distinct unit circles, each of which is tangent to the other two, find the radii of the circles which are tangent to all three circles.

2023 Lusophon Mathematical Olympiad, 6

Tags: algebra

A calculator has two operations $A$ and $B$ and initially shows the number $1$. Operation $A$ turns $x$ into $x+1$ and operation B turns $x$ into $\dfrac{x}{x+1}$. a) Show all the ways we can get the number $\dfrac{20}{23}$. b) For every rational $r \neq 1$, determine if it is possible to get $r$ using only operations $A$ and $B$.

2020 CCA Math Bonanza, I9

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A sequence $a_n$ of real numbers satisfies $a_1=1$, $a_2=0$, and $a_n=(S_{n-1}+1)S_{n-2}$ for all integers $n\geq3$, where $S_k=a_1+a_2+\dots+a_k$ for positive integers $k$. What is the smallest integer $m>2$ such that $127$ divides $a_m$? [i]2020 CCA Math Bonanza Individual Round #9[/i]

2019 Online Math Open Problems, 6

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An ant starts at the origin of the Cartesian coordinate plane. Each minute it moves randomly one unit in one of the directions up, down, left, or right, with all four directions being equally likely; its direction each minute is independent of its direction in any previous minutes. It stops when it reaches a point $(x,y)$ such that $|x|+|y|=3$. The expected number of moves it makes before stopping can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$. [i]Proposed by Yannick Yao[/i]

1990 National High School Mathematics League, 11

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$\frac{1}{2^{1990}}(1-3\text{C}_{1990}^2+3^2\text{C}_{1990}^4-3^3\text{C}_{1990}^6+\cdots+3^{994}\text{C}_{1990}^{1988}-3^{995}\text{C}_{1990}^{1990})=$________.

2008 JBMO Shortlist, 4

Tags: geometry

Let $ABC$ be a triangle, ($BC < AB$). The line $l$ passing trough the vertices $C$ and orthogonal to the angle bisector $BE$ of $\angle B$, meets $BE$ and the median $BD$ of the side $AC$ at points $F$ and $G$, respectively. Prove that segment $DF$ bisects the segment $EG$.

2015 Junior Balkan Team Selection Tests - Moldova, 1

Tags: algebra

Ler $a$ be the number $123456789$. Compare the numbers $$2014^{9^{9^a}}, 2015^{a^{a^9}}$$

2021 Malaysia IMONST 1, 8

Tags: algebra

A tree grows in the following manner. On the first day, one branch grows out of the ground. On the second day, a leaf grows on the branch and the branch tip splits up into two new branches. On each subsequent day, a new leaf grows on every existing branch, and each branch tip splits up into two new branches. How many leaves does the tree have at the end of the tenth day?

2015 Romania Team Selection Tests, 2

Tags: graph theory , combinatorics

Let $n$ be an integer greater than $1$, and let $p$ be a prime divisor of $n$. A confederation consists of $p$ states, each of which has exactly $n$ airports. There are $p$ air companies operating interstate flights only such that every two airports in different states are joined by a direct (two-way) flight operated by one of these companies. Determine the maximal integer $N$ satisfying the following condition: In every such confederation it is possible to choose one of the $p$ air companies and $N$ of the $np$ airports such that one may travel (not necessarily directly) from any one of the $N$ chosen airports to any other such only by flights operated by the chosen air company.

1992 IMO Shortlist, 2

Tags: recurrence relation , functional equation , quadratic , algebra

Let $ \mathbb{R}^\plus{}$ be the set of all non-negative real numbers. Given two positive real numbers $ a$ and $ b,$ suppose that a mapping $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{}$ satisfies the functional equation: \[ f(f(x)) \plus{} af(x) \equal{} b(a \plus{} b)x.\] Prove that there exists a unique solution of this equation.

2005 iTest, 10

Tags: probability

The probability of U2 dismantling an atomic bomb is $11\%$. The probability of Coldplay finding X & Y is $23\%$. If the probability of both events occurring is $ 6\%,$ find the probability that neither occurs.