Olympiad problems

2022 Brazil Team Selection Test, 1

Let $a, b, c$ be positive real numbers. Show that $$a^5+b^5+c^5 \geq 5abc(b^2-ac)$$ and determine when the equality occurs.

2019 Jozsef Wildt International Math Competition, W. 12

Tags: calculus , integration , inequalities

If $0 < a < b$ then: $$\frac{\int \limits^{\frac{a+b}{2}}_{a}\left(\tan^{-1}t\right)dt}{\int \limits_{a}^{b}\left(\tan^{-1}t\right)dt}<\frac{1}{2}$$

2019 Online Math Open Problems, 2

Tags:

Let $A$, $B$, $C$, and $P$ be points in the plane such that no three of them are collinear. Suppose that the areas of triangles $BPC$, $CPA$, and $APB$ are 13, 14, and 15, respectively. Compute the sum of all possible values for the area of triangle $ABC$. [i]Proposed by Ankan Bhattacharya[/i]

1953 Moscow Mathematical Olympiad, 253

Tags: inequalities , algebra , root , trinomial

Given the equations (1) $ax^2 + bx + c = 0$ (2)$ -ax^2 + bx + c = 0$ prove that if $x_1$ and $x_2$ are some roots of equations (1) and (2), respectively, then there is a root $x_3$ of the equation $$\frac{a}{2}x^2 + bx + c = 0$$ such that either $x_1 \le x_3 \le x_2$ or $x_1 \ge x_3 \ge x_2$.

2003 Bosnia and Herzegovina Junior BMO TST, 2

Tags: algebra

Solve in the set of rational numbers the equation $$2\sqrt{3(x + 1)^2} -3 \sqrt{2(y - 2)^2}= 4\sqrt2 + 5|\sqrt2 - \sqrt3|$$

1999 Bundeswettbewerb Mathematik, 1

Tags: combinatorics , combinatorial geometry

The vertices of a regular $2n$-gon (with $n > 2$ an integer) are labelled with the numbers $1,2,...,2n$ in some order. Assume that the sum of the labels at any two adjacent vertices equals the sum of the labels at the two diametrically opposite vertices. Prove that this is possible if and only if $n$ is odd.

2010 Contests, 4

Tags: geometry

Point $O$ is chosen in a triangle $ABC$ such that ${d_a},{d_b},{d_c}$ are distance from point $O$ to sides $BC,CA,AB$, respectively. Find position of point $O$ so that product ${d_a} \cdot {d_b} \cdot {d_c}$ becomes maximum.

2021 Iberoamerican, 2

Tags: circumcircle , geometry

Consider an acute-angled triangle $ABC$, with $AC>AB$, and let $\Gamma$ be its circumcircle. Let $E$ and $F$ be the midpoints of the sides $AC$ and $AB$, respectively. The circumcircle of the triangle $CEF$ and $\Gamma$ meet at $X$ and $C$, with $X\neq C$. The line $BX$ and the tangent to $\Gamma$ through $A$ meet at $Y$. Let $P$ be the point on segment $AB$ so that $YP = YA$, with $P\neq A$, and let $Q$ be the point where $AB$ and the parallel to $BC$ through $Y$ meet each other. Show that $F$ is the midpoint of $PQ$.

2019 Jozsef Wildt International Math Competition, W. 30

Tags: function , limit , zeta function , summation

[list=1] [*] Prove that $$\lim \limits_{n \to \infty} \left(n+\frac{1}{4}-\zeta(3)-\zeta(5)-\cdots -\zeta(2n+1)\right)=0$$ [*] Calculate $$\sum \limits_{n=1}^{\infty} \left(n+\frac{1}{4}-\zeta(3)-\zeta(5)-\cdots -\zeta(2n+1)\right)$$ [/list]

1990 IberoAmerican, 5

Tags: combinatorics , analytic geometry

$A$ and $B$ are two opposite vertices of an $n \times n$ board. Within each small square of the board, the diagonal parallel to $AB$ is drawn, so that the board is divided in $2n^{2}$ equal triangles. A coin moves from $A$ to $B$ along the grid, and for every segment of the grid that it visits, a seed is put in each triangle that contains the segment as a side. The path followed by the coin is such that no segment is visited more than once, and after the coins arrives at $B$, there are exactly two seeds in each of the $2n^{2}$ triangles of the board. Determine all the values of $n$ for which such scenario is possible.

2023 International Zhautykov Olympiad, 4

Tags: algebra

The sum of $n > 2$ nonzero real numbers (not necessarily distinct) equals zero. For each of the $2^n - 1$ ways to choose one or more of these numbers, their sums are written in non-increasing order in a row. The first number in the row is $S$. Find the smallest possible value of the second number.

