Olympiad problems

2016 Czech-Polish-Slovak Junior Match, 1

Let $ABC$ be a right-angled triangle with hypotenuse $AB$. Denote by $D$ the foot of the altitude from $C$. Let $Q, R$, and $P$ be the midpoints of the segments $AD, BD$, and $CD$, respectively. Prove that $\angle AP B + \angle QCR = 180^o$. Czech Republic

1999 Junior Balkan Team Selection Tests - Moldova, 3

Tags: number theory

On the board is written a number with nine non-zero and distinct digits. Prove that we can delete at most seven digits so that the number formed by the digits left to be a perfect square.

1995 Tournament Of Towns, (478) 2

Tags: irrational number , number theory , algebra

Let $p$ be the product of $n$ real numbers $x_1$, $x_2$,$...$, $x_n$. Prove that if $p - x_k$ is an odd integer for $k = 1, 2,..., n$, then each of the numbers $x_1$, $x_2$,$...$, $x_n$is irrational. (G Galperin)

1991 IMO Shortlist, 23

Tags: function , algebra , polynomial , functional equation

Let $ f$ and $ g$ be two integer-valued functions defined on the set of all integers such that [i](a)[/i] $ f(m \plus{} f(f(n))) \equal{} \minus{}f(f(m\plus{} 1) \minus{} n$ for all integers $ m$ and $ n;$ [i](b)[/i] $ g$ is a polynomial function with integer coefficients and g(n) = $ g(f(n))$ $ \forall n \in \mathbb{Z}.$

2022 IFYM, Sozopol, 2

Tags: algebra

We say that a rectangle and a triangle are [i]similar[/i], if they have the same area and the same perimeter. Let $P$ be a rectangle for which the ratio of the longer to the shorter side is at least $\lambda -1+\sqrt{\lambda (\lambda -2)}$ where $\lambda =\frac{3\sqrt{3}}{2}$. Prove that there exists a tringle that is [i]similar[/i] to $P$.

2006 Germany Team Selection Test, 3

Tags: induction , inequalities

Let $n$ be a positive integer, and let $b_{1}$, $b_{2}$, ..., $b_{n}$ be $n$ positive reals. Set $a_{1}=\frac{b_{1}}{b_{1}+b_{2}+...+b_{n}}$ and $a_{k}=\frac{b_{1}+b_{2}+...+b_{k}}{b_{1}+b_{2}+...+b_{k-1}}$ for every $k>1$. Prove the inequality $a_{1}+a_{2}+...+a_{n}\leq\frac{1}{a_{1}}+\frac{1}{a_{2}}+...+\frac{1}{a_{n}}$.

2024 Tuymaada Olympiad, 1

Tags: number theory , nt construction

Prove that a positive integer of the form $n^4 +1$ can have more than $1000$ divisors of the form $a^4 +1$ with integral $a$.

2013 Online Math Open Problems, 25

Tags: geometry , perimeter , function , calculus , inequalities , geometric transformation

Let $ABCD$ be a quadrilateral with $AD = 20$ and $BC = 13$. The area of $\triangle ABC$ is $338$ and the area of $\triangle DBC$ is $212$. Compute the smallest possible perimeter of $ABCD$. [i]Proposed by Evan Chen[/i]

2000 Romania Team Selection Test, 2

Tags: geometry , circumcircle , perimeter , geometric inequality , triangle

Let ABC be a triangle and $M$ be an interior point. Prove that \[ \min\{MA,MB,MC\}+MA+MB+MC<AB+AC+BC.\]

2020 MIG, 22

Tags:

Jane's uncle gives her a "$4$-balance." The $4$-balance acts like a normal balance scale, but it compares four masses instead of two, tilting towards the weight that is heaviest (if all four are equal, it stays balanced). He then gives her $25$ coins, one of which is a counterfeit heavier than the rest. What is the minimum number of uses of the $4$-balance needed to ensure she identifies the counterfeit? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$

1994 AIME Problems, 8

Tags: rotation , geometry , geometric transformation , linear algebra , matrix

The points $(0,0),$ $(a,11)$, and $(b,37)$ are the vertices of an equilateral triangle. Find the value of $ab$.

2022 Kyiv City MO Round 1, Problem 3

Tags: geometry

In triangle $ABC$ $\angle B > 90^\circ$. Tangents to this circle in points $A$ and $B$ meet at point $P$, and the line passing through $B$ perpendicular to $BC$ meets the line $AC$ at point $K$. Prove that $PA = PK$. [i](Proposed by Danylo Khilko)[/i]

1949-56 Chisinau City MO, 2

Tags: number theory , last digit

What is the last digit of $777^{777}$?

