Olympiad problems

2021 LMT Spring, B14

Tags: combinatorics

In the expansion of $(2x +3y)^{20}$, find the number of coefficients divisible by $144$. [i]Proposed by Hannah Shen[/i]

2002 National High School Mathematics League, 15

Tags: function

Quadratic function $f(x)=ax^2+bx+c$ satisfies that: $(1)\forall x\in\mathbb{R},f(x-4)=f(2-x),f(x)\geq x$; $(2)\forall x\in(0,2),f(x)\leq\left(\frac{x+1}{2}\right)^2$; $(3)\min\limits_{x\in\mathbb{R}}f(x)=0$ Find the maximum of $m(m>1)$, satisfying: There exists $t\in\mathbb{R}$, as long as $x\in[1,m]$, then $f(x+t)\leq x$.

2006 Stanford Mathematics Tournament, 15

Tags:

The odometer of a family car shows 15,951 miles. The driver noticed that this number is palindromic: it reads the same backward as forwards. "Curious," the driver said to himself, "it will be a long time before that happens again." Surprised, he saw his third palindromic odometer reading (not counting 15,951) exactly five hours later. How many miles per hour was the car traveling in those 5 hours (assuming speed was constant)?

1995 All-Russian Olympiad, 2

Tags: geometry proposed , geometry

A chord $CD$ of a circle with center $O$ is perpendicular to a diameter $AB$. A chord $AE$ bisects the radius $OC$. Show that the line $DE$ bisects the chord $BC$ [i]V. Gordon[/i]

2000 Spain Mathematical Olympiad, 1

Tags: modular arithmetic , floor function , number theory proposed , number theory

Find the largest integer $N$ satisfying the following two conditions: [b](i)[/b] $\left[ \frac N3 \right]$ consists of three equal digits; [b](ii)[/b] $\left[ \frac N3 \right] = 1 + 2 + 3 +\cdots + n$ for some positive integer $n.$

2021 Canadian Junior Mathematical Olympiad, 3

Tags: geometry

Let $ABCD$ be a trapezoid with $AB$ parallel to $CD$, $|AB|>|CD|$, and equal edges $|AD|=|BC|$. Let $I$ be the center of the circle tangent to lines $AB$, $AC$ and $BD$, where $A$ and $I$ are on opposite sides of $BD$. Let $J$ be the center of the circle tangent to lines $CD$, $AC$ and $BD$, where $D$ and $J$ are on opposite sides of $AC$. Prove that $|IC|=|JB|$.

2023 USAJMO Solutions by peace09, 3

Tags: rotation , function , AMC , USA(J)MO , USAJMO , combinatorics , Hi

Consider an $n$-by-$n$ board of unit squares for some odd positive integer $n$. We say that a collection $C$ of identical dominoes is a [i]maximal grid-aligned configuration[/i] on the board if $C$ consists of $(n^2-1)/2$ dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: $C$ then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let $k(C)$ be the number of distinct maximal grid-aligned configurations obtainable from $C$ by repeatedly sliding dominoes. Find the maximum value of $k(C)$ as a function of $n$. [i]Proposed by Holden Mui[/i]

2017 Abels Math Contest (Norwegian MO) Final, 1b

Tags: function , functional equation , algebra

Find all functions $f : R \to R$ which satisfy $f(x)f(y) = f(x + y) + xy$ for all $x, y \in R$.

