Olympiad problems

2006 Estonia Team Selection Test, 2

Tags: projection , circumcenter , concyclic , cyclic , independent , geometry

The center of the circumcircle of the acute triangle $ABC$ is $O$. The line $AO$ intersects $BC$ at $D$. On the sides $AB$ and $AC$ of the triangle, choose points $E$ and $F$, respectively, so that the points $A, E, D, F$ lie on the same circle. Let $E'$ and $F'$ projections of points $E$ and $F$ on side $BC$ respectively. Prove that length of the segment $E'F'$ does not depend on the position of points $E$ and $F$.

2015 AMC 12/AHSME, 3

Tags:

Isaac has written down one integer two times and another integer three times. The sum of the five numbers is $100$, and one of the numbers is $28$. What is the other number? $\textbf{(A) }8\qquad\textbf{(B) }11\qquad\textbf{(C) }14\qquad\textbf{(D) }15\qquad\textbf{(E) }18$

Gheorghe Țițeica 2025, P1

Tags: function , functional equation , real analysis

Find all continuous functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(x+y)=f(x+f(y))$ for all $x,y\in\mathbb{R}$.

2013 Cuba MO, 1

Tags: trinomial , algebra

Cris has the equation $-2x^2 + bx + c = 0$, and Cristian increases the coefficients of the Cris equation by $1$, obtaining the equation $-x^2 + (b + 1) x + (c + 1) = 0$. Mariloli notices that the real solutions of the Cristian's equation are the squares of the real solutions of the Cris equation. Find all possible values that can take the coefficients $b$ and $c$.

Ukrainian TYM Qualifying - geometry, IV.10

Tags: incircle , geometric inequality , geometry

Given a triangle $ABC$ and points $D, E, F$, which are points of contact of the inscribed circle to the sides of the triangle. i) Prove that $\frac{2pr}{R} \le DE + EF + DF \le p$ ($p$ is the semiperimeter, $r$ and $R$ are respectively the radius of the inscribed and circumscribed circle of $\vartriangle ABC$). ii). Find out when equality is achieved.

1997 Greece National Olympiad, 2

Tags: algebra , function

Let a function $f : \Bbb{R}^+ \to \Bbb{R}$ satisfy: (i) $f$ is strictly increasing, (ii) $f(x) > -1/x$ for all $x > 0$, (iii)$ f(x)f (f(x) + 1/x) = 1$ for all $x > 0$. Determine $f(1)$.

2019 Saudi Arabia JBMO TST, 1

Tags: sum , radical , inequalities

Let $a, b$ and $c$ be positive real numbers such that $a + b + c = 1$. Prove that $$\frac{a}{b}+\frac{b}{a}+\frac{b}{c}+\frac{c}{b}+\frac{c}{a}+\frac{a}{c} \ge 2\sqrt2 \left( \sqrt{\frac{1-a}{a}}+\sqrt{\frac{1-b}{b}}+\sqrt{\frac{1-c}{c}}\right)$$

2014 Estonia Team Selection Test, 1

Tags: prime , combinatorics

In Wonderland, the government of each country consists of exactly $a$ men and $b$ women, where $a$ and $b$ are fixed natural numbers and $b > 1$. For improving of relationships between countries, all possible working groups consisting of exactly one government member from each country, at least $n$ among whom are women, are formed (where $n$ is a fixed non-negative integer). The same person may belong to many working groups. Find all possibilities how many countries can be in Wonderland, given that the number of all working groups is prime.

1996 India Regional Mathematical Olympiad, 7

Tags:

If $A$ is a fifty element subset of the set $1,2,\ldots 100$ such that no two numbers from $A$ add up to $100$, show that $A$ contains a square.

IV Soros Olympiad 1997 - 98 (Russia), 9.10

Tags: geometry , incenter

A circle is drawn through vertices $A$ and $B$ of triangle $ABC$, intersecting sides $AC$ and $BC$ at points $M$ and $P$. It is known that the segment $MP$ contains the center of the circle inscribed in $ABC$. Find $MP$ if $AB = c$, $BC = a$, $CA=b$.

2011 NZMOC Camp Selection Problems, 4

Tags: geometry , parallelogram

Let a point $P$ inside a parallelogram $ABCD$ be given such that $\angle APB +\angle CPD = 180^o$. Prove that $AB \cdot AD = BP \cdot DP + AP \cdot CP$.

2006 Victor Vâlcovici, 2

Tags: algebra , complex number , complex number geometry

Prove that the affixes of three pairwise distinct complex numbers $ z_0,z_1,z_2 $ represent an isosceles triangle with right angle at $ z_0 $ if and only if $ \left( z_1-z_0 \right)^2 =-\left( z_2-z_0 \right)^2. $

2023-24 IOQM India, 7

Tags: dice , permutation , counting , combinatorics

Unconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$ with $1$ and $2$ appearing on opposite faces. Further, each face is colored either red or yellow with opposite faces always of the same color. Two dice are considered to have the same design if one of them can be rotated to obtain a dice that has the same numbers and colors on the corresponding faces as the other one. Find the number of distinct dice that can be designed.

