Olympiad problems

2001 Federal Math Competition of S&M, Problem 1

Solve in positive integers \[ x^y + y = y^x + x \]

2016 Mathematical Talent Reward Programme, MCQ: P 9

Tags: function

$f$ be a function satisfying $2f(x)+3f(-x)=x^2+5x$. Find $f(7)$ [list=1] [*] $-\frac{105}{4}$ [*] $-\frac{126}{5}$ [*] $-\frac{120}{7}$ [*] $-\frac{132}{7}$ [/list]

2011 Moldova Team Selection Test, 4

Tags: combinatorics

Initially, on the blackboard are written all natural numbers from $1$ to $20$. A move consists of selecting $2$ numbers $a<b$ written on the blackboard such that their difference is at least $2$, erasing these numbers and writting $a+1$ and $b-1$ instead. What is the maximum numbers of moves one can perform?

2007 Today's Calculation Of Integral, 238

Tags: calculus , integration , limit , calculus computations , logarithm

Find $ \lim_{a\to\infty} \frac {1}{a^2}\int_0^a \log (1 \plus{} e^x)\ dx.$

2009 Iran MO (3rd Round), 1

Tags: geometric transformation , reflection , trigonometry , perpendicular bisector , geometry

1-Let $ \triangle ABC$ be a triangle and $ (O)$ its circumcircle. $ D$ is the midpoint of arc $ BC$ which doesn't contain $ A$. We draw a circle $ W$ that is tangent internally to $ (O)$ at $ D$ and tangent to $ BC$.We draw the tangent $ AT$ from $ A$ to circle $ W$.$ P$ is taken on $ AB$ such that $ AP \equal{} AT$.$ P$ and $ T$ are at the same side wrt $ A$.PROVE $ \angle APD \equal{} 90^\circ$.

2007 Stanford Mathematics Tournament, 2

Tags: probability , analytic geometry

If $a$ and $b$ are each randomly and independently chosen in the interval $[-1, 1]$, what is the probability that $|a|+|b|<1$?

2020 MBMT, 19

Tags:

In a regular hexagon $ABCDEF$ of side length $8$ and center $K$, points $W$ and $U$ are chosen on $\overline{AB}$ and $\overline{CD}$ respectively such that $\overline{KW} = 7$ and $\angle WKU = 120^{\circ}$. Find the area of pentagon $WBCUK$. [i]Proposed by Bradley Guo[/i]

2000 Greece JBMO TST, 1

Tags: irreducible , number theory , digit

a) Prove that the fraction $\frac{3n+5}{2n+3}$ is irreducible for every $n \in N$ b) Let $x,y$ be digits of decimal representation system with $x>0$, and $\frac{\overline{xy}+12}{\overline{xy}-3}\in N$, prove that $x+y=9$. Is the converse true?

2023 Brazil National Olympiad, 3

Tags: diophantine equation , system of equations , number theory

Let $n$ be a positive integer. Show that there are integers $x_1, x_2, \ldots , x_n$, [i]not all equal[/i], satisfying $$\begin{cases} x_1^2+x_2+x_3+\ldots+x_n=0 \\ x_1+x_2^2+x_3+\ldots+x_n=0 \\ x_1+x_2+x_3^2+\ldots+x_n=0 \\ \vdots \\ x_1+x_2+x_3+\ldots+x_n^2=0 \end{cases}$$ if, and only if, $2n-1$ is not prime.

2016 Estonia Team Selection Test, 5

Tags: geometry , angle chasing

Let $O$ be the circumcentre of the acute triangle $ABC$. Let $c_1$ and $c_2$ be the circumcircles of triangles $ABO$ and $ACO$. Let $P$ and $Q$ be points on $c_1$ and $c_2$ respectively, such that OP is a diameter of $c_1$ and $OQ$ is a diameter of $c_2$. Let $T$ be the intesection of the tangent to $c_1$ at $P$ and the tangent to $c_2$ at $Q$. Let $D$ be the second intersection of the line $AC$ and the circle $c_1$. Prove that the points $D, O$ and $T$ are collinear

2024 India IMOTC, 3

Tags: number theory

Let $P(x) \in \mathbb{Q}[x]$ be a polynomial with rational coefficients and degree $d\ge 2$. Prove there is no infinite sequence $a_0, a_1, \ldots$ of rational numbers such that $P(a_i)=a_{i-1}+i$ for all $i\ge 1$. [i]Proposed by Pranjal Srivastava and Rohan Goyal[/i]

Russian TST 2016, P3

Tags: number theory , rational number

Prove that any rational number can be represented as a product of four rational numbers whose sum is zero.

1999 Hungary-Israel Binational, 1

Tags: polynomial , algebra

$ f(x)$ is a given polynomial whose degree at least 2. Define the following polynomial-sequence: $ g_1(x)\equal{}f(x), g_{n\plus{}1}(x)\equal{}f(g_n(x))$, for all $ n \in N$. Let $ r_n$ be the average of $ g_n(x)$'s roots. If $ r_{19}\equal{}99$, find $ r_{99}$.

2023 Harvard-MIT Mathematics Tournament, 17

Tags: guts

An equilateral triangle lies in the Cartesian plane such that the $x$-coordinates of its vertices are pairwise distinct and all satisfy the equation $x^3-9x^2 + 10x + 5 = 0.$ Compute the side length of the triangle.

