Olympiad problems

2025 Israel TST, P1

Let $ f(N) $ denote the maximum number of $ T $-tetrominoes that can be placed on an $ N \times N $ board such that each $ T $-tetromino covers at least one cell that is not covered by any other $ T $-tetromino. Find the smallest real number $ c $ such that \[ f(N) \leq cN^2 \] for all positive integers $ N $.

2015 AMC 10, 21

Tags: geometry , 3d geometry , tetrahedron , analytic geometry , calculus , vector

Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\tfrac{12}5\sqrt2$. What is the volume of the tetrahedron? $\textbf{(A) }3\sqrt2\qquad\textbf{(B) }2\sqrt5\qquad\textbf{(C) }\dfrac{24}5\qquad\textbf{(D) }3\sqrt3\qquad\textbf{(E) }\dfrac{24}5\sqrt2$

2018 Estonia Team Selection Test, 6

Tags: number theory , digit

We call a positive integer $n$ whose all digits are distinct [i]bright[/i], if either $n$ is a one-digit number or there exists a divisor of $n$ which can be obtained by omitting one digit of $n$ and which is bright itself. Find the largest bright positive integer. (We assume that numbers do not start with zero.)

1991 Bundeswettbewerb Mathematik, 4

Tags: combinatorics , combinatorial geometry , geometry , rectangle

A strip of width $1$ is to be divided by rectangular panels of common width $1$ and denominations long $a_1$, $a_2$, $a_3$, $. . .$ be paved without gaps ($a_1 \ne 1$). From the second panel on, each panel is similar but not congruent to the already paved part of the strip. When the first $n$ slabs are laid, the length of the paved part of the strip is $sn$. Given $a_1$, is there a number that is not surpassed by any $s_n$? The accuracy answer has to be proven.

1986 Federal Competition For Advanced Students, P2, 4

Tags: number theory

Find the largest $ n$ for which there is a natural number $ N$ with $ n$ decimal digits which are all different such that $ n!$ divides $ N$. Furthermore, for this largest $ n$ find all possible numbers $ N$.

2018 Kyiv Mathematical Festival, 3

Tags: inequalities

For every $x,y\ge0$ prove that $(x+1)^2+(y-1)^2\ge2\sqrt{2xy}.$

2023 LMT Fall, 12

Tags: geometry

In triangle $ABC$ with $AB = 7$, $AC = 8$, and $BC = 9$, the $A$-excircle is tangent to $BC$ at point $D$ and also tangent to lines $AB$ and $AC$ at points $ $ and $F$, respectively. Find $[DEF]$. (The $A$-excircle is the circle tangent to segment $BC$ and the extensions of rays $AB$ and $AC$. Also, $[XY Z]$ denotes the area of triangle $XY Z$.)

2015 Junior Regional Olympiad - FBH, 3

Tags: geometry , midpoint

Let $D$ be a midpoint of $BC$ of triangle $ABC$. On side $AB$ is given point $E$, and on side $AC$ is given point $F$ such that $\angle EDF = 90^{\circ}$. Prove that $BE+CF>EF$

V Soros Olympiad 1998 - 99 (Russia), 10.5

Tags: geometry , locus

An isosceles triangle $ABC$ ($AB = BC$) is given on the plane. Find the locus of points $M$ of the plane such that $ABCM$ is a convex quadrilateral and $\angle MAC + \angle CMB = 90^o$.

2009 Princeton University Math Competition, 8

Tags: geometry , circumcircle

Consider $\triangle ABC$ and a point $M$ in its interior so that $\angle MAB = 10^\circ$, $\angle MBA = 20^\circ$, $\angle MCA = 30^\circ$ and $\angle MAC = 40^\circ$. What is $\angle MBC$?

VII Soros Olympiad 2000 - 01, 10.3

Tags: function , algebra

Let $y = f (x)$ be a convex function defined on $[0,1]$, $f (0) = 0,$ $f (1) = 0$. It is also known that the area of the segment bounded by this function and the segment $[0, 1]$ is equal to $1$. Find and draw the set of points of the coordinate plane through which the graph of such a function can pass. (A function is called convex if all points of the line segment connecting any two points on its graph are located no higher than the graph of this function.)

2015 India Regional MathematicaI Olympiad, 2

Tags: quadratic , algebra , polynomial

Let $P(x) = x^2 + ax + b$ be a quadratic polynomial with real coefficients. Suppose there are real numbers $ s \neq t$ such that $P(s) = t$ and $P(t) = s$. Prove that $b-st$ is a root of $x^2 + ax + b - st$.

2014 ELMO Shortlist, 3

Tags: parameterization , inequalities

Let $a,b,c,d,e,f$ be positive real numbers. Given that $def+de+ef+fd=4$, show that \[ ((a+b)de+(b+c)ef+(c+a)fd)^2 \geq\ 12(abde+bcef+cafd). \][i]Proposed by Allen Liu[/i]

2011 JBMO Shortlist, 5

Tags: number theory , sum of digits

Find the least positive integer such that the sum of its digits is $2011$ and the product of its digits is a power of $6$.

