Olympiad problems

2024 Oral Moscow Geometry Olympiad, 4

Tags: geometry

Straight lines are drawn containing the sides of an unequal triangle $ABC$, its incircle $I$ circle and a its circumcircle, the center of which is not marked. Using only a ruler (without divisions), construct the symedian of the triangle (a straight line symmetrical to the median relative to the corresponding bisector), drawing no more than six lines.

1999 Slovenia National Olympiad, Problem 3

Tags: geometry , incenter

The incircle of a right triangle $ABC$ touches the hypotenuse $AB$ at a point $D$. Show that the area of $\triangle ABC$ equals $AD\cdot DB$.

1999 National High School Mathematics League, 3

Tags: logarithm

If $(\log_2 3)^x-(\log_5 3)^x\geq (\log_2 3)^{-y}-(\log_5 3)^{-y}$, then $\text{(A)}x-y\geq0\qquad\text{(B)}x+y\geq0\qquad\text{(C)}x-y\leq0\qquad\text{(D)}x+y\leq0$

1969 IMO Shortlist, 64

Tags: inequality , factorial , algebra

$(USS 1)$ Prove that for a natural number $n > 2, (n!)! > n[(n - 1)!]^{n!}.$

2001 Federal Math Competition of S&M, Problem 1

Tags: calculus , derivative , number theory

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$.