Olympiad problems

2004 Finnish National High School Mathematics Competition, 2

$a, b$ and $c$ are positive integers and \[\frac{a\sqrt{3} + b}{b\sqrt{3} + c}\] is a rational number. Show that \[\frac{a^2 + b^2 + c^2}{a + b + c}\] is an integer.

2021 Science ON all problems, 2

Tags: combinatorics

There is a football championship with $6$ teams involved, such that for any $2$ teams $A$ and $B$, $A$ plays with $B$ and $B$ plays with $A$ ($2$ such games are distinct). After every match, the winning teams gains $3$ points, the loosing team gains $0$ points and if there is a draw, both teams gain $1$ point each.\\ \\ In the end, the team standing on the last place has $12$ points and there are no $2$ teams that scored the same amount of points.\\ \\ For all the remaining teams, find their final scores and provide an example with the outcomes of all matches for at least one of the possible final situations. $\textit{(Andrei Bâra)}$

2024 ISI Entrance UGB, P4

Tags: function , calculus , limit

Let $f: \mathbb R \to \mathbb R$ be a function which is differentiable at $0$. Define another function $g: \mathbb R \to \mathbb R$ as follows: $$g(x) = \begin{cases} f(x)\sin\left(\frac 1x\right) ~ &\text{if} ~ x \neq 0 \\ 0 &\text{if} ~ x = 0. \end{cases}$$ Suppose that $g$ is also differentiable at $0$. Prove that \[g'(0) = f'(0) = f(0) = g(0) = 0.\]

2002 Switzerland Team Selection Test, 2

Tags: parallelogram , equal angles , geometry

A point$ O$ inside a parallelogram $ABCD$ satisfies $\angle AOB + \angle COD = \pi$. Prove that $\angle CBO = \angle CDO$.

1972 IMO Longlists, 46

Tags: linear algebra , matrix , number theory

Numbers $1, 2,\cdots, 16$ are written in a $4\times 4$ square matrix so that the sum of the numbers in every row, every column, and every diagonal is the same and furthermore that the numbers $1$ and $16$ lie in opposite corners. Prove that the sum of any two numbers symmetric with respect to the center of the square equals $17$.

2008 Greece National Olympiad, 3

Tags: trigonometry , geometry

A triangle $ABC$ with orthocenter $H$ is inscribed in a circle with center $K$ and radius $1$, where the angles at $B$ and $C$ are non-obtuse. If the lines $HK$ and $BC$ meet at point $S$ such that $SK(SK -SH) = 1$, compute the area of the concave quadrilateral $ABHC$.

2016 Israel Team Selection Test, 4

Tags: greatest common divisor , number theory

Find the greatest common divisor of all numbers of the form $(2^{a^2}\cdot 19^{b^2} \cdot 53^{c^2} + 8)^{16} - 1$ where $a,b,c$ are integers.

2023 Turkey Team Selection Test, 9

Tags: geometry

The points $ A,B,K,L,X$ lies of the circle $\Gamma$ in that order such that the arcs $\widehat{BK}$ and $\widehat{KL}$ are equal. The circle that passes through $A$ and tangent to $BK$ at $B$ intersects the line segment $KX$ at $P$ and $Q$. The circle that passes through $A$ and tangent to $BL$ at $B$ intersect the line segment $BX$ for the second time at $T$. Prove that $\angle{PTB} = \angle{XTQ}$

2022 MMATHS, 9

Tags: algebra

Suppose sequence $\{a_i\} = a_1, a_2, a_3, ....$ satisfies $a_{n+1} = \frac{1}{a_n+1}$ for all positive integers $n$. Define $b_k$ for positive integers $k \ge 2$ to be the minimum real number such that the product $a_1 \cdot a_2 \cdot ...\cdot a_k$ does not exceed $b_k$ for any positive integer choice of $a_1$. Find $\frac{1}{b_2}+\frac{1}{b_3}+\frac{1}{b_4}+...+\frac{1}{b_{10}}.$ .

2019 Belarus Team Selection Test, 6.1

Tags: geometry , geometric transformation , reflection

Two circles $\Omega$ and $\Gamma$ are internally tangent at the point $B$. The chord $AC$ of $\Gamma$ is tangent to $\Omega$ at the point $L$, and the segments $AB$ and $BC$ intersect $\Omega$ at the points $M$ and $N$. Let $M_1$ and $N_1$ be the reflections of $M$ and $N$ about the line $BL$; and let $M_2$ and $N_2$ be the reflections of $M$ and $N$ about the line $AC$. The lines $M_1M_2$ and $N_1N_2$ intersect at the point $K$. Prove that the lines $BK$ and $AC$ are perpendicular. [i](M. Karpuk)[/i]

2015 Turkey Team Selection Test, 7

Tags: functional equation , algebra , function , sophie germain identity

Find all the functions $f:R\to R$ such that \[f(x^2) + 4y^2f(y) = (f(x-y) + y^2)(f(x+y) + f(y))\] for every real $x,y$.

2021 Sharygin Geometry Olympiad, 8.8

Tags: combinatorial geometry , geometry , obtuse triangle , convex polygon , equal sides

Does there exist a convex polygon such that all its sidelengths are equal and all triangle formed by its vertices are obtuse-angled?

