Olympiad problems

1986 Polish MO Finals, 1

Tags: geometry , rectangle , perimeter , combinatorial geometry , combinatorics , inequalities

A square of side $1$ is covered with $m^2$ rectangles. Show that there is a rectangle with perimeter at least $\frac{4}{m}$.

2011 Morocco TST, 1

Tags: binomial coefficient , number theory , summation , divisibility , modular arithmetic

Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \] cannot be divided by $5$.

$\Delta ABC$ is isosceles with a circumscribed circle $\omega (O)$. Let $H$ be the foot of the altitude from $C$ to $AB$ and let $M$ be the middle point of $AB$. We define a point $X$ as the second intersection point of the circle with diameter $CM$ and $\omega$ and let $XH$ intersect $\omega$ for a second time in $Y$. If $CO\cap AB=D$, then prove that the circumscribed circle of $\Delta YHD$ is tangent to $\omega$.

2018 AMC 8, 15

Tags: geometry , circles

In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of $1$ square unit, then what is the area of the shaded region, in square units? [asy] size(4cm); filldraw(scale(2)*unitcircle,gray,black); filldraw(shift(-1,0)*unitcircle,white,black); filldraw(shift(1,0)*unitcircle,white,black); [/asy] $\textbf{(A) } \frac{1}{4} \qquad \textbf{(B) } \frac{1}{3} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } 1 \qquad \textbf{(E) } \frac{\pi}{2}$

1951 Polish MO Finals, 2

Tags: number theory , digit , divide

What digits should be placed instead of zeros in the third and fifth places in the number $3000003$ to obtain a number divisible by $13$?

2002 India IMO Training Camp, 16

Tags: sum , modular arithmetic , number theory , additive number theory

Is it possible to find $100$ positive integers not exceeding $25,000$, such that all pairwise sums of them are different?

1991 India National Olympiad, 5

Tags: geometry , incenter

Triangle $ABC$ has an incenter $I$. Let points $X$, $Y$ be located on the line segments $AB$, $AC$ respectively, so that $BX \cdot AB = IB^2$ and $CY \cdot AC = IC^2$. Given that the points $X, I, Y$ lie on a straight line, find the possible values of the measure of angle $A$.

2017 Dutch IMO TST, 1

Tags: number theory

Let $a, b,c$ be distinct positive integers, and suppose that $p = ab+bc+ca$ is a prime number. $(a)$ Show that $a^2,b^,c^2$ give distinct remainders after division by $p$. (b) Show that $a^3,b^3,c^3$ give distinct remainders after division by $p$.

Novosibirsk Oral Geo Oly IX, 2023.6

Tags: geometry , angle

Two quarter-circles touch as shown. Find the angle $x$. [img]https://cdn.artofproblemsolving.com/attachments/b/4/e70d5d69e46d6d40368f143cb83cf10b7d6d98.png[/img]

2007 Bulgarian Autumn Math Competition, Problem 9.4

Tags: number theory , divisibility

Find the smallest natural number, which divides $2^{n}+15$ for some natural number $n$ and can be expressed in the form $3x^2-4xy+3y^2$ for some integers $x$ and $y$.

2018 Korea Winter Program Practice Test, 3

Tags: number theory

Let $n$ be a "Good Number" if sum of all divisors of $n$ is less than $2n$ for $n\in \mathbb{Z}.$ Does there exist an infinite set $M$ that satisfies the following? For all $a,b\in M,$ $a+b$ is good number. ($a=b$ is allowed.)

1977 Miklós Schweitzer, 8

Tags: function , real analysis

Let $ p \geq 1$ be a real number and $ \mathbb{R}_\plus{}\equal{}(0, \infty)$. For which continuous functions $ g : \mathbb{R}_\plus{} \rightarrow \mathbb{R}_\plus{}$ are following functions all convex? \[ M_n(x)\equal{}\left[ \frac{\sum_{i\equal{}1}^n g(\frac{x_i}{x_{i\plus{}1}}) x_{i\plus{}1}^p}{\sum_{i\equal{}1}^n g(\frac{x_i}{x_{i\plus{}1}})} \right ]^\frac 1p ,\] \[ x\equal{}(x_1,\ldots, x_{n\plus{}1}) \in \mathbb{R}_\plus{} ^ {n\plus{}1} , \; n\equal{}1,2,\ldots\] [i]L. Losonczi[/i]

2002 Polish MO Finals, 3

Tags: induction , number theory , strong induction

$k$ is a positive integer. The sequence $a_1, a_2, a_3, ...$ is defined by $a_1 = k+1$, $a_{n+1} = a_n ^2 - ka_n + k$. Show that $a_m$ and $a_n$ are coprime (for $m \not = n$).

2017 Peru IMO TST, 10

Tags: squarefree , polynomial , integer , number theory , prime

Let $P (n)$ and $Q (n)$ be two polynomials (not constant) whose coefficients are integers not negative. For each positive integer $n$, define $x_n = 2016^{P (n)} + Q (n)$. Prove that there exist infinite primes $p$ for which there is a positive integer $m$, squarefree, such that $p | x_m$. Clarification: A positive integer is squarefree if it is not divisible by the square of any prime number.