2011 Kosovo National Mathematical Olympiad, 3

Tags: function , trigonometry , calculus , algebra , system of equations , inequalities

Find maximal value of the function $f(x)=8-3\sin^2 (3x)+6 \sin (6x)$

2016 Iran MO (2nd Round), 1

Tags: inequalities , algebra

If $0<a\leq b\leq c$ prove that $$\frac{(c-a)^2}{6c}\leq \frac{a+b+c}{3}-\frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$$

2017 International Zhautykov Olympiad, 3

Tags: combinatorics , tiling , geometry , rectangle

Rectangle on a checked paper with length of a unit square side being $1$ Is divided into domino figures( two unit square sharing a common edge). Prove that you colour all corners of squares on the edge of rectangle and inside rectangle with $3$ colours such that for any two corners with distance $1$ the following conditions hold: they are coloured in different colour if the line connecting the two corners is on the border of two domino figures and coloured in same colour if the line connecting the two corners is inside a domino figure.

2015 Canadian Mathematical Olympiad Qualification, 4

Tags: geometry , cyclic quadrilateral

Given an acute-angled triangle $ABC$ whose altitudes from $B$ and $C$ intersect at $H$, let $P$ be any point on side $BC$ and $X, Y$ be points on $AB, AC$, respectively, such that $PB = PX$ and $PC = PY$. Prove that the points $A, H, X, Y$ lie on a common circle.

2003 CentroAmerican, 6

Tags: number theory

Say a number is [i]tico[/i] if the sum of it's digits is a multiple of $2003$. $\text{(i)}$ Show that there exists a positive integer $N$ such that the first $2003$ multiples, $N,2N,3N,\ldots 2003N$ are all tico. $\text{(ii)}$ Does there exist a positive integer $N$ such that all it's multiples are tico?

2005 AMC 12/AHSME, 14

Tags: geometry , symmetry , power of a point

A circle having center $ (0,k)$, with $ k > 6$, is tangent to the lines $ y \equal{} x, y \equal{} \minus{} x$ and $ y \equal{} 6$. What is the radius of this circle? $ \textbf{(A)}\ 6 \sqrt 2 \minus{} 6\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 6 \sqrt 2\qquad \textbf{(D)}\ 12\qquad \textbf{(E)}\ 6 \plus{} 6 \sqrt 2$

2023 Romanian Master of Mathematics, 1

Tags: number theory , prime number , diophantine equation

Determine all prime numbers $p$ and all positive integers $x$ and $y$ satisfying $$x^3+y^3=p(xy+p).$$

1997 Belarusian National Olympiad, 3

Tags: algebra

$$Problem3;$$If distinct real numbers x,y satisfy $\{x\} = \{y\}$ and $\{x^3\}=\{y^3\}$ prove that $x$ is a root of a quadratic equation with integer coefficients.

2009 All-Russian Olympiad Regional Round, 11.5

Tags: point , combinatorics , combinatorial geometry , line

We drew several straight lines on the plane and marked all of them intersection points. How many lines could be drawn? if one point is marked on one of the drawn lines, on the other - three, and on the third - five? Find all possible options and prove that there are no others.

2002 Turkey Team Selection Test, 2

Tags: geometric transformation , homothety , reflection , rotation , inversion , geometry

Two circles are internally tangent at a point $A$. Let $C$ be a point on the smaller circle other than $A$. The tangent line to the smaller circle at $C$ meets the bigger circle at $D$ and $E$; and the line $AC$ meets the bigger circle at $A$ and $P$. Show that the line $PE$ is tangent to the circle through $A$, $C$, and $E$.

1999 ITAMO, 5

Tags: combinatorics , grid

There is a village of pile-built dwellings on a lake, set on the gridpoints of an $m \times n$ rectangular grid. Each dwelling is connected by exactly $p$ bridges to some of the neighboring dwellings (diagonal connections are not allowed, two dwellings can be connected by more than one bridge). Determine for which values $m,n, p$ it is possible to place the bridges so that from any dwelling one can reach any other dwelling.

1997 All-Russian Olympiad, 1

Tags: quadratic , polynomial , inequalities , algebra

Let $P(x)$ be a quadratic polynomial with nonnegative coeficients. Show that for any real numbers $x$ and $y$, we have the inequality $P(xy)^2 \leqslant P(x^2)P(y^2)$. [i]E. Malinnikova[/i]