2019 AMC 8, 15

Tags:

On a beach 50 people are wearing sunglasses and 35 people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is also wearing sunglasses is $\frac{2}{5}$. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap? $\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}$

Ukrainian From Tasks to Tasks - geometry, 2010.5

Tags: geometry , right triangle

In a right triangle $ABC$ ($\angle C = 90^o$) it is known that $AC = 4$ cm, $BC = 3$ cm. The points $A_1, B_1$ and $C_1$ are such that $AA_1 \parallel BC$, $BB_1\parallel A_1C$, $CC_1\parallel A_1B_1$, $A_1B_1C_1= 90^o$, $A_1B_1= 1$ cm. Find $B_1C_1$.

2024 Austrian MO National Competition, 1

Tags: algebra , parameter , parameterization , functional equation

Let $\alpha$ and $\beta$ be real numbers with $\beta \ne 0$. Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[f(\alpha f(x)+f(y))=\beta x+f(y)\] holds for all real $x$ and $y$. [i](Walther Janous)[/i]

2018 MIG, 8

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A marathon runner has a very peculiar way of training for a marathon. On the first day of week $1$, the runner runs a distance equivalent to the first prime number. On the second day, the runner runs a distance equal to the second prime number, continuing this pattern until the $7$th day of the week. Each successive week, the runner runs one more mile per day than they did on the same day of the previous week. The runner continues this process until the average distance run each week exceeds the distance of a marathon ($26.2$ miles). How many weeks does the marathoner train?

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]

2012 Puerto Rico Team Selection Test, 1

Tags: inequalities , algebra

Let $x, y$ and $z$ be consecutive integers such that \[\frac 1x+\frac 1y+\frac 1z >\frac{1}{45}.\] Find the maximum value of $x + y + z$.

KoMaL A Problems 2020/2021, A. 798

Tags: expected value , combinatorics , probability

Let $0<p<1$ be given. Initially, we have $n$ coins, all of which have probability $p$ of landing on heads, and probability $1-p$ of landing on tails (the results of the tosses are independent of each other). In each round, we toss our coins and remove those that result in heads. We keep repeating this until all our coins are removed. Let $k_n$ denote the expected number of rounds that are needed to get rid of all the coins. Prove that there exists $c>0$ for which the following inequality holds for all $n>0$ \[c\bigg(1+\frac{1}{2}+\cdots+\frac{1}{n}\bigg)<k_n<1+c\bigg(1+\frac{1}{2}+\cdots+\frac{1}{n}\bigg).\]

1969 IMO Shortlist, 18

Tags: coprime , frobenius , additive number theory , number theory

$(FRA 1)$ Let $a$ and $b$ be two nonnegative integers. Denote by $H(a, b)$ the set of numbers $n$ of the form $n = pa + qb,$ where $p$ and $q$ are positive integers. Determine $H(a) = H(a, a)$. Prove that if $a \neq b,$ it is enough to know all the sets $H(a, b)$ for coprime numbers $a, b$ in order to know all the sets $H(a, b)$. Prove that in the case of coprime numbers $a$ and $b, H(a, b)$ contains all numbers greater than or equal to $\omega = (a - 1)(b -1)$ and also $\frac{\omega}{2}$ numbers smaller than $\omega$

2017 NIMO Problems, 2

Tags: trapezoid , geometry

Trapezoid $ABCD$ is an isosceles trapezoid with $AD=BC$. Point $P$ is the intersection of the diagonals $AC$ and $BD$. If the area of $\triangle ABP$ is $50$ and the area of $\triangle CDP$ is $72$, what is the area of the entire trapezoid? [i]Proposed by David Altizio

2020 Iran Team Selection Test, 1

Tags: combinatorics , graph theory

A weighted complete graph with distinct positive wights is given such that in every triangle is [i]degenerate [/i] that is wight of an edge is equal to sum of two other. Prove that one can assign values to the vertexes of this graph such that the wight of each edge is the difference between two assigned values of the endpoints. [i]Proposed by Morteza Saghafian [/i]

2000 Harvard-MIT Mathematics Tournament, 7

Tags: algebra

Find $[ \sqrt{19992000}]$ where $[a]$ is the greatest integer less than or equal to $x$.

Geometry Mathley 2011-12, 3.1

Tags: geometry , equal segments , circumcircle , circles

$AB,AC$ are tangent to a circle $(O)$, $B,C$ are the points of tangency. $Q$ is a point iside the angle $BAC$, on the ray $AQ$, take a point $P$ suc that $OP$ is perpendicular to $AQ$. The line $OP$ meets the circumcircles triangles $BPQ$ and $CPQ$ at $I, J$. Prove that $OI = OJ$. Hồ Quang Vinh

2019 Swedish Mathematical Competition, 4

Tags: circles , combinatorial geometry , combinatorics

Let $\Omega$ be a circle disk with radius $1$. Determine the minimum $r$ that has the following property: You can select three points on $\Omega$ so that each circle disk located in $\Omega$ and has a radius greater than $r$ contains at least one of the three points.

1967 Polish MO Finals, 6

Tags: geometry , 3d geometry , sphere

Given a sphere and a plane that has no common points with the sphere. Find the geometric locus of the centers of the circles of tangency with the sphere of those cones circumcribed on the sphere whose vertices lie on the given plane.