1985 Tournament Of Towns, (100) 4

Tags: combinatorics , time

Two chess players play each other at chess using clocks (when a player makes a move , the player stops his clock and starts the clock of his opponent) . It is known that when both players have just completed their $40$th move , both of their clocks read exactly $2$ hr $30$ min . Prove that there was a moment in the game when the clock of one player registered $1$ min $51$ sec less than that of the other . Furthermore , can one assert that the difference between the two clock readings was ever equal to $2$ minutes? (S . Fomin , Leningrad)

2014 NIMO Problems, 3

Tags: trigonometry , analytic geometry , symmetry , geometry , number theory , relatively prime , similar triangles

Let $ABCD$ be a square with side length $2$. Let $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{CD}$ respectively, and let $X$ and $Y$ be the feet of the perpendiculars from $A$ to $\overline{MD}$ and $\overline{NB}$, also respectively. The square of the length of segment $\overline{XY}$ can be written in the form $\tfrac pq$ where $p$ and $q$ are positive relatively prime integers. What is $100p+q$? [i]Proposed by David Altizio[/i]

2015 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: combinatorics , set

Alice and Mary were searching attic and found scale and box with weights. When they sorted weights by mass, they found out there exist $5$ different groups of weights. Playing with the scale and weights, they discovered that if they put any two weights on the left side of scale, they can find other two weights and put on to the right side of scale so scale is in balance. Find the minimal number of weights in the box

1997 Pre-Preparation Course Examination, 1

Tags: polynomial , greatest common divisor , algebra

Let $n$ be a positive integer. Prove that there exist polynomials$f(x)$and $g(x$) with integer coefficients such that \[f(x)\left(x + 1 \right)^{2^n}+ g(x) \left(x^{2^n}+ 1 \right) = 2.\]

2007 Tournament Of Towns, 5

Tags: combinatorics

Attached to each of a number of objects is a tag which states the correct mass of the object. The tags have fallen off and have been replaced on the objects at random. We wish to determine if by chance all tags are in fact correct. We may use exactly once a horizontal lever which is supported at its middle. The objects can be hung from the lever at any point on either side of the support. The lever either stays horizontal or tilts to one side. Is this task always possible?

2023 CMIMC Geometry, 5

Tags: geometry

In trapezoid $ABCD, AB=3, BC=2, CD=5,$ and $\angle B = \angle C = 90^{\circ}.$ The angle bisectors of $\angle A$ and $\angle D$ intersect at a point $P$ in the interior of $ABCD.$ Compute $BP^2+CP^2.$ [i]Proposed by Kyle Lee[/i]

2013 Balkan MO, 1

Tags: geometry , incenter , trigonometry , perimeter , circumcircle

In a triangle $ABC$, the excircle $\omega_a$ opposite $A$ touches $AB$ at $P$ and $AC$ at $Q$, while the excircle $\omega_b$ opposite $B$ touches $BA$ at $M$ and $BC$ at $N$. Let $K$ be the projection of $C$ onto $MN$ and let $L$ be the projection of $C$ onto $PQ$. Show that the quadrilateral $MKLP$ is cyclic. ([i]Bulgaria[/i])

2010 Princeton University Math Competition, 1

Tags:

Let the operation $\bigstar$ be defined by $x\bigstar y=y^x-xy$. Calculate $(3\bigstar4)-(4\bigstar3)$.

2021 European Mathematical Cup, 2

Tags: trisector , geometry , circumcircle

Let $ABC$ be an acute-angled triangle such that $|AB|<|AC|$. Let $X$ and $Y$ be points on the minor arc ${BC}$ of the circumcircle of $ABC$ such that $|BX|=|XY|=|YC|$. Suppose that there exists a point $N$ on the segment $\overline{AY}$ such that $|AB|=|AN|=|NC|$. Prove that the line $NC$ passes through the midpoint of the segment $\overline{AX}$. \\ \\ (Ivan Novak)

2017 China Team Selection Test, 6

Tags: combinatorics

Every cell of a $2017\times 2017$ grid is colored either black or white, such that every cell has at least one side in common with another cell of the same color. Let $V_1$ be the set of all black cells, $V_2$ be the set of all white cells. For set $V_i (i=1,2)$, if two cells share a common side, draw an edge with the centers of the two cells as endpoints, obtaining graphs $G_i$. If both $G_1$ and $G_2$ are connected paths (no cycles, no splits), prove that the center of the grid is one of the endpoints of $G_1$ or $G_2$.

Today's calculation of integrals, 884

Tags: calculus , integration , trigonometry , inequalities , calculus computations , integral inequality

Prove that : \[\pi (e-1)<\int_0^{\pi} e^{|\cos 4x|}dx<2(e^{\frac{\pi}{2}}-1)\]