2016 Hanoi Open Mathematics Competitions, 2

Tags: algebra , combinatorics

Given an array of numbers $A = (672, 673, 674, ..., 2016)$ on table. Three arbitrary numbers $a,b,c \in A$ are step by step replaced by number $\frac13 min(a,b,c)$. After $672$ times, on the table there is only one number $m$, such that (A): $0 < m < 1$ (B): $m = 1$ (C): $1 < m < 2$ (D): $m = 2$ (E): None of the above.

1991 AMC 12/AHSME, 9

Tags:

From time $t = 0$ to time $t = 1$ a population increased by $i\%$, and from time $t = 1$ to time $t = 2$ the population increased by $j\%$. Therefore, from time $t = 0$ to time $t = 2$ the population increased by $ \textbf{(A)}\ (i + j)\%\qquad\textbf{(B)}\ ij\%\qquad\textbf{(C)}\ (i+ij)\%\qquad\textbf{(D)}\ \left(i + j + \frac{ij}{100}\right)\%\qquad\textbf{(E)}\left( i + j + \frac{i + j}{100}\right)\% $

2016 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: geometry , circles

Circle of radius $R_1$ is inscribed in an acute angle $\alpha$. Second circle with radius $R_2$ touches one of the sides forming the angle $\alpha$ in same point as first circle and intersects the second side in points $A$ and $B$, such that centers of both circles lie inside angle $\alpha$. Prove that $$AB=4\cos{\frac{\alpha}{2}}\sqrt{(R_2-R_1)\left(R_1 \cos^2 \frac{\alpha}{2}+R_2 \sin^2 \frac{\alpha}{2}\right)}$$

2015 239 Open Mathematical Olympiad, 1

Tags: combinatorics

There are 10 stones of different weights with distinct pairwise sums. We have a special two-tiered balance scale such that only two stones can be put on each cup and then we understand which cup is heavier. Prove that having this scale you can either find the heaviest or the lightest stone.

TNO 2023 Senior, 6

Tags: combinatorial geometry , coloring , algebra , combinatorics

The points inside a circle $ \Gamma $ are painted with $ n \geq 1 $ colors. A color is said to be dense in a circle $ \Omega $ if every circle contained within $ \Omega $ has points of that color in its interior. Prove that there exists at least one color that is dense in some circle contained within $ \Gamma $.

2012 Indonesia TST, 2

Tags: combinatorics

Let $T$ be the set of all 2-digit numbers whose digits are in $\{1,2,3,4,5,6\}$ and the tens digit is strictly smaller than the units digit. Suppose $S$ is a subset of $T$ such that it contains all six digits and no three numbers in $S$ use all six digits. If the cardinality of $S$ is $n$, find all possible values of $n$.

Ukrainian TYM Qualifying - geometry, X.13

Tags: geometry , geometric inequality , folding , hexagon

A paper square is bent along the line $\ell$, which passes through its center, so that a non-convex hexagon is formed. Investigate the question of the circle of largest radius that can be placed in such a hexagon.

III Soros Olympiad 1996 - 97 (Russia), 9.6

Tags: geometry , collinear

Let $ABC$ be an isosceles right triangle with hypotenuse $AB$, $D$ be some point in the plane such that $2CD = AB$ and point $C$ inside the triangle $ABD$. We construct two rays with a start in $C$, intersecting $AD$ and $BD$ and perpendicular to them. On the first one, intersecting $AD$, we will plot the segment $CK = AD$, and on the second one - $CM = BD$. Prove that points $M$, $D$ and $K$ lie on the same line.

2009 India IMO Training Camp, 4

Tags: trigonometry , geometry , circumcircle

Let $ \gamma$ be circumcircle of $ \triangle ABC$.Let $ R_a$ be radius of circle touching $ AB,AC$&$ \gamma$ internally.Define $ R_b,R_c$ similarly. Prove That $ \frac {1}{aR_a} \plus{} \frac {1}{bR_b} \plus{} \frac {1}{cR_c} \equal{} \frac {s^2}{rabc}$.

2017 CCA Math Bonanza, T8

Tags:

A group of $25$ CCA students decide they want to go to Disneyland, which is $105$ miles away. To save some time, they rent a bus with capacity $10$ people which can travel up to $60$ miles per hour. On the other hand, a student will run up to $9$ miles per hour. However, because a complicated plan of getting on and off the bus may be confusing to some students, a student may only board the bus once. What is the least number of minutes it will take for all students to reach Disneyland? Note: both the bus and students may travel backwards. [i]2017 CCA Math Bonanza Team Round #8[/i]

2001 Saint Petersburg Mathematical Olympiad, 11.6

Tags: algebra , functional equation , function

Find all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that for any $x,y$ the following is true: $$f(x+y+f(y))=f(x)+2y$$ [I]proposed by F. Petrov[/i]

2014 Junior Balkan Team Selection Tests - Romania, 2

Tags: pigeonhole principle , symmetry , combinatorics

Let $x_1, x_2 \ldots , x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p+x_q+x_r$ (with $1\le p<q<r\le 5$) are equal to $0$, then $x_1=x_2=\cdots=x_5=0$.

2004 France Team Selection Test, 2

Tags: trigonometry , angle bisector , geometry

Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$. Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$. [i]Alternative formulation.[/i] Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines $AN$ and $CM$ intersect at a point $Q$. Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$.