2007 Iran Team Selection Test, 2

Tags: ceiling function , pigeonhole principle , modular arithmetic , search , combinatorics proposed , combinatorics

Let $A$ be the largest subset of $\{1,\dots,n\}$ such that for each $x\in A$, $x$ divides at most one other element in $A$. Prove that \[\frac{2n}3\leq |A|\leq \left\lceil \frac{3n}4\right\rceil. \]

2010 Contests, 3

Tags: inequalities , inequalities proposed , Kazakhstan

Positive real $A$ is given. Find maximum value of $M$ for which inequality $ \frac{1}{x}+\frac{1}{y}+\frac{A}{x+y} \geq \frac{M}{\sqrt{xy}} $ holds for all $x, y>0$

2012 Kazakhstan National Olympiad, 3

Tags: inequalities , inequalities unsolved , Kazakhstan , Kanat Satylkhanov

Let $ a,b,c,d>0$ for which the following conditions:: $a)$ $(a-c)(b-d)=-4$ $b)$ $\frac{a+c}{2}\geq\frac{a^{2}+b^{2}+c^{2}+d^{2}}{a+b+c+d}$ Find the minimum of expression $a+c$

2011 LMT, 7

Tags:

A triangle $ABC$ has side lengths $AB=8$ and $BC=10.$ Given that the altitude to side $BC$ has length $4,$ what is the length of the altitude to side $AB?$

2010 Indonesia Juniors, day 2

Tags: algebra , geometry , combinatorics , number theory , indonesia juniors

p1. If $x + y + z = 2$, show that $\frac{1}{xy+z-1}+\frac{1}{yz+x-1}+\frac{1}{xz+y-1}=\frac{-1}{(x-1)(y-1)(z-1)}$. p2. Determine the simplest form of $\frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!}+\frac{5}{3!+4!+5!}+...+\frac{100}{98!+99!+100!}$ p3. It is known that $ABCD$ and $DEFG$ are two parallelograms. Point $E$ lies on $AB$ and point $C$ lie on $FG$. The area of $ABCD$ is $20$ units. $H$ is the point on $DG$ so that $EH$ is perpendicular to $DG$. If the length of $DG$ is $5$ units, determine the length of $EH$. [img]https://cdn.artofproblemsolving.com/attachments/b/e/42453bf6768129ed84fbdc81ab7235e780b0e1.png[/img] p4. Each room in the following picture will be painted so that every two rooms which is directly connected to the door is given a different color. If $10$ different colors are provided and $4$ of them can not be used close together for two rooms that are directly connected with a door, determine how many different ways to color the $4$ rooms. [img]https://cdn.artofproblemsolving.com/attachments/4/a/e80a464a282b3fe3cdadde832b2fd38b51a41a.png[/img] 5. The floor of a hall is rectangular $ABCD$ with $AB = 30$ meters and $BC = 15$ meters. A cat is in position $A$. Seeing the cat, the mouse in the midpoint of $AB$ ran and tried to escape from cat. The mouse runs from its place to point $C$ at a speed of $3$ meters/second. The trajectory is a straight line. Watching the mice run away, at the same time from point $A$ the cat is chasing with a speed of $5$ meters/second. If the cat's path is also a straight line and the mouse caught before in $C$, determine an equation that can be used for determine the position and time the mouse was caught by the cat.

2023 BMT, Tie 2

Tags: geometry

Triangle $\vartriangle ABC$ has $\angle ABC = \angle BCA = 45^o$ and $AB = 1$. Let $D$ be on $\overline{AC}$ such that $\angle ABD =30^o$. Let $\overleftrightarrow{BD}$ and the line through $A$ parallel to $\overleftrightarrow{BC}$ intersect at $E$. Compute the area of $\vartriangle ADE$.

2021 Belarusian National Olympiad, 10.3

Tags: number theory

Odd numbers $x,y,z$ such that $gcd(x,y,z)=1$ are given. It turned out that $x^2+y^2+z^2 \vdots x+y+z$ Prove that $x+y+z-2$ is not divisible by $3$

2008 All-Russian Olympiad, 2

Tags: ratio , algebra proposed , algebra

Petya and Vasya are given equal sets of $ N$ weights, in which the masses of any two weights are in ratio at most $ 1.25$. Petya succeeded to divide his set into $ 10$ groups of equal masses, while Vasya succeeded to divide his set into $ 11$ groups of equal masses. Find the smallest possible $ N$.

