Olympiad problems

2023 Romania National Olympiad, 4

Tags: combinatorics , coloring

In an art museum, $n$ paintings are exhibited, where $n \geq 33.$ In total, $15$ colors are used for these paintings such that any two paintings have at least one common color, and no two paintings have exactly the same colors. Determine all possible values of $n \geq 33$ such that regardless of how we color the paintings with the given properties, we can choose four distinct paintings, which we can label as $T_1, T_2, T_3,$ and $T_4,$ such that any color that is used in both $T_1$ and $T_2$ can also be found in either $T_3$ or $T_4$.

2022 Czech and Slovak Olympiad III A, 1

Tags: algebra

In a sequence of $71$ nonzero real numbers, each number (apart from the fitrst one and the last one) is one less than the product of its two neighbors. Prove that the first and the last number are equal. [i](Josef Tkadlec)[/i]

2012 Indonesia TST, 4

Tags: quadratic , floor function , search , modular arithmetic , induction , pigeonhole principle , number theory

Determine all natural numbers $n$ such that for each natural number $a$ relatively prime with $n$ and $a \le 1 + \left\lfloor \sqrt{n} \right\rfloor$ there exists some integer $x$ with $a \equiv x^2 \mod n$. Remark: "Natural numbers" is the set of positive integers.

2014 Romania National Olympiad, 2

Tags: algebra

Let $ a $ be an odd natural that is not a perfect square, and $ m,n\in\mathbb{N} . $ Then [b]a)[/b] $ \left\{ m\left( a+\sqrt a \right) \right\}\neq\left\{ n\left( a-\sqrt a \right) \right\} $ [b]b)[/b] $ \left[ m\left( a+\sqrt a \right) \right]\neq\left[ n\left( a-\sqrt a \right) \right] $ Here, $ \{\},[] $ denotes the fractionary, respectively the integer part.

2009 Federal Competition For Advanced Students, P1, 3

Tags: combinatorics

There are $n$ bus stops placed around the circular lake. Each bus stop is connected by a road to the two adjacent stops (we call a [i]segment [/i] the entire road between two stops). Determine the number of bus routes that start and end in the fixed bus stop A, pass through each bus stop at least once and travel through exactly $n+1$ [i]segments[/i].

2020 Germany Team Selection Test, 2

Tags: geometry

Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$. (Nigeria)

1994 AMC 8, 25

Tags:

Find the sum of the digits in the answer to $\underbrace{9999\cdots 99}_{94\text{ nines}} \times \underbrace{4444\cdots 44}_{94\text{ fours}}$ where a string of $94$ nines is multiplied by a string of $94$ fours. $\text{(A)}\ 846 \qquad \text{(B)}\ 855 \qquad \text{(C)}\ 945 \qquad \text{(D)}\ 954 \qquad \text{(E)}\ 1072$

2016 LMT, 5

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An isosceles triangle has angles of $50^\circ,x^\circ,$ and $y^\circ$. Find the maximum possible value of $x-y$. [i]Proposed by Nathan Ramesh

2022 Bulgarian Spring Math Competition, Problem 9.1

Tags: algebra , quadratic trinomial , quadratic , function

Let $f(x)$ be a quadratic function with integer coefficients. If we know that $f(0)$, $f(3)$ and $f(4)$ are all different and elements of the set $\{2, 20, 202, 2022\}$, determine all possible values of $f(1)$.

2018 Yasinsky Geometry Olympiad, 6

Tags: projection , geometry , angle chasing , equal angles , altitude

$AH$ is the altitude of the acute triangle $ABC$, $K$ and $L$ are the feet of the perpendiculars, from point $H$ on sides $AB$ and $AC$ respectively. Prove that the angles $BKC$ and $BLC$ are equal.

2023 Yasinsky Geometry Olympiad, 6

Tags: geometry , equilateral

An acute triangle $ABC$ is surrounded by equilateral triangles $KLM$ and $PQR$ such that its vertices lie on the sides of these equilateral triangle as shown on the picture. Lines $PK$ and $QL$ intersect at point $D$. Prove that $\angle ABC + \angle PDQ = 120^o$. (Yurii Biletskyi) [img]https://cdn.artofproblemsolving.com/attachments/4/6/32d3f74f07ca6a8edcabe4a08aa321eb3a5010.png[/img]

1986 Traian Lălescu, 2.1

Tags: combinatorics , algebra

Consider the numbers $ a_n=1-\binom{n}{3} +\binom{n}{6} -\cdots, b_n= -\binom{n}{1} +\binom{n}{4}-\binom{n}{7} +\cdots $ and $ c_n=\binom{n}{2} -\binom{n}{5} +\binom{n}{8} -\cdots , $ for a natural number $ n\ge 2. $ Prove that $$ a_n^2+b_n^2+c_n^2-a_nb_n-b_nc_n-c_na_n =3^{n-1}. $$

2005 Georgia Team Selection Test, 7

Tags: number theory

Determine all positive integers $ n$, for which $ 2^{n\minus{}1}n\plus{}1$ is a perfect square.