2017 Spain Mathematical Olympiad, 4

Tags: combinatorial game , combinatorics

You are given a row made by $2018$ squares, numbered consecutively from $0$ to $2017$. Initially, there is a coin in the square $0$. Two players $A$ and $B$ play alternatively, starting with $A$, on the following way: In his turn, each player can either make his coin advance $53$ squares or make the coin go back $2$ squares. On each move the coin can never go to a number less than $0$ or greater than $2017$. The player who puts the coin on the square $2017$ wins. ¿Who is the one with the wining strategy and how should he play to win?

2021 AMC 10 Spring, 18

Tags: prob

A fair 6-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number? $\textbf{(A)}\ \frac{1}{120} \qquad\textbf{(B)}\ \frac{1}{32} \qquad\textbf{(C)}\frac{1}{20} \qquad\textbf{(D)}\ \frac{3}{20} \qquad\textbf{(E)}\ \frac{1}{6}$

2010 Romanian Masters In Mathematics, 3

Tags: geometric transformation , trigonometry , geometry , ratio , homothety , projective geometry , trig identities

Let $A_1A_2A_3A_4$ be a quadrilateral with no pair of parallel sides. For each $i=1, 2, 3, 4$, define $\omega_1$ to be the circle touching the quadrilateral externally, and which is tangent to the lines $A_{i-1}A_i, A_iA_{i+1}$ and $A_{i+1}A_{i+2}$ (indices are considered modulo $4$ so $A_0=A_4, A_5=A_1$ and $A_6=A_2$). Let $T_i$ be the point of tangency of $\omega_i$ with the side $A_iA_{i+1}$. Prove that the lines $A_1A_2, A_3A_4$ and $T_2T_4$ are concurrent if and only if the lines $A_2A_3, A_4A_1$ and $T_1T_3$ are concurrent. [i]Pavel Kozhevnikov, Russia[/i]

2001 Romania National Olympiad, 3

Tags: trapezoid , geometry

We consider a right trapezoid $ABCD$, in which $AB||CD,AB>CD,AD\perp AB$ and $AD>CD$. The diagonals $AC$ and $BD$ intersect at $O$. The parallel through $O$ to $AB$ intersects $AD$ in $E$ and $BE$ intersects $CD$ in $F$. Prove that $CE\perp AF$ if and only if $AB\cdot CD=AD^2-CD^2$ .

STEMS 2023 Math Cat A, 3

Tags: geometry , ratio

Given a triangle $ABC$ with angles $\angle A = 60^{\circ}, \angle B = 75^{\circ}, \angle C = 45^{\circ}$, let $H$ be its orthocentre, and $O$ be its circumcenter. Let $F$ be the midpoint of side $AB$, and $Q$ be the foot of the perpendicular from $B$ onto $AC$. Denote by $X$ the intersection point of the lines $FH$ and $QO$. Suppose the ratio of the length of $FX$ and the circumradius of the triangle is given by $\dfrac{a + b \sqrt{c}}{d}$, then find the value of $1000a + 100b + 10c + d$.

2018 Turkey Team Selection Test, 5

Tags: combinatorics

We say that a group of $25$ students is a [i]team[/i] if any two students in this group are friends. It is known that in the school any student belongs to at least one team but if any two students end their friendships at least one student does not belong to any team. We say that a team is [i]special[/i] if at least one student of the team has no friend outside of this team. Show that any two friends belong to some special team.

VI Soros Olympiad 1999 - 2000 (Russia), 8.5

Tags: number theory , system of equations

Solve the following system of equations in natural numbers $$\begin{cases} a^4+14ab+1=n^4 \\ b^4+14bc+1=m^4 \\ c^4+14ca+1=k^4 \end{cases}$$

2010 Princeton University Math Competition, 8

Tags: geometry , 3d geometry , sphere , tetrahedron

There is a point source of light in an empty universe. What is the minimum number of solid balls (of any size) one must place in space so that any light ray emanating from the light source intersects at least one ball?

1949-56 Chisinau City MO, 1

Tags: algebra

The numbers $1, 2, ..., 1000$ are written out in a row along a circle. Starting from the first, every fifteenth number in the circle is crossed out $(1, 16, 31, ...)$, in this case, the crossed out numbers are still taken into account at each new round of the circle. How many numbers are left uncrossed?

1938 Moscow Mathematical Olympiad, 041

Tags: geometric construction , triangle construction , geometry

Given the base, height and the difference between the angles at the base of a triangle, construct the triangle.

2021 Argentina National Olympiad Level 2, 2

Tags: geometry

In a semicircle with center $O$, let $C$ be a point on the diameter $AB$ different from $A, B$ and $O.$ Draw through $C$ two rays such that the angles that these rays form with the diameter $AB$ are equal and that they intersect at the semicircle at $D$ and at $E$. The line perpendicular to $CD$ through $D$ intersects the semicircle at $K.$ Prove that if $D\neq E,$ then $KE$ is parallel to $AB.$