2010 Kazakhstan National Olympiad, 1

Tags: ellipse , rectangle , geometry , conic

Triangle $ABC$ is given. Consider ellipse $ \Omega _1$, passes through $C$ with focuses in $A$ and $B$. Similarly define ellipses $ \Omega _2 , \Omega _3$ with focuses $B,C$ and $C,A$ respectively. Prove, that if all ellipses have common point $D$ then $A,B,C,D$ lies on the circle. Ellipse with focuses $X,Y$, passes through $Z$- locus of point $T$, such that $XT+YT=XZ+YZ$

2025 India STEMS Category A, 3

Tags: geometry , a-hm point , reflection

Let $ABC$ be an acute scalene triangle with orthocenter $H$. Let $M$ be the midpoint of $BC$. $N$ is the point on line $AM$ such that $(BMN)$ is tangent to $AB$. Finally, let $H'$ be the reflection of $H$ in $B$. Prove that $\angle ANH'=90^{\circ}$. [i]Proposed by Malay Mahajan and Siddharth Choppara[/i]

2003 Alexandru Myller, 2

Tags: linear algebra , matrix

Let be two $ 3\times 3 $ real matrices that have the property that $$ AX=\begin{pmatrix}0\\0\\0\end{pmatrix}\implies BX=\begin{pmatrix}0\\0\\0\end{pmatrix} , $$ for any three-dimensional vectors $ X. $ Prove that there exists a $ 3\times 3 $ real matrix $ C $ such that $ B=CA. $

2023 BMT, 4

Tags: number theory

Given positive integers $a \ge 2$ and $k$, let $m_a(k)$ denote the remainder when $k$ is divided by $a$. Compute the number of positive integers, $n$, less than 500 such that $m_2(m_5(m_{11}(n))) = 1$.

2024 Miklos Schweitzer, 6

Tags:

During heat diffusion, we say that the evolution of temperature at a point $x \in \mathbb{R}^n$ is astonishing if it changes monotonicity infinitely many times. Can it happen that the temperature evolves astonishingly at every point $x \in \mathbb{R}^n$? More precisely, does there exist a nonnegative $u \in C^2((0, +\infty) \times \mathbb{R}^n)$ solving the heat equation $\partial_t u = \Delta u$, such that $u(t,x) \to 0$ for every $x$ as $t \to \infty$, and for every $x \in \mathbb{R}^n$, the function $t \mapsto u(t,x)$ changes monotonicity infinitely many times on $(0, \infty)$?

1988 Romania Team Selection Test, 1

Tags: 3d geometry , sphere , geometric transformation , rotation , geometry

Consider a sphere and a plane $\pi$. For a variable point $M \in \pi$, exterior to the sphere, one considers the circular cone with vertex in $M$ and tangent to the sphere. Find the locus of the centers of all circles which appear as tangent points between the sphere and the cone. [i]Octavian Stanasila[/i]

2001 All-Russian Olympiad, 3

Tags: geometry , circumcircle , ratio , trigonometry , geometric transformation , homothety , power of a point

Let the circle $ {\omega}_{1}$ be internally tangent to another circle $ {\omega}_{2}$ at $ N$.Take a point $ K$ on $ {\omega}_{1}$ and draw a tangent $ AB$ which intersects $ {\omega}_{2}$ at $ A$ and $ B$. Let $M$ be the midpoint of the arc $ AB$ which is on the opposite side of $ N$. Prove that, the circumradius of the $ \triangle KBM$ doesnt depend on the choice of $ K$.

2001 Bulgaria National Olympiad, 2

Tags: parallelogram , circumcircle , power of a point , radical axis , geometry

Suppose that $ABCD$ is a parallelogram such that $DAB>90$. Let the point $H$ to be on $AD$ such that $BH$ is perpendicular to $AD$. Let the point $M$ to be the midpoint of $AB$. Let the point $K$ to be the intersecting point of the line $DM$ with the circumcircle of $ADB$. Prove that $HKCD$ is concyclic.

2023/2024 Tournament of Towns, 2

Tags: combinatorics

2. There are three hands on a clock. Each of them rotates in a normal direction at some non-zero speed, which can be wrong. In the morning the long and the short hands coincided. Just in three hours after that moment the long and the mid-length hands coincided. After next four hours the short and the mid-length hands coincided. Will it necessarily occur that all three hands will coincide? Alexandr Yuran

2023 AMC 10, 12

Tags:

How many three-digit positive integers $N$ satisfy the following properties? - The number $N$ is divisible by $7$. - The number formed by reversing the digits of $N$ is divisible by $5$. $\textbf{(A) }13\qquad\textbf{(B) }14\qquad\textbf{(C) }15\qquad\textbf{(D) }16\qquad\textbf{(E) }17$

2005 Miklós Schweitzer, 4

Tags: abstract algebra , free group

Let F be a countable free group and let $F = H_1> H_2> H_3> \cdots$ be a descending chain of finite index subgroups of group F. Suppose that $\cap H_i$ does not contain any nontrivial normal subgroups of F. Prove that there exist $g_i\in F$ for which the conjugated subgroups $H_i^{g_i}$ also form a chain, and $\cap H_i^{g_i}=\{1\}$. [hide=Note]Nielsen-Schreier Theorem might be useful.[/hide]