2018 Balkan MO Shortlist, N3

Tags: number theory

Find all primes $p$ and $q$ such that $3p^{q-1}+1$ divides $11^p+17^p$ Proposed by Stanislav Dimitrov,Bulgaria

1998 Dutch Mathematical Olympiad, 5

Tags: function

Find all real solutions of the following equation: \[ (x + 1995)(x + 1997)(x + 1999)(x + 2001) + 16 = 0. \]

2024 China National Olympiad, 2

Tags: inequalities , algebra

Find the largest real number $c$ such that $$\sum_{i=1}^{n}\sum_{j=1}^{n}(n-|i-j|)x_ix_j \geq c\sum_{j=1}^{n}x^2_i$$ for any positive integer $n $ and any real numbers $x_1,x_2,\dots,x_n.$

2002 Rioplatense Mathematical Olympiad, Level 3, 4

Tags: inequality , 3-variable inequality , inequalities , algebra

Let $a, b$ and $c$ be positive real numbers. Show that $\frac{a+b}{c^2}+ \frac{c+a}{b^2}+ \frac{b+c}{a^2}\ge \frac{9}{a+b+c}+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

2017 Ukraine Team Selection Test, 4

Tags: algebra , polynomial

Whether exist set $A$ that contain 2016 real numbers (some of them may be equal) not all of which equal 0 such that next statement holds. For arbitrary 1008-element subset of $A$ there is a monic polynomial of degree 1008 such that elements of this subset are roots of the polynomial and other 1008 elements of $A$ are coefficients of this polynomial's degrees from 0 to 1007.

2021 Francophone Mathematical Olympiad, 4

Tags: functional , algebra , number theory , functional equation

Let $\mathbb{N}_{\ge 1}$ be the set of positive integers. Find all functions $f \colon \mathbb{N}_{\ge 1} \to \mathbb{N}_{\ge 1}$ such that, for all positive integers $m$ and $n$: (a) $n = \left(f(2n)-f(n)\right)\left(2 f(n) - f(2n)\right)$, (b)$f(m)f(n) - f(mn) = \left(f(2m)-f(m)\right)\left(2 f(n) - f(2n)\right) + \left(f(2n)-f(n)\right)\left(2 f(m) - f(2m)\right)$, (c) $m-n$ divides $f(2m)-f(2n)$ if $m$ and $n$ are distinct odd prime numbers.

2019 Sharygin Geometry Olympiad, 15

Tags: geometry

The incircle $\omega$ of triangle $ABC$ touches the sides $BC$, $CA$ and $AB$ at points $D$, $E$ and $F$ respectively. The perpendicular from $E$ to $DF$ meets $BC$ at point $X$, and the perpendicular from $F$ to $DE$ meets $BC$ at point $Y$. The segment $AD$ meets $\omega$ for the second time at point $Z$. Prove that the circumcircle of the triangle $XYZ$ touches $\omega$.

2011 Bulgaria National Olympiad, 3

Tags: limit , function , geometry

Triangle $ABC$ and a function $f:\mathbb{R}^+\to\mathbb{R}$ have the following property: for every line segment $DE$ from the interior of the triangle with midpoint $M$, the inequality $f(d(D))+f(d(E))\le 2f(d(M))$, where $d(X)$ is the distance from point $X$ to the nearest side of the triangle ($X$ is in the interior of $\triangle ABC$). Prove that for each line segment $PQ$ and each point interior point $N$ the inequality $|QN|f(d(P))+|PN|f(d(Q))\le |PQ|f(d(N))$ holds.

2016 District Olympiad, 1

Tags: matrices , algebra , linear algebra

Let $ A\in M_2\left( \mathbb{C}\right) $ such that $ \det \left( A^2+A+I_2\right) =\det \left( A^2-A+I_2\right) =3. $ Prove that $ A^2\left( A^2+I_2\right) =2I_2. $

2018 Sharygin Geometry Olympiad, 5

Tags: geometry

The side $AB$ of a square $ABCD$ is the base of an isosceles triangle $ABE$ such that $AE=BE$ lying outside the square. Let $M$ be the midpoint of $AE$, $O$ be the intersection of $AC$ and $BD$. $K$ is the intersection of $OM$ and $ED$. Prove that $EK=KO$.

2014 Irish Math Olympiad, 6

Tags: number theory

Each of the four positive integers $N,N +1,N +2,N +3$ has exactly six positive divisors. There are exactly$ 20$ dierent positive numbers which are exact divisors of at least one of the numbers. One of these is $27$. Find all possible values of $N$.(Both $1$ and $m$ are counted as divisors of the number $m$.)

2021 Oral Moscow Geometry Olympiad, 2

Tags: perimeter , congruent , geometry

Two quadrangles have equal areas, perimeters and corresponding angles. Are such quadrilaterals necessarily congurent ?