2020 Sharygin Geometry Olympiad, 17

Tags: geometry

Chords $A_1A_2$ and $B_1B_2$ meet at point $D$. Suppose $D'$ is the inversion image of $D$ and the line $A_1B_1$ meets the perpendicular bisector to $DD'$ at a point $C$. Prove that $CD\parallel A_2B_2$.

2020 CHMMC Winter (2020-21), 7

Tags: combinatorics

Given $10$ points on a plane such that no three are collinear, we connect each pair of points with a segment and color each segment either red or blue. Assume that there exists some point $A$ among the $10$ points such that: 1. There is an odd number of red segments connected to $A$} 2. The number of red segments connected to each of the other points are all different Find the number of red triangles (i.e, a triangle whose three sides are all red segments) on the plane.

2000 Harvard-MIT Mathematics Tournament, 10

Tags: geometry , algebra

How many times per day do at least two of the three hands on a clock coincide?

2024 Korea Junior Math Olympiad, 4

Tags: number theory

find all positive integer n such that there exists positive integers (a,b) such that (a^n + b^n)/n! is a positive integer smaller than 101

2000 Putnam, 5

Tags: induction

Let $S_0$ be a finite set of positive integers. We define finite sets $S_1, S_2, \cdots$ of positive integers as follows: the integer $a$ in $S_{n+1}$ if and only if exactly one of $a-1$ or $a$ is in $S_n$. Show that there exist infinitely many integers $N$ for which $S_N = S_0 \cup \{ N + a: a \in S_0 \}$.

2005 SNSB Admission, 3

Tags: complex analysis , function , complex number , algebra , polynomial

Let $ f:\mathbb{C}\longrightarrow\mathbb{C} $ be an holomorphic function which has the property that there exist three positive real numbers $ a,b,c $ such that $ |f(z)|\geqslant a|z|^b , $ for any complex numbers $ z $ with $ |z|\geqslant c. $ Prove that $ f $ is polynomial with degree at least $ \lceil b\rceil . $

2021 AMC 10 Fall, 14

Tags: algebra , system of equations , absolute value

How many ordered pairs $(x,y)$ of real numbers satisfy the following system of equations? \begin{align*} x^2+3y&=9\\ (|x|+|y|-4)^2&=1\\ \end{align*} $\textbf{(A)}\: 1\qquad\textbf{(B)} \: 2\qquad\textbf{(C)} \: 3\qquad\textbf{(D)} \: 5\qquad\textbf{(E)} \: 7$

2005 Purple Comet Problems, 1

Tags:

A cubic inch of the newly discovered material madelbromium weighs $5$ ounces. How many pounds will a cubic yard of madelbromium weigh?

1992 National High School Mathematics League, 14

Tags: 3d geometry , geometric transformation , reflection , geometry

$l,m$ are skew lines. Three points $A,B,C$ on line $l$ satisfy that $AB=BC$. Projection of $A,B,C$ on $m$ are $D,E,F$. If $|AD|=\sqrt{15},|BE|=\frac{7}{2}|CF|=\sqrt{10}$, find the distance between $l$ and $m$.

2020 USMCA, 5

Tags:

Call a positive integer $n$ an $A-B$ number if the base $A$ and base $B$ representations of $n$ are three-digit numbers that are reverses of each other. For example, $87$ is a $5-6$ number because $87 = 223_6 = 322_5$. Compute the sum of all $7-11$ numbers.

2009 Italy TST, 1

Tags: polynomial , algebra

Let $n$ be an even positive integer. An $n$-degree monic polynomial $P(x)$ has $n$ real roots (not necessarily distinct). Suppose $y$ is a positive real number such that for any real number $t<y$, we have $P(t)>0$. Prove that \[P(0)^{\frac{1}{n}}-P(y)^{\frac{1}{n}}\ge y.\]

1994 India National Olympiad, 5

Tags: rectangle , geometry

A circle passes through the vertex of a rectangle $ABCD$ and touches its sides $AB$ and $AD$ at $M$ and $N$ respectively. If the distance from $C$ to the line segment $MN$ is equal to $5$ units, find the area of rectangle $ABCD$.