Olympiad problems

2021 AMC 10 Spring, 10

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An inverted cone with base radius $12 \text{ cm}$ and height $18 \text{ cm}$ is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of $24 \text{ cm}$. What is the height in centimeters of the water in the cylinder? $\textbf{(A) }1.5 \qquad \textbf{(B) }3 \qquad \textbf{(C) }4 \qquad \textbf{(D) }4.5 \qquad \textbf{(E) }6$

2016 China Northern MO, 1

Tags: algebra

$a_1,a_2,\cdots,a_n$ are positive real numbers, $a_1+a_2+\cdots,a_n=1$. Prove that $$\sum_{m=1}^n\frac{a_m}{\prod\limits_{k=1}^m(1+a_k)}\leq1-\frac{1}{2^n}.$$

2004 Tournament Of Towns, 1

Tags: geometric transformation , reflection , incenter , circumcircle , geometry

Let us call a triangle rational if each of its angles is a rational number when measured in degrees. Let us call a point inside triangle rational if joining it to the three vertices of the triangle we get three rational triangles. Show that any acute rational triangle contains at least three distinct rational points.

1986 IMO, 2

Tags: triangle , geometry , rotation , fixed point , equilateral triangle

Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$. Define $A_s=A_{s-3}$ for all $s\ge4$. Construct a set of points $P_1,P_2,P_3,\ldots$ such that $P_{k+1}$ is the image of $P_k$ under a rotation center $A_{k+1}$ through an angle $120^o$ clockwise for $k=0,1,2,\ldots$. Prove that if $P_{1986}=P_0$, then the triangle $A_1A_2A_3$ is equilateral.

2020-21 KVS IOQM India, 16

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If $x$ and $y$ are positive integers such that $(x-4)(x-10)=2^y$, then Find maximum value of $x+y$

2021 ASDAN Math Tournament, 3

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Let $V$ be a set of eight points in $3\text{D}$ space that are the vertices of a cube with side length $1$. Compute the number of ways we can color the vertices in $V$ yellow or blue such that [list] [*] each vertex receives exactly one color, and [/*] [*] there exists a point in $3\text{D}$ space whose distance to each yellow vertex is less than $1$ and whose distance to each blue vertex is greater than $1$. [/*] [/list]

2014 Greece Team Selection Test, 2

Tags: polynomial , algebra

Find all real non-zero polynomials satisfying $P(x)^3+3P(x)^2=P(x^{3})-3P(-x)$ for all $x\in\mathbb{R}$.

2013 Czech-Polish-Slovak Junior Match, 3

Tags: equal segments , geometry , pentagon , cyclic

The $ABCDE$ pentagon is inscribed in a circle and $AB = BC = CD$. Segments $AC$ and $BE$ intersect at $K$, and Segments $AD$ and $CE$ intersect at point$ L$. Prove that $AK = KL$.

1992 Poland - First Round, 9

Tags: inequalities

Prove that for all real numbers $a,b,c$ the inequality $(a^2+b^2-c^2)(b^2+c^2-a^2)(c^2+a^2-b^2) \leq (a+b-c)^2(b+c-a)^2(c+a-b)^2$ holds.

2010 Mathcenter Contest, 4

Tags: geometry , functional , convex

Let $P$ be a plane. Prove that there is no function $f :P\rightarrow P$ where, for any convex quadrilateral $ABCD$, the points $f(A)$, $f(B)$, $f(C)$, $f (D)$ are the vertices of a concave quadrilateral. [i](tatari/nightmare)[/i]

1958 AMC 12/AHSME, 34

Tags: inequalities

The numerator of a fraction is $ 6x \plus{} 1$, then denominator is $ 7 \minus{} 4x$, and $ x$ can have any value between $ \minus{}2$ and $ 2$, both included. The values of $ x$ for which the numerator is greater than the denominator are: $ \textbf{(A)}\ \frac{3}{5} < x \le 2\qquad \textbf{(B)}\ \frac{3}{5} \le x \le 2\qquad \textbf{(C)}\ 0 < x \le 2\qquad \\ \textbf{(D)}\ 0 \le x \le 2\qquad \textbf{(E)}\ \minus{}2 \le x \le 2$

1962 IMO, 3

Tags: geometry , 3d geometry , perimeter , vector , cube

Consider the cube $ABCDA'B'C'D'$ ($ABCD$ and $A'B'C'D'$ are the upper and lower bases, repsectively, and edges $AA', BB', CC', DD'$ are parallel). The point $X$ moves at a constant speed along the perimeter of the square $ABCD$ in the direction $ABCDA$, and the point $Y$ moves at the same rate along the perimiter of the square $B'C'CB$ in the direction $B'C'CBB'$. Points $X$ and $Y$ begin their motion at the same instant from the starting positions $A$ and $B'$, respectively. Determine and draw the locus of the midpionts of the segments $XY$.

2008 Balkan MO Shortlist, A7

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Let $x,y,z,t \in \mathbb{R}_{\geq 0}$. Show \begin{align*} \sqrt{xy}+\sqrt{xz}+\sqrt{xt}+\sqrt{yz}+\sqrt{yt}+\sqrt{zt} \geq 3 \sqrt[3]{xyz+xyt+xzt+yzt} \end{align*} and determine the equality cases.