2014 Federal Competition For Advanced Students, 3

Tags: recurrence relation , Sequence , number theory

Let $a_n$ be a sequence defined by some $a_0$ and the recursion $a_{n+1} = a_n + 2 \cdot 3^n$ for $n \ge 0$. Determine all rational values of $a_0$ such that $a^j_k / a^k_j$ is an integer for all integers $j$ and $k$ with $0 < j < k$.

2018 China Western Mathematical Olympiad, 2

Tags: algebra , inequalities , fractional part , n-variable inequality

Let $n \geq 2$ be an integer. Positive reals $x_1, x_2, \cdots, x_n$ satisfy $x_1x_2 \cdots x_n = 1$. Show: $$\{x_1\} + \{x_2\} + \cdots + \{x_n\} < \frac{2n-1}{2}$$ Where $\{x\}$ denotes the fractional part of $x$.

2018 Iran Team Selection Test, 3

Tags: geometry , Pole and Polar , Pappus

In triangle $ABC$ let $M$ be the midpoint of $BC$. Let $\omega$ be a circle inside of $ABC$ and is tangent to $AB,AC$ at $E,F$, respectively. The tangents from $M$ to $\omega$ meet $\omega$ at $P,Q$ such that $P$ and $B$ lie on the same side of $AM$. Let $X \equiv PM \cap BF $ and $Y \equiv QM \cap CE $. If $2PM=BC$ prove that $XY$ is tangent to $\omega$. [i]Proposed by Iman Maghsoudi[/i]

2008 Purple Comet Problems, 5

Tags:

What is the measurement in degrees of the angle formed by the minute and hour hands when a clock reads $12:12$?

2024 Chile National Olympiad., 5

Tags: combinatorics , Chile

You have a collection of at least two tokens where each one has a number less than or equal to 10 written on it. The sum of the numbers on the tokens is $ S $. Find all possible values of $ S $ that guarantee that the tokens can be separated into two groups such that the sum of each group does not exceed 80.

2020 Mexico National Olympiad, 3

Tags: combinatorics , Circular array , vector

Let $n\ge 3$ be an integer. Two players, Ana and Beto, play the following game. Ana tags the vertices of a regular $n$- gon with the numbers from $1$ to $n$, in any order she wants. Every vertex must be tagged with a different number. Then, we place a turkey in each of the $n$ vertices. These turkeys are trained for the following. If Beto whistles, each turkey moves to the adjacent vertex with greater tag. If Beto claps, each turkey moves to the adjacent vertex with lower tag. Beto wins if, after some number of whistles and claps, he gets to move all the turkeys to the same vertex. Ana wins if she can tag the vertices so that Beto can't do this. For each $n\ge 3$, determine which player has a winning strategy. [i]Proposed by Victor and Isaías de la Fuente[/i]

2021 DIME, 5

Tags: DIME P5

Let $\mathcal{S}$ be the set of all positive integers which are both a multiple of $3$ and have at least one digit that is a $1$. For example, $123$ is in $\mathcal{S}$ and $450$ is not. The probability that a randomly chosen $3$-digit positive integer is in $\mathcal{S}$ can be written as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by GammaZero[/i]

1997 Korea National Olympiad, 1

Tags: partition , combinatorics

Let $f(n)$ be the number of ways to express positive integer $n$ as a sum of positive odd integers. Compute $f(n).$ (If the order of odd numbers are different, then it is considered as different expression.)

1969 All Soviet Union Mathematical Olympiad, 125

Tags: Cubic , algebra

Given an equation $$x^3 + ?x^2 + ?x + ? = 0$$ First player substitutes an integer on the place of one of the interrogative marks, than the same do the second with one of the two remained marks, and, finally, the first puts the integer instead of the last mark. Explain how can the first provide the existence of three integer roots in the obtained equation. (The roots may coincide.)