2005 Tournament of Towns, 2

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Two ants crawl along the perimeter of a polygonal table, so that the distance between them is always 10 cm. Each side of the table is more than 1 meter long. At the initial moment both ants are on the same side of the table. (a) [i](2 points)[/i] Suppose that the table is a convex polygon. Is it always true that both ants can visit each point on the perimeter? (b) [i](3 points)[/i] Is it always true (this time without assumption of convexity) that each point on the perimeter can be visited by at least one ant?

2008 Germany Team Selection Test, 3

Tags: polynomial , inequalities , algebra

Find all real polynomials $ f$ with $ x,y \in \mathbb{R}$ such that \[ 2 y f(x \plus{} y) \plus{} (x \minus{} y)(f(x) \plus{} f(y)) \geq 0. \]

2019 New Zealand MO, 4

Tags: number theory , divide , divisible , multiple

Show that the number $122^n - 102^n - 21^n$ is always one less than a multiple of $2020$, for any positive integer $n$.

1957 AMC 12/AHSME, 12

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Comparing the numbers $ 10^{\minus{}49}$ and $ 2\cdot 10^{\minus{}50}$ we may say: $ \textbf{(A)}\ \text{the first exceeds the second by }{8\cdot 10^{\minus{}1}}\qquad\\ \textbf{(B)}\ \text{the first exceeds the second by }{2\cdot 10^{\minus{}1}}\qquad \\ \textbf{(C)}\ \text{the first exceeds the second by }{8\cdot 10^{\minus{}50}}\qquad \\ \textbf{(D)}\ \text{the second is five times the first}\qquad \\ \textbf{(E)}\ \text{the first exceeds the second by }{5}$

2017 IOM, 2

Tags: graph , combinatorics

In a country there are two-way non-stopflights between some pairs of cities. Any city can be reached from any other by a sequence of at most $100$ flights. Moreover, any city can be reached from any other by a sequence of an even number of flights. What is the smallest $d$ for which one can always claim that any city can be reached from any other by a sequence of an even number of flights not exceeding $d$?

1984 Putnam, A4

Tags: geometry , area

A convex pentagon $P=ABCDE$ is inscribed in a circle of radius $1$. Find the maximum area of $P$ subject to the condition that the chords $AC$ and $BD$ are perpendicular.

2017 239 Open Mathematical Olympiad, 5

Tags: graph theory , combinatorics

A school has three classes. Some pairs of children from different classes are enemies (there are no enemies in a class). It is known that every child from the first class has as many enemies in the second class as in the third; the same is true for other classes. Prove that the number of pairs of children from classes having a common enemy is not less than the number of pairs of children being enemies.

2023 Pan-American Girls’ Mathematical Olympiad, 2

Tags: square grid , combinatorics

In each cell of an $n \times n$ grid, one of the numbers $0$, $1,$ or $2$ must be written. Determine all positive integers $n$ for which there exists a way to fill the $n \times n$ grid such that, when calculating the sum of the numbers in each row and each column, the numbers $1, 2, \ldots, 2n$ are obtained in some order.

2022 Baltic Way, 17

Tags: number theory

Let $n$ be a positive integer such that the sum of its positive divisors is at least $2022n$. Prove that $n$ has at least $2022$ distinct prime factors.

2017 Hanoi Open Mathematics Competitions, 5

Tags: fraction , algebra , combinatorics

Write $2017$ following numbers on the blackboard: $-\frac{1008}{1008}, -\frac{1007}{1008}, ..., -\frac{1}{1008}, 0,\frac{1}{1008},\frac{2}{1008}, ... ,\frac{1007}{1008},\frac{1008}{1008}$ . One processes some steps as: erase two arbitrary numbers $x, y$ on the blackboard and then write on it the number $x + 7xy + y$. After $2016$ steps, there is only one number. The last one on the blackboard is (A): $-\frac{1}{1008}$ (B): $0$ (C): $\frac{1}{1008}$ (D): $-\frac{144}{1008}$ (E): None of the above

2004 Romania National Olympiad, 3

Tags: trigonometry , inequalities , power of a point , radical axis , geometry

Let $H$ be the orthocenter of the acute triangle $ABC$. Let $BB'$ and $CC'$ be altitudes of the triangle ($B^{\prime} \in AC$, $C^{\prime} \in AB$). A variable line $\ell$ passing through $H$ intersects the segments $[BC']$ and $[CB']$ in $M$ and $N$. The perpendicular lines of $\ell$ from $M$ and $N$ intersect $BB'$ and $CC'$ in $P$ and $Q$. Determine the locus of the midpoint of the segment $[ PQ]$. [i]Gheorghe Szolosy[/i]

2001 Tournament Of Towns, 2

Tags: combinatorics

At the end of the school year it became clear that for any arbitrarily chosen group of no less than 5 students, 80% of the marks “F” received by this group were given to no more than 20% of the students in the group. Prove that at least 3/4 of all “F” marks were given to the same student.